Mean squared error

In statistics the mean squared error of an estimator T of an unobservable parameter θ is

<math>\operatorname{MSE}(T)=\operatorname{E}((T-\theta)^2),</math>

i.e., it is the expected value of the square of the "error". The "error" is the amount by which the estimator differs from the quantity to be estimated. The mean squared error satisfies the identity

<math>\operatorname{MSE}(T)=\operatorname{var}(T)+(\operatorname{bias}(T))^2</math>

where

<math>\operatorname{bias}(T)=E(T)-\theta,</math>

i.e., the bias is the amount by which the expected value of the estimator differs from the unobservable quantity to be estimated.

Here is a concrete example. Suppose

<math>X_1,\dots,X_n\sim\operatorname{N}(\mu,\sigma^2),</math>

i.e., this is a random sample of size n from a normally distributed population. Two estimators of σ² are sometimes used (as are others):

<math>\frac{1}{n}\sum_{i=1}^n\left(X_i-\overline{X}\,\right)^2\ {\rm and}\ \frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}\,\right)^2 </math>

where

<math>\overline{X}=(X_1+\cdots+X_n)/n</math>

is the "sample mean". The first of these estimators is the maximum likelihood estimator, and is biased, i.e., its bias is not zero, but has a smaller variance than the second, which is unbiased. The smaller variance compensates somewhat for the bias, so that the mean squared error of the biased estimator is slightly smaller than that of the unbiased estimator.