Question

1.4 In reference to Example 18 below, show that n-1 as stated there Example 18 Let X1, , X, be a random sample from the N(Aơ distribution, where both u and ơ are unknown. By an application of either theorem, it is seen that X is the UMVU estimate of u. Working with the Cramer-Rao inequality regarding σ'as the estimated parameter, it is seen that the CR lower bound is equal to 2ơ4 / n. Next, Theorem 7 leads to the UMVU estimate of Var,2 (S2 ) Furthermore, it has been seen (Exercise 1.4) that n-1, which is strictly larger than n . This is the reason the Cramér-Rao inequality approach fails. For a little more extensive discussion on this example, see, e.g., Example 9, Section 12,4, in the reference cited in Remark 10.

Answer 1

here we will apply the defination of chi square variate and nean of chi square distribution. here we take chi square n-1 d.f since mu is unknown therefore replace with its unbiased estimate sample mean x bar.