mean square error

The mean square error of an estimator $\widehat{\theta }$ of a parameter $\theta$ in a statistical model is defined as:

$$\mathrm{MSE}(\widehat{\theta }):=\mathrm{E}\left[{(\widehat{\theta }-\theta )}^2\right].$$

From the definition of the variance $\mathrm{Var}[X]=\mathrm{E}[X^2]-\mathrm{E}{[X]}^2$, we can express the mean square error in terms of the bias by expanding the right hand side above:

$$\mathrm{MSE}(\widehat{\theta })=\mathrm{Var}\left[\widehat{\theta }\right]+\mathrm{Bias}{(\widehat{\theta })}^2.$$

If $\widehat{\theta }$ is an unbiased estimator, then its mean square error is identical to its variance: $\mathrm{MSE}(\widehat{\theta })=\mathrm{Var}[\widehat{\theta }]$. An unbiased estimator such that $\mathrm{MSE}(\widehat{\theta })$ 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 $X_1,X_2,\mathrm{\dots },X_n$ 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

$$\mathrm{MSE}(\overline{X})=\mathrm{Var}\left[\overline{X}\right]=\mathrm{Var}\left[\frac{1}{n}\sum _{i=1}^n{X}_i\right]=\frac{1}{n^2}\left(\sum _{i=1}^n{\sigma }^2\right)=\frac{{\sigma }^2}{n}.$$

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