Estimators

As I have mentioned MANY times before inferential statistics are a BIG deal, recall they these are statistics which extend results from a sample to a population. This week we will be talking about how confident we can be when making these inferences.

The basic idea of this process is that we would like to estimate parameters such as mean and standard deviation of a population.

A point estimate is a specific numerical value estimate of a parameter.

The best point estimate of the population mean \(\mu\) is the sample means \(\overline{x}\).

Properties of a Good Estimator:

The estimator should be an unbiased estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.
The estimator should be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.
The estimator should be a relatively efficient estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.

This is a great wish list but :

"How good is a point estimate?"

That is how do we know how good an estimate really is? So to combat this we make "broader" estimates instead of just a point.

An Interval Estimate of a parameter is an interval or a range of values used to estimate the parameter. This estimate may or may not contain the value of the parameter being estimated.

But of course your parameter might not lie in the interval you have chosen. What can we do?