What is the meaning of "expected value of an estimator"?

Question

I am studying statistics and i am having trouble understanding some proofs because i don't quite understand what the concept of "expected value of an estimator" means and what is the difference with the value of the esimator itself.

Say i got a sample and I take the variance v of that sample. That variance v is my estimator. What is the meaning of the expected value of the variance?

Thanks a lot!

Answer 1

The sample that you take is a random sample from your population, so the sample variance $v$ is (at least before you actually take the sample of the population and compute the sample variance) itself a random variable. If you can figure out the distribution of the sample variance, then you can find its expected value.

In general, once we have the sample in place, the estimator that we compute is a fixed value that depends on the actual sample that we got. Until we've taken the sample, it's a random variable that we can analyze in terms of expected value, variance, etc.

Answer 2

A statistic is a function of sample values. It may be used either in estimating a population parameter or testing for the significance of a hypothesis made about a population parameter. A statistic, when used to estimate a population parameter is called an estimator and is called test statistic in hypothesis testing. The value obtained for an estimator for a given sample is called estimate. The estimate will be considered as the value of the parameter, which is unknown. For different samples, an estimator will result in different estimates.

Estimator is to a random variable and estimate is to a value of the random variable. Every estimator will have a probability distribution of its own. This is called the sampling distribution and is the basis for statistical inference.

We would like to know on the average what could be the value of the estimator?. In other words, what is the value around which the distribution of the estimator is centered? Expected value of an estimator gives the center of the sampling distribution of the estimator. Ideally, we would like this center to coincide with the unknown parameter. It may or may not. If it does, we say the estimator is unbiased; else, biased. Unbiasedness is one of the desirable quality of an estimator. For this reason, we would like to know the expected value of an estimator.