Estimator bias

Mathematical statement

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.

Representative formula (exact TeX)

B_theta(T)=E_theta(T)-theta

Assumptions

• All target theta values, complete exact parameter laws and the common total statistic are explicit.
• This slice fixes g(theta)=theta; any other target must be specified separately.

Symbol scope

• B_theta — estimator bias
• T — estimator

Intuition, definition, and misconceptions

Bias is a parameter-indexed family.

Unbiasedness requires zero bias over the entire declared parameter space.

An observed estimate is not bias; every other statistical property has its own evidence obligation.

One estimate equal to theta does not establish an unbiased estimator.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Sampling distribution — Each parameter has its estimator law.
• Expectation in finite models — The bias uses E_theta[T].
• Parametric statistical model — The full parameter space and laws are supplied.

Verification boundary

Only complete finite parameter models and exact rational expectations are handled; no asymptotic property is claimed.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add MSE and the bias-variance decomposition.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.