Question

To show that an estimator can be consistent without being unbiased or even asymptotically unbiased, consider the following estimation procedure: to estimate the mean of a population with the finite variance sigma^2, we first take a random sample of size n. Then we randomly draw one of n slips of paper numbered from 1 through n, and if the number we draw is 2,3,…, or n, we use as our estimator the mean of the random sample; otherwise, we use the estimate n^2. Show that this estimation procedure is

a) consistent;
b) neither unbiased nor asymptotically unbiased.

Answer 1

The estimation procedure is consistent but not unbiased or asymptotically unbiased.

Step-by-step explanation:

The estimation procedure described is consistent, meaning that as the sample size increases, the estimator will converge to the true population mean. This can be shown by considering two cases. If we draw a number between 2 and n, the estimator will be the sample mean, which is an unbiased and consistent estimator.

If we draw 1, the estimator will be n^2, which is a constant and does not depend on the sample. Therefore, as n becomes large, the probability of drawing 1 becomes negligible, and the estimator converges to the sample mean.

However, the estimation procedure is neither unbiased nor asymptotically unbiased. The estimator assigns a non-zero probability to the value n^2, which introduces bias. As n increases, the probability of drawing 1 becomes negligible, but the bias introduced by using n^2 remains. Therefore, the estimator is biased and does not converge to the population mean as n increases.