Parameter-Indexed Expectations, Bias and Full-Model Unbiasedness
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batch-20261014-advance-20261106-r03-estimator-bias-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 2, 3/2, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 2, 3/2, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
2. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 1/2, 13/5, 21/10, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 1/2, 13/5, 21/10, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
3. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 11/3, 19/6, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 11/3, 19/6, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
4. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 2/3, 71/17, 179/51, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 2/3, 71/17, 179/51, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
5. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 38/7, 69/14, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 38/7, 69/14, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
6. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 3/4, 111/19, 387/76, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 3/4, 111/19, 387/76, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
7. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 29/4, 27/4, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 29/4, 27/4, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
8. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 4/5, 53/7, 237/35, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 4/5, 53/7, 237/35, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
9. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
10. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 5/6, 215/23, 1175/138, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 5/6, 215/23, 1175/138, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
11. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 11, 21/2, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 11, 21/2, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
12. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 6/7, 279/25, 1803/175, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 6/7, 279/25, 1803/175, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
13. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 142/11, 273/22, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 142/11, 273/22, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
14. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 7/8, 13, 97/8, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 7/8, 13, 97/8, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
15. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 89/6, 43/3, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 89/6, 43/3, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
16. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 8/9, 431/29, 3647/261, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 8/9, 431/29, 3647/261, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
17. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 218/13, 423/26, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 218/13, 423/26, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
18. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 9/10, 519/31, 4911/310, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 9/10, 519/31, 4911/310, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
19. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 131/7, 255/14, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 131/7, 255/14, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
20. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 10/11, 205/11, 195/11, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 10/11, 205/11, 195/11, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
21. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 62/3, 121/6, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 62/3, 121/6, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
22. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 11/12, 719/35, 8243/420, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 11/12, 719/35, 8243/420, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
23. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 181/8, 177/8, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 181/8, 177/8, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
24. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(1, 12/13, 831/37, 10359/481, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 12/13, 831/37, 10359/481, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
25. Subtract the Target from the Expectation under That Same Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).
View answer and explanation
(0, 1/2, 418/17, 819/34, expectation_from_same_theta_sampling_distribution)
Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 418/17, 819/34, expectation_from_same_theta_sampling_distribution).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-at-parameter · candidate
26. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 2, 3/2], [1, 1/3, 13/7, 32/21]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 2, 3/2], [1, 1/3, 13/7, 32/21]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
27. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 31/11, 51/22], [1, 1/2, 13/5, 21/10]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 31/11, 51/22], [1, 1/2, 13/5, 21/10]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
28. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 11/3, 19/6], [1, 3/5, 27/8, 111/40]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 11/3, 19/6], [1, 3/5, 27/8, 111/40]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
29. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 59/13, 105/26], [1, 2/3, 71/17, 179/51]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 59/13, 105/26], [1, 2/3, 71/17, 179/51]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
30. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 38/7, 69/14], [1, 5/7, 5, 30/7]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 38/7, 69/14], [1, 5/7, 5, 30/7]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
31. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 19/3, 35/6], [1, 3/4, 111/19, 387/76]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 19/3, 35/6], [1, 3/4, 111/19, 387/76]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
32. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 29/4, 27/4], [1, 7/9, 67/10, 533/90]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 29/4, 27/4], [1, 7/9, 67/10, 533/90]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
33. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 139/17, 261/34], [1, 4/5, 53/7, 237/35]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 139/17, 261/34], [1, 4/5, 53/7, 237/35]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
34. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 82/9, 155/18], [1, 9/11, 93/11, 84/11]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 82/9, 155/18], [1, 9/11, 93/11, 84/11]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
35. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 191/19, 363/38], [1, 5/6, 215/23, 1175/138]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 191/19, 363/38], [1, 5/6, 215/23, 1175/138]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
36. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 11, 21/2], [1, 11/13, 41/4, 489/52]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 11, 21/2], [1, 11/13, 41/4, 489/52]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
37. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 251/21, 481/42], [1, 6/7, 279/25, 1803/175]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 251/21, 481/42], [1, 6/7, 279/25, 1803/175]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
38. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 142/11, 273/22], [1, 13/15, 157/13, 2186/195]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 142/11, 273/22], [1, 13/15, 157/13, 2186/195]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
39. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 319/23, 615/46], [1, 7/8, 13, 97/8]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 319/23, 615/46], [1, 7/8, 13, 97/8]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
40. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 89/6, 43/3], [1, 15/17, 195/14, 3105/238]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 89/6, 43/3], [1, 15/17, 195/14, 3105/238]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
41. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 79/5, 153/10], [1, 8/9, 431/29, 3647/261]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 79/5, 153/10], [1, 8/9, 431/29, 3647/261]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
42. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 218/13, 423/26], [1, 17/19, 79/5, 1416/95]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 218/13, 423/26], [1, 17/19, 79/5, 1416/95]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
43. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 479/27, 931/54], [1, 9/10, 519/31, 4911/310]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 479/27, 931/54], [1, 9/10, 519/31, 4911/310]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
44. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 131/7, 255/14], [1, 19/21, 283/16, 5639/336]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 131/7, 255/14], [1, 19/21, 283/16, 5639/336]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
45. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 571/29, 1113/58], [1, 10/11, 205/11, 195/11]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 571/29, 1113/58], [1, 10/11, 205/11, 195/11]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
46. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 62/3, 121/6], [1, 21/23, 333/17, 7302/391]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 62/3, 121/6], [1, 21/23, 333/17, 7302/391]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
47. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 671/31, 1311/62], [1, 11/12, 719/35, 8243/420]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 671/31, 1311/62], [1, 11/12, 719/35, 8243/420]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
48. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 181/8, 177/8], [1, 23/25, 43/2, 1029/50]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 181/8, 177/8], [1, 23/25, 43/2, 1029/50]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
49. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 779/33, 1525/66], [1, 12/13, 831/37, 10359/481]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 779/33, 1525/66], [1, 12/13, 831/37, 10359/481]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
50. Bias Is an Indexed Family, Not One Observed Error
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).
View answer and explanation
([[0, 1/2, 418/17, 819/34], [1, 25/27, 445/19, 11540/513]], bias_is_parameter_indexed_family)
For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 418/17, 819/34], [1, 25/27, 445/19, 11540/513]], bias_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-family-readback · candidate
51. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 3/2], [1, 32/21]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 3/2], [1, 32/21]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
52. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 51/22], [1, 21/10]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 51/22], [1, 21/10]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
53. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 19/6], [1, 111/40]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 19/6], [1, 111/40]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
54. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 105/26], [1, 179/51]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 105/26], [1, 179/51]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
55. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 69/14], [1, 30/7]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 69/14], [1, 30/7]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
56. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 35/6], [1, 387/76]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 35/6], [1, 387/76]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
57. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 27/4], [1, 533/90]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 27/4], [1, 533/90]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
58. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 261/34], [1, 237/35]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 261/34], [1, 237/35]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
59. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 155/18], [1, 84/11]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 155/18], [1, 84/11]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
60. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 363/38], [1, 1175/138]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 363/38], [1, 1175/138]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
61. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 21/2], [1, 489/52]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 21/2], [1, 489/52]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
62. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 481/42], [1, 1803/175]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 481/42], [1, 1803/175]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
63. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 273/22], [1, 2186/195]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 273/22], [1, 2186/195]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
64. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 615/46], [1, 97/8]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 615/46], [1, 97/8]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
65. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 43/3], [1, 3105/238]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 43/3], [1, 3105/238]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
66. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 153/10], [1, 3647/261]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 153/10], [1, 3647/261]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
67. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 423/26], [1, 1416/95]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 423/26], [1, 1416/95]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
68. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 931/54], [1, 4911/310]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 931/54], [1, 4911/310]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
69. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 255/14], [1, 5639/336]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 255/14], [1, 5639/336]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
70. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 1113/58], [1, 195/11]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1113/58], [1, 195/11]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
71. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 121/6], [1, 7302/391]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 121/6], [1, 7302/391]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
72. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 1311/62], [1, 8243/420]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1311/62], [1, 8243/420]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
73. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 177/8], [1, 1029/50]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 177/8], [1, 1029/50]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
74. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 1525/66], [1, 10359/481]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1525/66], [1, 10359/481]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
75. Unbiasedness Requires Zero Bias at Every Declared Parameter
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).
View answer and explanation
([[0, 819/34], [1, 11540/513]], No, unbiased_requires_every_theta)
Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 819/34], [1, 11540/513]], No, unbiased_requires_every_theta).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-unbiased-audit · candidate
76. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 0, 0, 3/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 3/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
77. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 1, 1, 21/10, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 21/10, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
78. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 2, 4, 19/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 4, 19/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
79. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 3, 6, 179/51, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 6, 179/51, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
80. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 0, 0, 69/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 69/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
81. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 1, 1, 387/76, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 387/76, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
82. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 2, 8, 27/4, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 8, 27/4, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
83. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 3, 10, 237/35, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 10, 237/35, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
84. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 0, 0, 155/18, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 155/18, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
85. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 1, 1, 1175/138, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 1175/138, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
86. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 2, 12, 21/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 12, 21/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
87. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 3, 14, 1803/175, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 14, 1803/175, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
88. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 0, 0, 273/22, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 273/22, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
89. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 1, 1, 97/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 97/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
90. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 2, 16, 43/3, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 16, 43/3, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
91. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 3, 18, 3647/261, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 18, 3647/261, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
92. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 0, 0, 423/26, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 423/26, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
93. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 1, 1, 4911/310, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 4911/310, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
94. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 2, 20, 255/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 20, 255/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
95. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 3, 22, 195/11, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 22, 195/11, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
96. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 0, 0, 121/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 121/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
97. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 1, 1, 8243/420, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 8243/420, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
98. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 2, 24, 177/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 24, 177/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
99. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(1, 3, 26, 10359/481, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 26, 10359/481, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
100. One Realized Estimate Does Not Establish Bias or Other Properties
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).
No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.
View answer and explanation
(0, 0, 0, 819/34, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)
Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 819/34, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).
Reviewed step-by-step derivation
Estimator bias · qf-mod37-bias-observed-boundary · candidate
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