Definition:Unbiased Estimator

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Definition

Let $\theta$ be a population parameter of some statistical model.

Let $\delta$ be an estimator of $\theta$.

We call $\delta$ an unbiased estimator if and only if its bias is equal to $0$ regardless of the true value of $\theta$.

Examples

Sample Variance

For a random sample of $n$ observations $x_i$ for $1 = 1, 2, \ldots, n$, an unbiased estimator for the population variance $\sigma^2$ is given by:

$\ds \dfrac 1 {n - 1} \sum_i \paren {x_i - \bar x}^2$

or presented as:

$\ds \dfrac n {n - 1} {s_x}^2$

where ${s_x}^2 is the sample variance.

Compare with the plug-in estimator of the same thing:

$\ds \sum_i \dfrac {\paren {x_i - \bar x}^2} n$

This is a biased estimator of $\sigma^2$.

Also see

• Results about unbiased estimators can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): unbiased estimator
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): unbiased estimator
• 2011: Morris H. DeGroot and Mark J. Schervish: Probability and Statistics (4th ed.): $8.7$: Unbiased Estimators: Definition $8.7.1$