Lecture 8. A) Point Estimation

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Point Estimation

Let [math]X_{1}..X_{n}[/math] be a random sample from a distribution with cdf [math]F\left(\left.\cdot\right|\theta\right)[/math] where [math]\theta\in\Theta[/math] is unknown.

A point estimator is any function [math]\omega\left(X_{1}..X_{n}\right)[/math].

Notice that a point estimator is a statistic. This does mean that it too is a random variable: For different random samples, we will obtain different point estimators.

We call the realized value of an estimator (i.e., the value of the statistic applied to the realized values of a random sample) as an estimate.

Clearly, a good estimator will be close to [math]\theta[/math] in some probabilistic sense. Finally, an estimator cannot use the true value of [math]\theta[/math] itself.

We consider two methods for point estimation: The method of moments and maximum likelihood.