Let X1,… , Xn be a simple random sample from a N(????, ????2) population. For any constant k > 0, define

????̂ 2 k =

∑n i=1(Xi − X)

2 k

. Consider ????̂ 2 k as an estimator of ????2.

a. Compute the bias of ????̂ 2 k in terms of k. [Hint:

The sample variance s2 is unbiased, and ????̂ 2 k =

(n − 1)s2∕k.]

b. Compute the variance of ????̂ 2 k in terms of k. [Hint:

????2 s2 = 2????4∕(n − 1), and ????̂ 2 k = (n − 1)s2∕k.]

c. Compute the mean squared error of ????̂ 2 k in terms of k.

d. For what value of k is the mean squared error of

????̂ 2 k minimized?