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Mean-squared Error
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DEFINITION AND BASIC PROPERTIES

The MSE of an Estimator \hat{ heta} with respect to the estimated parameter heta is defined as

:\operatorname{MSE}(\hat{ heta})=\operatorname{E}((\hat{ heta}- heta)^2).

It can be shown that the MSE is the sum of the Variance and the square of Bias of the estimator
:\operatorname{MSE}(\hat{ heta})=\operatorname{Var}\left(\hat{ heta} ight)+ \left(\operatorname{Bias}(\hat{ heta}, heta) ight)^2.
In that sense, the MSE assesses the quality of the estimator in terms of its variation and unbiasedness. Note that the MSE is not equivalent to the expected value of the Absolute Error .

The root mean squared error (RMSE) (or '''root mean squared deviation''' (RMSD)) is then simply defined as the square root of the MSE.
:\operatorname{RMSE}(\hat{ heta}) = \sqrt{\operatorname{MSE}(\hat{ heta})}.

The defined MSE (as well as the RMSE) is a Random Variable , that needs to be estimated itself. This is usually done by the Sample Mean
::\operatorname{\widehat{MSE}}(\hat{ heta}) = rac{1}{n} \sum_{j=1}^n \left( heta_j- heta ight)^2
with heta_j being realizations of the estimator \hat{ heta} of size n.


EXAMPLES


Suppose we have a random sample of size ''n'' from a Normally Distributed population, X_1,\dots,X_n\sim\operatorname{N}(\mu,\sigma^2).

Some commonly-used estimators of the true parameters of the population, μ and σ2, are:
  θ σ<sup>2</sup> <math>\hat{ heta}</math> = the unbiased estimator of the Sample Variance , <math>S^2 = rac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}\, ight)^2</math> <math>\operatorname{MSE}(S^2)=\operatorname{E}((S^2-\sigma^2)^2)=\operatorname{var}(S^2)</math>