47. Bayes character and admissibilityof Hotelling's r: (i) Let (X"I"'" X"p), a = 1, . . . ; n, be a sample from a p-variate normal distribution with unknown mean = (~I" .. , ~p) and covariance matrix = A-I, and with p n - 1. Then the one-sample r 2 -test of H : = 0 against K: "* 0 is a Bayes test with respect to prior distributions Ao and Al which generalize those of Chapter 6, Example 13 (continued). (ii) The test of part (i) is admissible for testing H against the alternatives 1/;2 c for any c > O. [If w is the subset of points (O,~) of 0H satisfying ~-I = A + 11''1} for some fixed positive definite p XP matrix A and arbitrary 'I} = ('I}I " .., 'l}p), and O~. is the subset of points ,~) of OK satisfying ~-I = A + 11''1}, ~' = b~11' for the same A and some fixed b> 0, let Ao and Al have densities defined over w and 0A. b respectively by Ao('I}) = ColA + 11''I}I- n/ 2 and { nb2 } AI('I}) =CdA + 11''I}I- n / 2 exp T['I}(A + 11''I})-I11'] . (Kiefer and Schwartz, 1965).]