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Homemean square error

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# 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.

## Mathematics Subject Classification

62J10*no label found*94A12

*no label found*

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