Mean squared error of an estimator

Mathematical statement

Each theta requires a complete normalized finite sampling law. Compute sum over listed t of P_theta(T=t)*(t−theta)^2 directly, without using the later bias-variance decomposition. A multi-theta table keeps each law separate. MSE0 is equivalent to T=theta almost surely under P_theta. One observed estimate/loss does not define the expected loss, though its value may coincide. Unbiasedness does not imply zero MSE, and MSE is neither automatically variance nor a consistency certificate. This slice uses scalar parameters and exact rational laws.

Representative formula (exact TeX)

M(T,t)=e

Assumptions

• Listed estimator values, theta and probabilities are exact rationals, with nonnegative probabilities summing to1.
• The multi-theta family lists every parametric sampling law completely.

Symbol scope

• M — mean-squared-error operator
• T — scalar estimator
• t — target parameter
• e — exact expected squared error

Intuition, definition, and misconceptions

Define risk separately at each theta.

Weight squared errors under the complete finite law.

Separate zero-MSE, observed-error and unbiasedness boundaries.

A single sample error is not the definition of MSE, and unbiasedness alone does not give MSE0.

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

• Estimator bias — Use boundary comparisons only.
• Estimator variance — Use boundary comparisons only.
• Expectation in finite models — Take the expected squared loss.
• Sampling distribution — The complete sampling law is required.
• Parametric statistical model — Keep theta-indexed laws separate.

Verification boundary

No early bias-variance decomposition, consistency or asymptotic-optimality conclusion is used.

not_formalized

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

Next obligation

Add 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.