mean square errorRmsError2002-01-05 00:44:092006-09-23 00:03:10Definitionminimum variance unbiased estimatorrms errorroot-mean-squareroot mean squarerms\usepackage{amssymb}
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\usepackage{xypic}The \emph{mean square error} of an estimator $\hat{\theta}$ of a
parameter $\theta$ in a statistical model is defined as:
$$\operatorname{MSE}(\hat{\theta})\colon=\operatorname{E}\big[(\hat{\theta}-\theta)^2\big].$$
From the definition of the variance
$\operatorname{Var}[X]=\operatorname{E}[X^2]-\operatorname{E}[X]^2$,
we can express the mean square error in terms of the bias by
expanding the right hand side above:
$$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}\big[\hat{\theta}\big]+
\operatorname{Bias}(\hat{\theta})^2.$$
If $\hat{\theta}$ is an unbiased estimator, then its mean square
error is identical to its variance:
$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}[\hat{\theta}]$.
An unbiased estimator such that $\operatorname{MSE}(\hat{\theta})$
is a minimum value among all unbiased estimators for $\theta$ is
called a \emph{minimum variance unbiased estimator}, abbreviated \emph{MVUE}, or \emph{uniformly minimum variance unbiased estimator}, abbreviated \emph{UMVU} estimator.
\textbf{Example}. Suppose $X_1,X_2,\ldots,X_n$ are iid random
variables ($n$ independent measurements of the radius of a coin,
etc...) from a normal distribution $N(\mu,\sigma^2)$ (for example,
$\mu$ would be the true radius of the coin, and $\sigma^2$ would be
the error component of the measurements). Suppose $\overline{X}$
($=\overline{X}_n$) is the sample mean. Then $\overline{X}$ is an
unbiased estimator, so that
$$\operatorname{MSE}(\overline{X})=\operatorname{Var}\left[\overline{X}\right]=
\operatorname{Var}\left[\frac{1}{n}\sum_{i=1}^n
X_i\right]=\frac{1}{n^2}\left(\sum_{i=1}^n \sigma^2\right)=\frac{\sigma^2}{n}.$$
\textbf{Remark}. The square root of MSE is called the ``root mean square error'', or \emph{rms error} for short.