# Estimators

Probability distributions depend upon parameters. For example, the normal distribution depends upon the parameters m and s^{2} (the mean and variance). In some situations, these parameters may be unknown and we may wish to estimate them. An **estimator** is a statistic which is used to estimate a parameter.

**Desirable Characteristics**

Good estimators are those which have a small variance and small bias. The **bias** of an estimator q which is estimating a parameter p is E(q) - p .

An estimator is **unbiased** if the bias is zero. The sample mean and sample variance are unbiased estimators of the mean and variance.

So the best estimator for a particular parameter is that for which B(q) + V(q) is smallest, where B(q) is the bias of q.

**Example**

X_{1}, X_{2}, ..., X_{n} is a random sample taken from a normal distribution with mean m and variance s^{2}, where m is unknown. Show that the sample mean is an unbiased estimator for m.

We calculated that the expectation of the sample mean is m. Hence E(

) - m = 0 . So the sample mean is unbiased.

NB: Var(

) = s^{2}/n , so as n gets large, Var(

) gets closer to zero and so the sample mean,

, has a small variance for large values of n. It turns out that

is the best estimator for m.