Estimator variance

Mathematical statement

Keep sampling law, positive-mass support and mean at the same theta. Check Var_theta(T)=E_theta[(T−E_theta[T])^2]=E_theta[T^2]−(E_theta[T])^2 exactly. Zero variance means almost-sure constancy under that law, not equality with theta or unbiasedness.

Representative formula (exact TeX)

V_theta(T)=E_theta(T^2)-E_theta(T)^2

Assumptions

• The common finite sample space, complete exact parameter laws and fixed total statistic are supplied.
• All second moments are finite and use exact rational arithmetic.

Symbol scope

• V_theta — estimator variance
• T — estimator

Intuition, definition, and misconceptions

Variance is a parameter-indexed family.

The second-moment identity provides an exact cross-check.

Variance alone does not establish unbiasedness, consistency or MSE.

Zero variance does not imply unbiasedness, and variance is not MSE without checking the target and bias.

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 a law of the estimator.
• Variance in finite probability models — The criterion is a second central moment.

Verification boundary

Only complete finite sampling laws are handled; consistency across sample sizes requires a sequence model.

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 efficiency comparisons.

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.