Question 4: (8 Total Marks) Let X1, , Xn be it'd N (p, a) and suppose we are interested in estimating 1'01, 0) = 02. The commonly used estimator is the sample variance: _ 2 _ 1 Ti _ 2 r1 _ s _ n _ 12mm X) For this case, recall we have shown that (n 1192/62 ~ xfn). a) It is well known that T1 is unbiased. Show that it is also a consistent estimator of 02. [3 marks] An alternative estimator is the MLE, which in this case is 1 n _ 73:52:; (XiX)2 i=1 b) Find the MSE of T2 in terms of n a_nd 0. Briey comment. [HINTZ T2 = (n 1)T1 / 71.] [3 marks] A bit of algebra shows that Z}; (X i X )2 = {:1 X 12 nXZ. Furthermore, it is readily be shown that S = (X, 2&1 X [2) is a sufficient statistic for iid normal observations. Dene a new estimator of 02 as 2 N 2 _2 T = ( X - NX ) 3 71 1 i=1 i where N ~ Bin(n, 0.5) is independent ofall the Xi's c) Show that T3 is unbiased but argue that its MSE is larger than that of T1. [NOTE: You do not have to actually calculate the MSE to get full marks] [2 marks]