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Version 6 |
| The \emph{mean square error} of an estimator $\hat{\theta}$ of a |
The \emph{mean square error} of an estimator $\hat{\theta}$ of a |
| parameter $\theta$ in a statistical model is defined as: |
parameter $\theta$ in a statistical model is defined as: |
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$$\operatorname{MSE}(\hat{\theta})\colon=\operatorname{E}\big[(\hat{\theta}-\theta)^2\big].$$
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$$\operatorname{MSE}(\hat{\theta})\colon=\operatorname{E}[(\hat{\theta}-\theta)^2].$$
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| From the definition of the variance |
From the definition of the variance |
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$\operatorname{Var}[X]=\operatorname{E}[X^2]-\operatorname{E}[X]^2$,
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$\operatorname{Var}(X)=\operatorname{E}(X^2)-\operatorname{E}(X)^2$,
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| we can express the mean square error in terms of the bias by |
we can express the mean square error in terms of the bias by |
| expanding the right hand side above: |
expanding the right hand side above: |
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$$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}\big[\hat{\theta}\big]+
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$$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+
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| \operatorname{Bias}(\hat{\theta})^2.$$ |
\operatorname{Bias}(\hat{\theta})^2.$$ |
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| If $\hat{\theta}$ is an unbiased estimator, then its mean square |
If $\hat{\theta}$ is an unbiased estimator, then its mean square |
| error is identical to its variance: |
error is identical to its variance: |
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$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}[\hat{\theta}]$.
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$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})$.
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| An unbiased estimator such that $\operatorname{MSE}(\hat{\theta})$ |
An unbiased estimator such that $\operatorname{MSE}(\hat{\theta})$ |
| is a minimum value among all unbiased estimators for $\theta$ is |
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. |
called a \emph{minimum variance unbiased estimator}, abbreviated \emph{MVUE}, or \emph{uniformly minimum variance unbiased estimator}, abbreviated \emph{UMVU} estimator. |
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| \textbf{Example}. Suppose $X_1,X_2,\ldots,X_n$ are iid random |
\textbf{Example}. Suppose $X_1,X_2,\ldots,X_n$ are iid random |
| variables ($n$ independent measurements of the radius of a coin, |
variables ($n$ independent measurements of the radius of a coin, |
| etc...) from a normal distribution $N(\mu,\sigma^2)$ (for example, |
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 |
$\mu$ would be the true radius of the coin, and $\sigma^2$ would be |
| the error component of the measurements). Suppose $\overline{X}$ |
the error component of the measurements). Suppose $\overline{X}$ |
| ($=\overline{X}_n$) is the sample mean. Then $\overline{X}$ is an |
($=\overline{X}_n$) is the sample mean. Then $\overline{X}$ is an |
| unbiased estimator, so that |
unbiased estimator, so that |
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$$\operatorname{MSE}(\overline{X})=\operatorname{Var}\left[\overline{X}\right]=
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$$\operatorname{MSE}(\overline{X})=\operatorname{Var}\left(\overline{X}\right)=
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\operatorname{Var}\left[\frac{1}{n}\sum_{i=1}^n
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\operatorname{Var}\left(\frac{1}{n}\sum_{i=1}^n
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X_i\right]=\frac{1}{n^2}\left(\sum_{i=1}^n \sigma^2\right)=\frac{\sigma^2}{n}.$$
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X_i\right)=\frac{1}{n^2}\left(\sum_{i=1}^n \sigma^2\right)=\frac{\sigma^2}{n}.$$
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| \textbf{Remark}. The square root of MSE is called the ``root mean square error'', or \emph{rms error} for short. |
\textbf{Remark}. The square root of MSE is called the ``root mean square error'', or \emph{rms error} for short. |
