Let X1, ...,Xn be a random sample from a N(,2) distribution ( and 2 are unknown), and let S2 = 1 n n i=1 (Xi

Let X1, ...,Xn be a random sample from a N(μ,σ2) distribution (μ and σ2 are unknown), and let S2 = 1 n

n i=1

(Xi − X)

2, S2 1 = 1 n − 1

n i=1

(Xi − X)

2 be two estimators of σ2. (i) Compare the MSE’s of S2 and S2 1 . (ii) Consider S2 k = k

n i=1

(Xi − X)

2 as estimators of σ2 and find k for which S2 k has smallest MSE. Explain why, in practice, the only values of k used are (suboptimal): k = 1/n and k = 1/(n − 1).