A bias table over the whole parameter set

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

U^{-1}(0),U^{-1}(1/4),U^{-1}(1/2)

\{00\},\{01,10\},\{11\}

P_{1/4}^U=(9/16,6/16,1/16)

E_{1/4}U=(1/4)(6/16)+(1/2)(1/16)=1/8

b(1/4)=E_{1/4}U-1/4

b(1/4)=1/8-1/4=-1/8

P_{3/4}^U=(1/16,6/16,9/16)

E_{3/4}U=(1/4)(6/16)+(1/2)(9/16)=3/8

b(3/4)=E_{3/4}U-3/4

b(3/4)=3/8-3/4=-3/8

\text{unbiased}\iff\forall\theta\in\Theta:\ b(\theta)=0

U\text{ is biased on this model}

\theta=1/4,\quad x=01

U(01)-\theta=1/4-1/4=0\ne b(1/4)

b=( -1/8,-3/8)

b(\theta)=-\theta/2\quad(\theta\in\Theta)

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

Target

Subtract theta, not an observation or another law's mean.

All parameters

Retain 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)