# mean square error

The 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 minimum variance unbiased estimator, abbreviated MVUE, or uniformly minimum variance unbiased estimator, abbreviated UMVU estimator.

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

 $\operatorname{MSE}(\overline{X})=\operatorname{Var}\left[\overline{X}\right]=% \operatorname{Var}\left[\frac{1}{n}\sum_{i=1}^{n}X_{i}\right]=\frac{1}{n^{2}}% \left(\sum_{i=1}^{n}\sigma^{2}\right)=\frac{\sigma^{2}}{n}.$

Remark. The square root of MSE is called the “root mean square error”, or rms error for short.

 Title mean square error Canonical name MeanSquareError Date of creation 2013-03-22 12:07:42 Last modified on 2013-03-22 12:07:42 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 11 Author CWoo (3771) Entry type Definition Classification msc 62J10 Classification msc 94A12 Synonym MSE Synonym MVUE Synonym UMVU Synonym UMVUE Synonym uniformly minimum variance unbiased Related topic MeanSquareDeviation Defines minimum variance unbiased estimator Defines rms error Defines root-mean-square Defines root mean square Defines rms