Question 19378: Unbiasedness Requires Zero Bias at Every Declared Parameter

Original question and givens

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).

• G001 — 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).

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.

• mko-estimator-bias

Step-by-step derivation

source operation=bias-unbiased-audit;original snapshot and mathematics retained

explicit ordered source output fields=[parameter_biases, unbiased_all_parameters, boundary]

bias parameter0 full law mass := sum(1/12, 1/6, 1/4, 1/2)

bias parameter0 full law mass = 1

bias:original parameter SLOT=0;same outcome carrier Fin(4);ALL aligned law masses=[1/12, 1/6, 1/4, 1/2]

full sum=1;all masses nonnegative=Yes;qualified probability law;zero-mass outcomes retained

bias parameter1 full law mass := sum(1/8, 3/16, 1/4, 7/16)

bias parameter1 full law mass = 1

bias:original parameter SLOT=1;same outcome carrier Fin(4);ALL aligned law masses=[1/8, 3/16, 1/4, 7/16]

full sum=1;all masses nonnegative=Yes;qualified probability law;zero-mass outcomes retained

bias:parameter SLOT carrier Fin(2);common outcome carrier Fin(4);ONE total original statistic array=[0, 1, 4, 5]

T takes only the observation index,not the unknown parameter;SAME function at all parameter slots;its sampling law may still vary with the slot

original parameter SLOT order0..1;aligned exact theta_values=[1/2, 3/5]

declared target g(theta)=theta;indices are not targets;claims range over THIS supplied finite parameter model only

bias parameter0 original outcome0 weighted statistic := mul(1/12, 0)

bias parameter0 original outcome0 weighted statistic = 0

bias parameter0 original outcome1 weighted statistic := mul(1/6, 1)

bias parameter0 original outcome1 weighted statistic = 1/6

bias parameter0 original outcome2 weighted statistic := mul(1/4, 4)

bias parameter0 original outcome2 weighted statistic = 1

bias parameter0 original outcome3 weighted statistic := mul(1/2, 5)

bias parameter0 original outcome3 weighted statistic = 5/2

bias parameter0 same-law expectation := sum(0, 1/6, 1, 5/2)

bias parameter0 same-law expectation = 11/3

bias parameter0 expectation minus actual target := sub(11/3, 1/2)

bias parameter0 expectation minus actual target = 19/6

fixed original parameter SLOT=0;actual aligned theta VALUE=1/2;SAME complete law=[1/12, 1/6, 1/4, 1/2];same statistic=[0, 1, 4, 5];weighted terms=[0, 1/6, 1, 5/2]

E_at_slot[T]=11/3;Bias=E_at_slot[T]-ACTUAL theta value=19/6

bias parameter1 original outcome0 weighted statistic := mul(1/8, 0)

bias parameter1 original outcome0 weighted statistic = 0

bias parameter1 original outcome1 weighted statistic := mul(3/16, 1)

bias parameter1 original outcome1 weighted statistic = 3/16

bias parameter1 original outcome2 weighted statistic := mul(1/4, 4)

bias parameter1 original outcome2 weighted statistic = 1

bias parameter1 original outcome3 weighted statistic := mul(7/16, 5)

bias parameter1 original outcome3 weighted statistic = 35/16

bias parameter1 same-law expectation := sum(0, 3/16, 1, 35/16)

bias parameter1 same-law expectation = 27/8

bias parameter1 expectation minus actual target := sub(27/8, 3/5)

bias parameter1 expectation minus actual target = 111/40

fixed original parameter SLOT=1;actual aligned theta VALUE=3/5;SAME complete law=[1/8, 3/16, 1/4, 7/16];same statistic=[0, 1, 4, 5];weighted terms=[0, 3/16, 1, 35/16]

E_at_slot[T]=27/8;Bias=E_at_slot[T]-ACTUAL theta value=111/40

ALL original parameter bias rows=[[0, 19/6], [1, 111/40]]

every declared parameter bias zero=No;unbiased_requires_every_theta

Reverse checks

• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute bias parameter1 expectation minus actual target with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute bias parameter1 same-law expectation with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute bias parameter1 original outcome3 weighted statistic with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute bias parameter1 original outcome2 weighted statistic with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute bias parameter1 original outcome1 weighted statistic with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute bias parameter1 original outcome0 weighted statistic with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute bias parameter0 expectation minus actual target with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute bias parameter0 same-law expectation with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute bias parameter0 original outcome3 weighted statistic with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute bias parameter0 original outcome2 weighted statistic with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute bias parameter0 original outcome1 weighted statistic with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute bias parameter0 original outcome0 weighted statistic with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 bias parameter1 full law mass with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute bias parameter0 full law mass 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

([[0, 19/6], [1, 111/40]], No, unbiased_requires_every_theta)

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.