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Question 25136: One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0049-25136 · q-20261015-advance-20261114-r05-estimator-mean-squared-error-0086

Original question and givens

Use scalar theta=-2, complete listed estimator values=[-3, -2, -1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).

  • G001 — Use scalar theta=-2, complete listed estimator values=[-3, -2, -1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [mse-observed-unbiased-boundary]

After

derived = [[theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary]]

Conditions: none

List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

MSE ORIGINAL single theta ORIGINAL probability cumulative0 := sum(0, 1/4)

After

MSE ORIGINAL single theta ORIGINAL probability cumulative0 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

MSE ORIGINAL single theta ORIGINAL probability cumulative1 := sum(1/4, 1/2)

After

MSE ORIGINAL single theta ORIGINAL probability cumulative1 = 3/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

MSE ORIGINAL single theta ORIGINAL probability cumulative2 := sum(3/4, 1/4)

After

MSE ORIGINAL single theta ORIGINAL probability cumulative2 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [MSE ORIGINAL single theta, -2, [-3, -2, -1], [[1, 4], [1, 2], [1, 4]]]

After

derived = [[[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1]

Conditions: none

Fix this same P_theta.Check original-order nonnegative masses summing to1,then sort values numerically with their aligned masses.Keep null rows,without renormalizing or treating estimator values as parameter indices.

Step identifiers and dependencies

Rule: MSE_same_theta_complete_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S004

S006 · calculation

Before

MSE ORIGINAL single theta exact mean product0 := mul(-3, 1/4)

After

MSE ORIGINAL single theta exact mean product0 = -3/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

MSE ORIGINAL single theta exact mean cumulative0 := sum(0, -3/4)

After

MSE ORIGINAL single theta exact mean cumulative0 = -3/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

MSE ORIGINAL single theta exact mean product1 := mul(-2, 1/2)

After

MSE ORIGINAL single theta exact mean product1 = -1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

MSE ORIGINAL single theta exact mean cumulative1 := sum(-3/4, -1)

After

MSE ORIGINAL single theta exact mean cumulative1 = -7/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

MSE ORIGINAL single theta exact mean product2 := mul(-1, 1/4)

After

MSE ORIGINAL single theta exact mean product2 = -1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

MSE ORIGINAL single theta exact mean cumulative2 := sum(-7/4, -1/4)

After

MSE ORIGINAL single theta exact mean cumulative2 = -2

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

MSE ORIGINAL single theta error row0 := sub(-3, -2)

After

MSE ORIGINAL single theta error row0 = -1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

MSE ORIGINAL single theta squared error row0 := mul(-1, -1)

After

MSE ORIGINAL single theta squared error row0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

MSE ORIGINAL single theta weighted error row0 := mul(1/4, 1)

After

MSE ORIGINAL single theta weighted error row0 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

MSE ORIGINAL single theta MSE cumulative0 := sum(0, 1/4)

After

MSE ORIGINAL single theta MSE cumulative0 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

MSE ORIGINAL single theta error row1 := sub(-2, -2)

After

MSE ORIGINAL single theta error row1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

MSE ORIGINAL single theta squared error row1 := mul(0, 0)

After

MSE ORIGINAL single theta squared error row1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

MSE ORIGINAL single theta weighted error row1 := mul(1/2, 0)

After

MSE ORIGINAL single theta weighted error row1 = 0

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

MSE ORIGINAL single theta MSE cumulative1 := sum(1/4, 0)

After

MSE ORIGINAL single theta MSE cumulative1 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

MSE ORIGINAL single theta error row2 := sub(-1, -2)

After

MSE ORIGINAL single theta error row2 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

MSE ORIGINAL single theta squared error row2 := mul(1, 1)

After

MSE ORIGINAL single theta squared error row2 = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

MSE ORIGINAL single theta weighted error row2 := mul(1/4, 1)

After

MSE ORIGINAL single theta weighted error row2 = 1/4

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

MSE ORIGINAL single theta MSE cumulative2 := sum(1/4, 1/4)

After

MSE ORIGINAL single theta MSE cumulative2 = 1/2

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

source roles = [MSE ORIGINAL single theta, -2, [[-3, 1/4, 1, 1/4], [-2, 1/2, 0, 0], [-1, 1/4, 1, 1/4]]]

After

derived = [-2, 1/2, No, No, Yes]

Conditions: none

MSE directly sums all weighted squared errors,without a bias-variance decomposition.Nonnegative terms sum to0 iff the error vanishes at every positive-mass value;null values may differ.Both sides False still give a True iff.

Step identifiers and dependencies

Rule: MSE_nonnegative_terms_zero_IFF_POSITIVE_mass_equality · Uses: G001, S023

S025 · calculation

Before

MSE ORIGINAL observed minus theta := sub(-3, -2)

After

MSE ORIGINAL observed minus theta = -1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

MSE ORIGINAL one observed squared error := mul(-1, -1)

After

MSE ORIGINAL one observed squared error = 1

Conditions: none

Substitute original roles and established intermediates exactly,without rounding or substituting a single observation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

source roles = [-2, -3, 1, -2, 1/2]

After

derived = [Yes, No, Yes, No]

Conditions: none

One observed loss may accidentally equal MSE and may be a listed null value;the risk still uses the whole law.E[T]=theta need not imply zero risk.There is no n-indexed asymptotic law;No only withholds consistency.

Step identifiers and dependencies

Rule: MSE_observed_loss_unbiasedness_risk_and_consistency_separate · Uses: G001, S026

Reverse checks

  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Recompute MSE ORIGINAL one observed squared error with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute MSE ORIGINAL observed minus theta with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative2 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row2 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row2 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row2 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative1 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row1 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row1 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative0 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row0 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row0 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row0 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative2 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product2 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product1 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative2 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative0 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(-2, -3, 1, -2, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: e5ec9b285b64dcd8f1f62a6582437207725ac943d13297508e3a197af0dfd18d
  • candidate record SHA-256: afdcbbf8c2d1c000613a5ce5c997a3c55a253b3dd55dd5c8cce50dff617339ac
  • semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
  • question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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