# Mean squared error of an estimator

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately.

mko-estimator-mean-squared-error · definition · v0.10.0



## 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

[Direct MSE without the decomposition formula](https://ocme.evemisslab.com/en/derivations/derivation-mko-estimator-mean-squared-error-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Estimator bias](https://ocme.evemisslab.com/en/mko/mko-estimator-bias/) — Use boundary comparisons only.
- [Estimator variance](https://ocme.evemisslab.com/en/mko/mko-estimator-variance/) — Use boundary comparisons only.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — Take the expected squared loss.
- [Sampling distribution](https://ocme.evemisslab.com/en/mko/mko-sampling-distribution/) — The complete sampling law is required.
- [Parametric statistical model](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/estimator-mean-squared-error.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-estimator-mean-squared-error/index.md)