Exact Theta-Indexed Expected Loss, Almost-Sure Equality and Observed-Error Limits
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batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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1. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-12, complete listed estimator values=[-13, -12, -11] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-12, [[-13, 1/4, 1, 1/4], [-12, 1/2, 0, 0], [-11, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-12, [[-13, 1/4, 1, 1/4], [-12, 1/2, 0, 0], [-11, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
2. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-11, complete listed estimator values=[-13, -11, -9] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-11, [[-13, 1/4, 4, 1], [-11, 1/2, 0, 0], [-9, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-11, [[-13, 1/4, 4, 1], [-11, 1/2, 0, 0], [-9, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
3. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-10, complete listed estimator values=[-13, -10, -7] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-10, [[-13, 1/4, 9, 9/4], [-10, 1/2, 0, 0], [-7, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-10, [[-13, 1/4, 9, 9/4], [-10, 1/2, 0, 0], [-7, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
4. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-9, complete listed estimator values=[-13, -9, -5] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-9, [[-13, 1/4, 16, 4], [-9, 1/2, 0, 0], [-5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-9, [[-13, 1/4, 16, 4], [-9, 1/2, 0, 0], [-5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
5. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-8, complete listed estimator values=[-13, -8, -3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-8, [[-13, 1/4, 25, 25/4], [-8, 1/2, 0, 0], [-3, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-8, [[-13, 1/4, 25, 25/4], [-8, 1/2, 0, 0], [-3, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
6. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-7, complete listed estimator values=[-8, -7, -6] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-7, [[-8, 1/4, 1, 1/4], [-7, 1/2, 0, 0], [-6, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-7, [[-8, 1/4, 1, 1/4], [-7, 1/2, 0, 0], [-6, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
7. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-6, complete listed estimator values=[-8, -6, -4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-6, [[-8, 1/4, 4, 1], [-6, 1/2, 0, 0], [-4, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-6, [[-8, 1/4, 4, 1], [-6, 1/2, 0, 0], [-4, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
8. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-5, complete listed estimator values=[-8, -5, -2] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-5, [[-8, 1/4, 9, 9/4], [-5, 1/2, 0, 0], [-2, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-5, [[-8, 1/4, 9, 9/4], [-5, 1/2, 0, 0], [-2, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
9. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-4, complete listed estimator values=[-8, -4, 0] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-4, [[-8, 1/4, 16, 4], [-4, 1/2, 0, 0], [0, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-4, [[-8, 1/4, 16, 4], [-4, 1/2, 0, 0], [0, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
10. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-3, complete listed estimator values=[-8, -3, 2] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-3, [[-8, 1/4, 25, 25/4], [-3, 1/2, 0, 0], [2, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-3, [[-8, 1/4, 25, 25/4], [-3, 1/2, 0, 0], [2, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
11. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-2, complete listed estimator values=[-3, -2, -1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-2, [[-3, 1/4, 1, 1/4], [-2, 1/2, 0, 0], [-1, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-2, [[-3, 1/4, 1, 1/4], [-2, 1/2, 0, 0], [-1, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
12. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=-1, complete listed estimator values=[-3, -1, 1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(-1, [[-3, 1/4, 4, 1], [-1, 1/2, 0, 0], [1, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-1, [[-3, 1/4, 4, 1], [-1, 1/2, 0, 0], [1, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
13. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=0, complete listed estimator values=[-3, 0, 3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(0, [[-3, 1/4, 9, 9/4], [0, 1/2, 0, 0], [3, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (0, [[-3, 1/4, 9, 9/4], [0, 1/2, 0, 0], [3, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
14. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=1, complete listed estimator values=[-3, 1, 5] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(1, [[-3, 1/4, 16, 4], [1, 1/2, 0, 0], [5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (1, [[-3, 1/4, 16, 4], [1, 1/2, 0, 0], [5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
15. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=2, complete listed estimator values=[-3, 2, 7] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(2, [[-3, 1/4, 25, 25/4], [2, 1/2, 0, 0], [7, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (2, [[-3, 1/4, 25, 25/4], [2, 1/2, 0, 0], [7, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
16. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=3, complete listed estimator values=[2, 3, 4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(3, [[2, 1/4, 1, 1/4], [3, 1/2, 0, 0], [4, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (3, [[2, 1/4, 1, 1/4], [3, 1/2, 0, 0], [4, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
17. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=4, complete listed estimator values=[2, 4, 6] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(4, [[2, 1/4, 4, 1], [4, 1/2, 0, 0], [6, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (4, [[2, 1/4, 4, 1], [4, 1/2, 0, 0], [6, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
18. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=5, complete listed estimator values=[2, 5, 8] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(5, [[2, 1/4, 9, 9/4], [5, 1/2, 0, 0], [8, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (5, [[2, 1/4, 9, 9/4], [5, 1/2, 0, 0], [8, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
19. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=6, complete listed estimator values=[2, 6, 10] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(6, [[2, 1/4, 16, 4], [6, 1/2, 0, 0], [10, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (6, [[2, 1/4, 16, 4], [6, 1/2, 0, 0], [10, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
20. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=7, complete listed estimator values=[2, 7, 12] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(7, [[2, 1/4, 25, 25/4], [7, 1/2, 0, 0], [12, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (7, [[2, 1/4, 25, 25/4], [7, 1/2, 0, 0], [12, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
21. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=8, complete listed estimator values=[7, 8, 9] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(8, [[7, 1/4, 1, 1/4], [8, 1/2, 0, 0], [9, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (8, [[7, 1/4, 1, 1/4], [8, 1/2, 0, 0], [9, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
22. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=9, complete listed estimator values=[7, 9, 11] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(9, [[7, 1/4, 4, 1], [9, 1/2, 0, 0], [11, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (9, [[7, 1/4, 4, 1], [9, 1/2, 0, 0], [11, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
23. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=10, complete listed estimator values=[7, 10, 13] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(10, [[7, 1/4, 9, 9/4], [10, 1/2, 0, 0], [13, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (10, [[7, 1/4, 9, 9/4], [10, 1/2, 0, 0], [13, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
24. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=11, complete listed estimator values=[7, 11, 15] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(11, [[7, 1/4, 16, 4], [11, 1/2, 0, 0], [15, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (11, [[7, 1/4, 16, 4], [11, 1/2, 0, 0], [15, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
25. MSE Is the Complete Expected Squared Error under One Theta Law
Use scalar theta=12, complete listed estimator values=[7, 12, 17] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).
View answer and explanation
(12, [[7, 1/4, 25, 25/4], [12, 1/2, 0, 0], [17, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)
Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (12, [[7, 1/4, 25, 25/4], [12, 1/2, 0, 0], [17, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate
26. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-12, [-13, -12, -11], [[1, 4], [1, 2], [1, 4]]], [-11, [-12, -11, -10], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-11, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], 1/2], [-12, [[-13, 1/4], [-12, 1/2], [-11, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-11, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], 1/2], [-12, [[-13, 1/4], [-12, 1/2], [-11, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
27. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-11, [-13, -11, -9], [[1, 4], [1, 2], [1, 4]]], [-10, [-11, -10, -9], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-10, [[-11, 1/4], [-10, 1/2], [-9, 1/4]], 1/2], [-11, [[-13, 1/4], [-11, 1/2], [-9, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-10, [[-11, 1/4], [-10, 1/2], [-9, 1/4]], 1/2], [-11, [[-13, 1/4], [-11, 1/2], [-9, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
28. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-10, [-13, -10, -7], [[1, 4], [1, 2], [1, 4]]], [-9, [-10, -9, -8], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-10, [[-13, 1/4], [-10, 1/2], [-7, 1/4]], 9/2], [-9, [[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-10, [[-13, 1/4], [-10, 1/2], [-7, 1/4]], 9/2], [-9, [[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
29. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-9, [-13, -9, -5], [[1, 4], [1, 2], [1, 4]]], [-8, [-9, -8, -7], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-8, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], 1/2], [-9, [[-13, 1/4], [-9, 1/2], [-5, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-8, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], 1/2], [-9, [[-13, 1/4], [-9, 1/2], [-5, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
30. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-8, [-13, -8, -3], [[1, 4], [1, 2], [1, 4]]], [-7, [-8, -7, -6], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2], [-8, [[-13, 1/4], [-8, 1/2], [-3, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2], [-8, [[-13, 1/4], [-8, 1/2], [-3, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
31. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-7, [-8, -7, -6], [[1, 4], [1, 2], [1, 4]]], [-6, [-7, -6, -5], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-6, [[-7, 1/4], [-6, 1/2], [-5, 1/4]], 1/2], [-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-6, [[-7, 1/4], [-6, 1/2], [-5, 1/4]], 1/2], [-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
32. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-6, [-8, -6, -4], [[1, 4], [1, 2], [1, 4]]], [-5, [-6, -5, -4], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-5, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], 1/2], [-6, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-5, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], 1/2], [-6, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
33. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-5, [-8, -5, -2], [[1, 4], [1, 2], [1, 4]]], [-4, [-5, -4, -3], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-4, [[-5, 1/4], [-4, 1/2], [-3, 1/4]], 1/2], [-5, [[-8, 1/4], [-5, 1/2], [-2, 1/4]], 9/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-4, [[-5, 1/4], [-4, 1/2], [-3, 1/4]], 1/2], [-5, [[-8, 1/4], [-5, 1/2], [-2, 1/4]], 9/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
34. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-4, [-8, -4, 0], [[1, 4], [1, 2], [1, 4]]], [-3, [-4, -3, -2], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-3, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1/2], [-4, [[-8, 1/4], [-4, 1/2], [0, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-3, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1/2], [-4, [[-8, 1/4], [-4, 1/2], [0, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
35. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-3, [-8, -3, 2], [[1, 4], [1, 2], [1, 4]]], [-2, [-3, -2, -1], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2], [-3, [[-8, 1/4], [-3, 1/2], [2, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2], [-3, [[-8, 1/4], [-3, 1/2], [2, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
36. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-2, [-3, -2, -1], [[1, 4], [1, 2], [1, 4]]], [-1, [-2, -1, 0], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-1, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1/2], [-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-1, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1/2], [-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
37. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[-1, [-3, -1, 1], [[1, 4], [1, 2], [1, 4]]], [0, [-1, 0, 1], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[-1, [[-3, 1/4], [-1, 1/2], [1, 1/4]], 2], [0, [[-1, 1/4], [0, 1/2], [1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-1, [[-3, 1/4], [-1, 1/2], [1, 1/4]], 2], [0, [[-1, 1/4], [0, 1/2], [1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
38. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[0, [-3, 0, 3], [[1, 4], [1, 2], [1, 4]]], [1, [0, 1, 2], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[0, [[-3, 1/4], [0, 1/2], [3, 1/4]], 9/2], [1, [[0, 1/4], [1, 1/2], [2, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[0, [[-3, 1/4], [0, 1/2], [3, 1/4]], 9/2], [1, [[0, 1/4], [1, 1/2], [2, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
39. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[1, [-3, 1, 5], [[1, 4], [1, 2], [1, 4]]], [2, [1, 2, 3], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[1, [[-3, 1/4], [1, 1/2], [5, 1/4]], 8], [2, [[1, 1/4], [2, 1/2], [3, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[1, [[-3, 1/4], [1, 1/2], [5, 1/4]], 8], [2, [[1, 1/4], [2, 1/2], [3, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
40. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[2, [-3, 2, 7], [[1, 4], [1, 2], [1, 4]]], [3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[2, [[-3, 1/4], [2, 1/2], [7, 1/4]], 25/2], [3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[2, [[-3, 1/4], [2, 1/2], [7, 1/4]], 25/2], [3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
41. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]], [4, [3, 4, 5], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2], [4, [[3, 1/4], [4, 1/2], [5, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2], [4, [[3, 1/4], [4, 1/2], [5, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
42. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[4, [2, 4, 6], [[1, 4], [1, 2], [1, 4]]], [5, [4, 5, 6], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[4, [[2, 1/4], [4, 1/2], [6, 1/4]], 2], [5, [[4, 1/4], [5, 1/2], [6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[4, [[2, 1/4], [4, 1/2], [6, 1/4]], 2], [5, [[4, 1/4], [5, 1/2], [6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
43. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[5, [2, 5, 8], [[1, 4], [1, 2], [1, 4]]], [6, [5, 6, 7], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[5, [[2, 1/4], [5, 1/2], [8, 1/4]], 9/2], [6, [[5, 1/4], [6, 1/2], [7, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[5, [[2, 1/4], [5, 1/2], [8, 1/4]], 9/2], [6, [[5, 1/4], [6, 1/2], [7, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
44. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[6, [2, 6, 10], [[1, 4], [1, 2], [1, 4]]], [7, [6, 7, 8], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[6, [[2, 1/4], [6, 1/2], [10, 1/4]], 8], [7, [[6, 1/4], [7, 1/2], [8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[6, [[2, 1/4], [6, 1/2], [10, 1/4]], 8], [7, [[6, 1/4], [7, 1/2], [8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
45. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[7, [2, 7, 12], [[1, 4], [1, 2], [1, 4]]], [8, [7, 8, 9], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[7, [[2, 1/4], [7, 1/2], [12, 1/4]], 25/2], [8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[7, [[2, 1/4], [7, 1/2], [12, 1/4]], 25/2], [8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
46. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[8, [7, 8, 9], [[1, 4], [1, 2], [1, 4]]], [9, [8, 9, 10], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2], [9, [[8, 1/4], [9, 1/2], [10, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2], [9, [[8, 1/4], [9, 1/2], [10, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
47. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[9, [7, 9, 11], [[1, 4], [1, 2], [1, 4]]], [10, [9, 10, 11], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[10, [[9, 1/4], [10, 1/2], [11, 1/4]], 1/2], [9, [[7, 1/4], [9, 1/2], [11, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[10, [[9, 1/4], [10, 1/2], [11, 1/4]], 1/2], [9, [[7, 1/4], [9, 1/2], [11, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
48. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[10, [7, 10, 13], [[1, 4], [1, 2], [1, 4]]], [11, [10, 11, 12], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[10, [[7, 1/4], [10, 1/2], [13, 1/4]], 9/2], [11, [[10, 1/4], [11, 1/2], [12, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[10, [[7, 1/4], [10, 1/2], [13, 1/4]], 9/2], [11, [[10, 1/4], [11, 1/2], [12, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
49. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[11, [7, 11, 15], [[1, 4], [1, 2], [1, 4]]], [12, [11, 12, 13], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[11, [[7, 1/4], [11, 1/2], [15, 1/4]], 8], [12, [[11, 1/4], [12, 1/2], [13, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[11, [[7, 1/4], [11, 1/2], [15, 1/4]], 8], [12, [[11, 1/4], [12, 1/2], [13, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
50. Each Theta Has Its Own Complete Sampling Law and Risk
Use complete laws=[[12, [7, 12, 17], [[1, 4], [1, 2], [1, 4]]], [13, [12, 13, 14], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).
View answer and explanation
([[12, [[7, 1/4], [12, 1/2], [17, 1/4]], 25/2], [13, [[12, 1/4], [13, 1/2], [14, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)
For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[12, [[7, 1/4], [12, 1/2], [17, 1/4]], 25/2], [13, [[12, 1/4], [13, 1/2], [14, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate
51. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-12, complete listed estimator values=[-12, -11] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-12, [[-12, 1, 0, 0], [-11, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-12, [[-12, 1, 0, 0], [-11, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
52. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-11, complete listed estimator values=[-11, -9] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-11, [[-11, 1, 0, 0], [-9, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-11, [[-11, 1, 0, 0], [-9, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
53. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-10, complete listed estimator values=[-10, -7] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-10, [[-10, 1, 0, 0], [-7, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-10, [[-10, 1, 0, 0], [-7, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
54. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-9, complete listed estimator values=[-9, -5] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-9, [[-9, 1, 0, 0], [-5, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-9, [[-9, 1, 0, 0], [-5, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
55. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-8, complete listed estimator values=[-8, -3] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-8, [[-8, 1, 0, 0], [-3, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-8, [[-8, 1, 0, 0], [-3, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
56. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-7, complete listed estimator values=[-7, -6] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-7, [[-7, 1, 0, 0], [-6, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-7, [[-7, 1, 0, 0], [-6, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
57. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-6, complete listed estimator values=[-6, -4] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-6, [[-6, 1, 0, 0], [-4, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-6, [[-6, 1, 0, 0], [-4, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
58. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-5, complete listed estimator values=[-5, -2] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-5, [[-5, 1, 0, 0], [-2, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-5, [[-5, 1, 0, 0], [-2, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
59. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-4, complete listed estimator values=[-4, 0] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-4, [[-4, 1, 0, 0], [0, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-4, [[-4, 1, 0, 0], [0, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
60. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-3, complete listed estimator values=[-3, 2] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-3, [[-3, 1, 0, 0], [2, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-3, [[-3, 1, 0, 0], [2, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
61. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-2, complete listed estimator values=[-2, -1] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-2, [[-2, 1, 0, 0], [-1, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-2, [[-2, 1, 0, 0], [-1, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
62. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=-1, complete listed estimator values=[-1, 1] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(-1, [[-1, 1, 0, 0], [1, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-1, [[-1, 1, 0, 0], [1, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
63. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=0, complete listed estimator values=[0, 3] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(0, [[0, 1, 0, 0], [3, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (0, [[0, 1, 0, 0], [3, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
64. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=1, complete listed estimator values=[1, 5] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(1, [[1, 1, 0, 0], [5, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (1, [[1, 1, 0, 0], [5, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
65. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=2, complete listed estimator values=[2, 7] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(2, [[2, 1, 0, 0], [7, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (2, [[2, 1, 0, 0], [7, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
66. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=3, complete listed estimator values=[3, 4] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(3, [[3, 1, 0, 0], [4, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (3, [[3, 1, 0, 0], [4, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
67. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=4, complete listed estimator values=[4, 6] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(4, [[4, 1, 0, 0], [6, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (4, [[4, 1, 0, 0], [6, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
68. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=5, complete listed estimator values=[5, 8] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(5, [[5, 1, 0, 0], [8, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (5, [[5, 1, 0, 0], [8, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
69. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=6, complete listed estimator values=[6, 10] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(6, [[6, 1, 0, 0], [10, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (6, [[6, 1, 0, 0], [10, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
70. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=7, complete listed estimator values=[7, 12] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(7, [[7, 1, 0, 0], [12, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (7, [[7, 1, 0, 0], [12, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
71. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=8, complete listed estimator values=[8, 9] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(8, [[8, 1, 0, 0], [9, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (8, [[8, 1, 0, 0], [9, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
72. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=9, complete listed estimator values=[9, 11] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(9, [[9, 1, 0, 0], [11, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (9, [[9, 1, 0, 0], [11, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
73. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=10, complete listed estimator values=[10, 13] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(10, [[10, 1, 0, 0], [13, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (10, [[10, 1, 0, 0], [13, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
74. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=11, complete listed estimator values=[11, 15] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(11, [[11, 1, 0, 0], [15, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (11, [[11, 1, 0, 0], [15, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
75. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row
Use scalar theta=12, complete listed estimator values=[12, 17] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).
Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.
View answer and explanation
(12, [[12, 1, 0, 0], [17, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)
Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (12, [[12, 1, 0, 0], [17, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate
76. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-12, complete listed estimator values=[-13, -12, -11] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-12, -13, 1, -12, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-12, -13, 1, -12, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
77. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-11, complete listed estimator values=[-13, -11, -9] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-11, -13, 4, -11, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-11, -13, 4, -11, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
78. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-10, complete listed estimator values=[-13, -10, -7] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-10, -13, 9, -10, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-10, -13, 9, -10, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
79. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-9, complete listed estimator values=[-13, -9, -5] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-9, -13, 16, -9, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-9, -13, 16, -9, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
80. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-8, complete listed estimator values=[-13, -8, -3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-8, -13, 25, -8, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-8, -13, 25, -8, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
81. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-7, complete listed estimator values=[-8, -7, -6] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-7, -8, 1, -7, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-7, -8, 1, -7, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
82. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-6, complete listed estimator values=[-8, -6, -4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-6, -8, 4, -6, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-6, -8, 4, -6, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
83. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-5, complete listed estimator values=[-8, -5, -2] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-5, -8, 9, -5, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-5, -8, 9, -5, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
84. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-4, complete listed estimator values=[-8, -4, 0] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-4, -8, 16, -4, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-4, -8, 16, -4, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
85. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-3, complete listed estimator values=[-8, -3, 2] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-3, -8, 25, -3, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-3, -8, 25, -3, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
86. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-2, complete listed estimator values=[-3, -2, -1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-2, -3, 1, -2, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-2, -3, 1, -2, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
87. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=-1, complete listed estimator values=[-3, -1, 1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(-1, -3, 4, -1, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-1, -3, 4, -1, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
88. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=0, complete listed estimator values=[-3, 0, 3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(0, -3, 9, 0, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (0, -3, 9, 0, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
89. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=1, complete listed estimator values=[-3, 1, 5] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(1, -3, 16, 1, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (1, -3, 16, 1, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
90. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=2, complete listed estimator values=[-3, 2, 7] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(2, -3, 25, 2, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (2, -3, 25, 2, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
91. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=3, complete listed estimator values=[2, 3, 4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(3, 2, 1, 3, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (3, 2, 1, 3, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
92. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=4, complete listed estimator values=[2, 4, 6] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(4, 2, 4, 4, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (4, 2, 4, 4, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
93. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=5, complete listed estimator values=[2, 5, 8] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(5, 2, 9, 5, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (5, 2, 9, 5, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
94. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=6, complete listed estimator values=[2, 6, 10] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(6, 2, 16, 6, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (6, 2, 16, 6, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
95. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=7, complete listed estimator values=[2, 7, 12] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(7, 2, 25, 7, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (7, 2, 25, 7, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
96. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=8, complete listed estimator values=[7, 8, 9] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(8, 7, 1, 8, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (8, 7, 1, 8, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
97. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=9, complete listed estimator values=[7, 9, 11] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(9, 7, 4, 9, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (9, 7, 4, 9, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
98. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=10, complete listed estimator values=[7, 10, 13] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(10, 7, 9, 10, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (10, 7, 9, 10, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
99. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=11, complete listed estimator values=[7, 11, 15] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(11, 7, 16, 11, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (11, 7, 16, 11, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
100. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency
Use scalar theta=12, complete listed estimator values=[7, 12, 17] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).
observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.
View answer and explanation
(12, 7, 25, 12, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)
Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (12, 7, 25, 12, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency).
Reviewed step-by-step derivation
Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate
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