# Question 19363: One Realized Estimate Does Not Establish Bias or Other Properties [Return to catalogue page 97](https://ocme.evemisslab.com/en/questions/page/97/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r03-estimator-bias-foundations/#q-20261014-advance-20261106-r03-estimator-bias-0088) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0037-19363 · q-20261014-advance-20261106-r03-estimator-bias-0088 ## Original question and givens 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). - G001 — 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). ## 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](https://ocme.evemisslab.com/en/mko/mko-estimator-bias/) ## Step-by-step derivation S001 · impliesBefore source operation=bias-observed-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore bias parameter0 full law mass := sum(1/22, 1/11, 3/22, 8/11) After bias parameter0 full law mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore bias:original parameter SLOT=0;same outcome carrier Fin(4);ALL aligned law masses=[1/22, 1/11, 3/22, 8/11] After full sum=1;all masses nonnegative=Yes;qualified probability law;zero-mass outcomes retained Conditions: none Each complete parameter law must separately be nonnegative and total1.Keep zero-mass outcomes in the original table;do not normalize across parameter rows to invent a prior or posterior. Step identifiers and dependenciesRule: finite_statistical_full_law_not_partial_or_prior · Uses: G001, S002 S004 · calculationBefore bias parameter1 full law mass := sum(1/13, 3/26, 2/13, 17/26) After bias parameter1 full law mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore bias:original parameter SLOT=1;same outcome carrier Fin(4);ALL aligned law masses=[1/13, 3/26, 2/13, 17/26] After full sum=1;all masses nonnegative=Yes;qualified probability law;zero-mass outcomes retained Conditions: none Each complete parameter law must separately be nonnegative and total1.Keep zero-mass outcomes in the original table;do not normalize across parameter rows to invent a prior or posterior. Step identifiers and dependenciesRule: finite_statistical_full_law_not_partial_or_prior · Uses: G001, S004 S006 · impliesBefore bias:parameter SLOT carrier Fin(2);common outcome carrier Fin(4);ONE total original statistic array=[0, 1, 14, 15] After 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 Conditions: none Functional parameter-independence is different from a parameter-free distribution of T(X).Use the same original statistic value at every outcome rather than changing T between laws. Step identifiers and dependenciesRule: finite_statistic_definition_not_distribution_free · Uses: G001, S005 S007 · impliesBefore original parameter SLOT order0..1;aligned exact theta_values=[1/2, 13/15] After declared target g(theta)=theta;indices are not targets;claims range over THIS supplied finite parameter model only Conditions: none Keep target values separate from parameter indices;theta values need not be distinct.No unprovided continuous parameter space,prior or sequence of statistics is supplied,so a universal claim within this finite model is not extended beyond it. Step identifiers and dependenciesRule: bias_declared_target_and_finite_parameter_scope · Uses: G001, S006 S008 · calculationBefore bias parameter0 original outcome0 weighted statistic := mul(1/22, 0) After bias parameter0 original outcome0 weighted statistic = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore bias parameter0 original outcome1 weighted statistic := mul(1/11, 1) After bias parameter0 original outcome1 weighted statistic = 1/11 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore bias parameter0 original outcome2 weighted statistic := mul(3/22, 14) After bias parameter0 original outcome2 weighted statistic = 21/11 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore bias parameter0 original outcome3 weighted statistic := mul(8/11, 15) After bias parameter0 original outcome3 weighted statistic = 120/11 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore bias parameter0 same-law expectation := sum(0, 1/11, 21/11, 120/11) After bias parameter0 same-law expectation = 142/11 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore bias parameter0 expectation minus actual target := sub(142/11, 1/2) After bias parameter0 expectation minus actual target = 273/22 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore fixed original parameter SLOT=0;actual aligned theta VALUE=1/2;SAME complete law=[1/22, 1/11, 3/22, 8/11];same statistic=[0, 1, 14, 15];weighted terms=[0, 1/11, 21/11, 120/11] After E_at_slot[T]=142/11;Bias=E_at_slot[T]-ACTUAL theta value=273/22 Conditions: none The target g(theta)=theta is now declared.Subtract the aligned theta_values entry,not its slot index or one observed estimate,and use no other parameter's law. Step identifiers and dependenciesRule: bias_same_law_expectation_minus_target_not_index · Uses: G001, S013 S015 · impliesBefore original parameter SLOT=0;original outcome=0;observed estimate=0;SEPARATE same-law bias=273/22 After single-observation property sufficiency=No;NOT a comparison of whether estimate numerically equals bias;no zero-variance/consistency/sufficiency/minimum-MSE claim Conditions: none The No withholds property claims from one observation;it does not assert numerical inequality of estimate and bias.Unbiasedness itself does not imply zero variance,consistency,sufficiency or minimum MSE;these require their own target and sampling framework. Step identifiers and dependenciesRule: bias_observed_No_nonclaim_not_numerical_inequality · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, 0, 0, 273/22, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE) 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: 51931dcc800f5f6aac2ad77eef546a1b8d6a4cd58db7166f5f857ae7813c7288 - candidate record SHA-256: 8513800ef3883875eb03437dec14e138aeeecc31f8f87e7d535375a620c79061 - semantic review receipt SHA-256: 065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80 - question-first witness SHA-256: 7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r03-estimator-bias-0088.json)