Question

Given estimator ?=cΣ(??−?̅)² for ?²,

where ?², represents variance of a normal distribution whose mean and variance are both unknown.

a. Find c that gives the minimum-MSE estimator ?∗for ?².

b. Is ?∗ MSE-consistent? Why or why not?

Answer 1

Given are observations from a distribution with population variance and mean . Given

tne estimator,

Note here ,

I is outside summation (constant) gives minimum variance which does not make sense.
a) The above expression can be rewritten as

The expectation of is,

is an unbiased estimator of , when

This is because population variance of a sample data is a biased estimator of variance.

b) Since as  ,  . The MLE is consistent.