# A bias table over the whole parameter set [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-estimator-bias/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-estimator-bias-01 · ocme-new142-reader-adoption-20261021 ## Goal and scope Compute EthetaU-theta at every theta, distinct from observed error. Target g(theta)=theta; no asymptotic claim. ## Givens and symbols - work - G1 — Theta={1/4,3/4}, Omega={00,01,10,11}; P_(1/4)=(9,3,3,1)/16 and P_(3/4)=(1,3,3,9)/16. - G2 — U(x)=(x1+x2)/4 has total table(0,1/4,1/4,1/2) in Omega order, with target g(theta)=theta. - θ — fixed unknown parameter - U — one total statistic ## Rules and sources - mko_definition · cited — Use one common finite sample space and total statistic independent of theta. For each theta compute E_theta[T] from its sampling distribution and subtract theta. One parameter, sample or observed estimate does not establish unbiasedness over the model. Unbiasedness alone does not give zero variance, consistency, sufficiency or minimum MSE. · mko-estimator-bias - work · prerequisite — Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes. Bias is b(theta)=Etheta[U]-theta; unbiasedness requires zero bias at every theta. ## Main derivation S01 · impliesBefore U^{-1}(0),U^{-1}(1/4),U^{-1}(1/2) After \{00\},\{01,10\},\{11\} Conditions: none Rule: work · Uses: G1, G2, work Leaf basis: work Both laws use the same three fibres of U. S02 · impliesBefore P_{1/4}^U=(9/16,6/16,1/16) After E_{1/4}U=(1/4)(6/16)+(1/2)(1/16)=1/8 Conditions: none Rule: work · Uses: G1, G2, S01, work Leaf basis: work The complete three-point sampling law gives the first mean. S03 · impliesBefore b(1/4)=E_{1/4}U-1/4 After b(1/4)=1/8-1/4=-1/8 Conditions: none Rule: work · Uses: G1, G2, S02, work Leaf basis: work Subtract the specified target theta, not an observed estimate. S04 · impliesBefore P_{3/4}^U=(1/16,6/16,9/16) After E_{3/4}U=(1/4)(6/16)+(1/2)(9/16)=3/8 Conditions: none Rule: work · Uses: G1, G2, S03, work Leaf basis: work Recompute the mean under the second law. S05 · impliesBefore b(3/4)=E_{3/4}U-3/4 After b(3/4)=3/8-3/4=-3/8 Conditions: none Rule: work · Uses: G1, G2, S04, work Leaf basis: work Bias at the second parameter is also nonzero. S06 · impliesBefore \text{unbiased}\iff\forall\theta\in\Theta:\ b(\theta)=0 After U\text{ is biased on this model} Conditions: none Rule: work · Uses: G1, G2, S05, work Leaf basis: work Any nonzero bias refutes unbiasedness; here both are nonzero. S07 · impliesBefore \theta=1/4,\quad x=01 After U(01)-\theta=1/4-1/4=0\ne b(1/4) Conditions: none Rule: work · Uses: G1, G2, S06, work Leaf basis: work One zero observed error can coexist with nonzero bias. S08 · impliesBefore b=( -1/8,-3/8) After b(\theta)=-\theta/2\quad(\theta\in\Theta) Conditions: none Rule: work · Uses: G1, G2, S07, work Leaf basis: work Verify the compact formula on the complete two-point parameter set without extending an unspecified model. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · G1, G2, S07, work — Consistency and minimum MSE remain unproved by this bias table. - S07 · premise_audit · G1, G2, S06, work — The observed error has not been averaged under the sampling law. - S06 · premise_audit · G1, G2, S05, work — Unbiasedness is a whole-parameter condition, not success at one chosen theta. - S05 · premise_audit · G1, G2, S04, work — These two values exhaust the parameter set. - S04 · premise_audit · G1, G2, S03, work — Do not reuse1/16 for the last mass instead of 9/16. - S03 · premise_audit · G1, G2, S02, work — Bias is signed; do not take an absolute value first. - S02 · premise_audit · G1, G2, S01, work — The zero-value mass remains part of the law even though its mean contribution vanishes. - S01 · premise_audit · G1, G2, work — U cannot change with theta while computing bias. ## Error controls and negative controls TargetSubtract theta, not an observation or another law's mean. All parametersRetain signed bias separately at both theta values. - Observed 01 hits theta=1/4, so U is unbiased. → reject — Under that law the mean is 1/8 and bias-1/8. - Bias must be nonnegative. → reject — Both biases here are negative. - One theta check proves modelwide unbiasedness. → reject — The universal condition covers every theta, including the other value here. ## Result and limitations Biases are -1/8 and -3/8, so U is biased; one exact observed estimate does not change that. ## Verification and source binding 2 declared checks for this guide; only the listed expressions are covered. - source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f - adopted source SHA-256: d114601f2cd77f86c632babb6a0f10f4d14465a1b73f2a05799b3a8f9c90cbeb - original source SHA-256: a5d42a968e40f0a69ed6cc147607b52fd1f7dbdc6be2f7247051d81167c6668f - Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW [Same-source full JSON (formulas, ASTs, controls and revision log)](https://ocme.evemisslab.com/data/derivations/coverage-20261021/guides/derivation-mko-estimator-bias-01.json)