|
|
|
|
mean square error
|
(Definition)
|
|
|
The mean square error of an estimator
 of a parameter  in a statistical model is defined as:
From the definition of the variance
![$ \operatorname{Var}[X]=\operatorname{E}[X^2]-\operatorname{E}[X]^2$](http://images.planetmath.org:8080/cache/objects/1289/l2h/img4.png "$ \operatorname{Var}[X]=\operatorname{E}[X^2]-\operatorname{E}[X]^2$") , we can express the mean square error in terms of the bias by expanding the right hand side above:
If
 is an unbiased estimator, then its mean square error is identical to its variance:
![$ \operatorname{MSE}(\hat{\theta})=\operatorname{Var}[\hat{\theta}]$](http://images.planetmath.org:8080/cache/objects/1289/l2h/img7.png "$ \operatorname{MSE}(\hat{\theta})=\operatorname{Var}[\hat{\theta}]$") . An unbiased estimator such that
 is a minimum value among all unbiased estimators for  is called a minimum variance unbiased estimator, abbreviated MVUE, or uniformly minimum variance unbiased estimator, abbreviated UMVU estimator.
Example. Suppose
 are iid random variables ( independent measurements of the radius of a coin, etc...) from a normal distribution
 (for example,  would be the true radius of the coin, and  would be the error component of the measurements). Suppose
 (
 ) is the sample mean. Then
 is an unbiased estimator, so that
Remark. The square root of MSE is called the “root mean square error”, or rms error for short.
|
"mean square error" is owned by CWoo. [ full author list (2) | owner history (1) ]
|
|
(view preamble)
| Other names: |
MSE, MVUE, UMVU, UMVUE, uniformly minimum variance unbiased |
| Also defines: |
minimum variance unbiased estimator, rms error, root-mean-square, root mean square, rms |
|
|
Cross-references: square root, sample mean, component, normal distribution, radius, independent, random variables, iid, unbiased estimator, right hand side, bias, terms, variance, statistical model, parameter, estimator
There are 3 references to this entry.
This is version 7 of mean square error, born on 2002-01-05, modified 2006-09-23.
Object id is 1289, canonical name is RmsError.
Accessed 60971 times total.
Classification:
| AMS MSC: | 94A12 (Information and communication, circuits :: Communication, information :: Signal theory ) | | | 62J10 (Statistics :: Linear inference, regression :: Analysis of variance and covariance) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![$\displaystyle \operatorname{MSE}(\hat{\theta})\colon=\operatorname{E}\big[(\hat{\theta}-\theta)^2\big].$](http://images.planetmath.org:8080/cache/objects/1289/l2h/img3.png "$\displaystyle \operatorname{MSE}(\hat{\theta})\colon=\operatorname{E}\big[(\hat{\theta}-\theta)^2\big].$"){kind=small}
![$\displaystyle \operatorname{MSE}(\hat{\theta})=\operatorname{Var}\big[\hat{\theta}\big]+ \operatorname{Bias}(\hat{\theta})^2.$](http://images.planetmath.org:8080/cache/objects/1289/l2h/img5.png "$\displaystyle \operatorname{MSE}(\hat{\theta})=\operatorname{Var}\big[\hat{\theta}\big]+ \operatorname{Bias}(\hat{\theta})^2.$"){kind=small}
![$\displaystyle \operatorname{MSE}(\overline{X})=\operatorname{Var}\left[\overlin... ... X_i\right]=\frac{1}{n^2}\left(\sum_{i=1}^n \sigma^2\right)=\frac{\sigma^2}{n}.$](http://images.planetmath.org:8080/cache/objects/1289/l2h/img18.png "$\displaystyle \operatorname{MSE}(\overline{X})=\operatorname{Var}\left[\overlin... ... X_i\right]=\frac{1}{n^2}\left(\sum_{i=1}^n \sigma^2\right)=\frac{\sigma^2}{n}.$")