Problem 1. Suppose that Y is a normal random variable with variance 1 and unknown mean e. It is desired to guess the value of unknown mean 0. Since the experimenter feels the loss is roughly like square error (d - 0)2 when the true e is small but is like squared relative error (0-1d - 1)? when is large, he or she chooses loss function (0 - d)?/(1 + 02) to reflect this behavior. (a) Specify S, 0, D, and L (i.e., the sample space, the set of all possible distribution functions, the decision space, and the loss function)- (b) Determine and plot on the same graph the risk function of the 6 procedures 6; defined by 6(Y) = Y; 62(Y) = (1+ Y)/2; 63(Y) = Y/2; 6(Y) = 2Y; 6 (Y) = 0; 6(Y) = 1; You can save time by working (e) first but may find it easier to work (b) first. Your calculation will be made simpler if you first compute the risk function of a general procedure of the form 6(Y) = a + by. A check: Rs. (0) = (02 + 4)/(1 + 02).] (c) From these calculations, can you assert that any of these six procedures is inadmissible? (d) On the basis of the risk functions, if one of these 6 procedures must be used, which procedure would you use, and why? (Note: Don't consult any references in answering this. Later you will find out the precise meaning of your present intuition.) (e) Suppose Y is replaced by the vector (Y1, ...; Ym) of iid normal N(0, 1) random variables, and we want to guess the value of unknown mean o based on (Y1, ..., Y,). In this case, specify S, 0, D, and L (i.e., the sample space, the set of all possible distribution functions, the decision space, and the loss function). (f) Under the setting of (e), the procedures corresponding to 61, 62, 63, 66 are 61,n ( Y1, . . ., Yn) = Yni 62, n ( Y1, . . .; Yn) = Ynn-1 63, n ( V1, . .., Ym) = Vn YT 1 + vn' 86. n ( Y1 , . . ., Ym) = 1. Compute the risk functions of these four procedures, and plot graphs of these four risk functions (or, rather, of nRan to make the results comparable to those of part (b)) for a large (e.g., for n = 10, 000). [Use the fact that Y, is N(0, n") distributed. Again, you may find it is easier first to find (1 + 02) -'Es(a + bY, - 0)2 for general a, b. ] (g) If n is large, which of the four procedures of part (e) would you use, and why? ( Your answer to this last may differ from the answer to part (d) for the case n = 1; does it?) (h) Suppose the statistician decides to restrict consideration to procedures da,b,n = a + by, of the form mentioned at the end of (e). He or she is concerned about the behavior of the risk function when is large. Show that the risk function approaches 0 as |0 - co if and only if b = 1. In addition, among procedures with b = 1, show that the choice a = 0 gives uniformly smallest risk function. This justification of the procedure 61,n = Y, under the restriction to procedures of the form on, b,n will seem more sensible to many people than a justification in terms of the "unbiasedness" criterion to be discussed later].(i) Show that the procedure 66,n, defined by dan (Y1, .... Y,) =1, is admissible for each n. [Hints: how can another procedure o' satisfy Ry(0)