Mean Squared Error equals Variance for Unbiased Estimator

Theorem

Let $T$ be an unbiased estimator for a population parameter of a population $P$.

Then the mean squared error for $T$ equals the variance of $T$.

Proof

Let $T$ be a general estimator.

From Mean Squared Error for Biased Estimator:

$M = \var T + \paren {\map B T}^2$

where:

$\var T$ denotes the variance of $T$

$\map B T$ denotes the bias on $T$.

By definition, if $T$ is unbiased:

$\map B T = 0$

The result follows.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): mean squared error
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): mean squared error