Mean Squared Error

Topics: Statistics

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(definition)

Let ${\hat{\theta }}_n$ be an estimator of parameter $\theta$.

The mean squared error of $\hat{\theta }$ is the expected value of the square of the difference between $\hat{\theta }$ and $\theta$.

That is:

$$\mathrm{MSE}(\hat{\theta })=\mathbb{E}\left[(\hat{\theta }-\theta {)}^2\right]$$

(theorem)

We also have that:

$$\mathrm{MSE}(\hat{\theta })=\mathrm{Var}(\hat{\theta })+{\left(\mathrm{B}(\hat{\theta })\right)}^2$$

(see bias)

Proof

Notice that:

$$\begin{array}{rl}\mathrm{MSE}({\hat{\theta }}_n)& =\mathbb{E}\left[(\hat{\theta }-\theta {)}^2\right]\\ & =\mathbb{E}[{\hat{\theta }}^2-2\theta \hat{\theta }+{\theta }^2]\\ & =\mathbb{E}[{\hat{\theta }}^2]-2\theta \mathbb{E}[\hat{\theta }]+{\theta }^2\\ & =\mathbb{E}[{\hat{\theta }}^2]-{\left(\mathbb{E}[\hat{\theta }]\right)}^2+{\left(\mathbb{E}[\hat{\theta }]\right)}^2-2\theta \mathbb{E}[\hat{\theta }]+{\theta }^2\\ & =\mathrm{Var}(\hat{\theta })+{\left(\mathbb{E}[\hat{\theta }]-\theta \right)}^2\\ & =\mathrm{Var}(\hat{\theta })-{\left(\mathrm{B}(\hat{\theta })\right)}^2\end{array}$$