To get information about unknown parameters, we need data whose distribution depends on these parameters. A function of the data is used to estimate each parameter. We start with models involving a single unknown parameter, \(\theta\).

Definition

If \({X_1, X_2, \dots, X_n}\) is a random sample from a distribution whose shape depends on an unknown parameter \(\theta\), then any function of the random sample,

\[ \hat{\theta}(X_1, X_2, \dots, X_n) \]

is a random variable and could potentially be used as an estimator of \(\theta\).

Possible estimators are:

Sample mean
\[\hat{\theta} = \frac {\sum_{i=1}^n {X_i}} n\]
Sample median
\[\hat{\theta} = median(X_1, X_2, \dots, X_n) \]
Sample maximum
\[\hat{\theta} = max(X_1, X_2, \dots, X_n) \]

Since there are various possible functions of the data that might be used as estimators, what makes a good estimator of a parameter?