Estimator and Estimate

An estimator of a population parameter is a function of the sample. If the sample is $\{x_i\}_{i=1}^n$, then an estimator is $f(x_1,\dots,x_n)$ which is also a random variable given that $x_i$ is random.

• point estimator: function that produces a single number
• interval estimator: function that produces an interval, e.g. confidence interval.

We can evaluate an estimator in terms of unbiasedness and efficiency.

Unbiasedness

A point estimator $\hat\theta$ is an unbiased estimator of a population parameter $\theta$ if $\mathbb{E}[\hat\theta]=\theta$.

Efficiency

The most efficient estimator among unbiased estimators has the minimum variance (MVUE). The relative efficiency of $\hat\theta_1$ with respect to $\hat\theta_2$ is

$$\frac{Var(\hat\theta_2)}{Var(\hat\theta_1)}$$

i.e., if $Var(\hat\theta_2)\gt Var(\hat\theta_1)$, then $\hat\theta_1$ is more efficient.