Question

To show 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 nite variance 2, we rst 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 estimator n2.
Show that this estimation procedure is (a) consistent; and (b) neither unbiased nor asymp-
totically unbiased.

Answer 1

Let Tn be an estimator when we use estimator the mean of the random sample, the probability is (n-1)/n otherwise we use the estimae n2 with probability 1/n Hence the expectation of the estimator is 7l 7l 7l For consistent requires that for all e >0 (b) Here X is the estimate the mean of a population 71 n 1+2...7n 71 n-1 (n( 2n 1) nna i 7l n(n 1 n2 is biased estimator of u