4. (i) The noncentral X2 and F distributions have strictly monotone likelihood ratio . (ii) Under the assumptions of Section 1, the hypothesis H' : 0/2 s o/~ (% > 0 given) remains invariant under the transformations G; (i = 1,2,3) that were used to reduce H : 0/ = 0, and there exists a UMP invariant test with rejection region W> C. The constant C' is determined by Po/-o { W > C} =

a, with the density of W given by (6). [(i): Let j(z) = E'f_obkz kIL'f_oakzk where the constants ab

b, are > 0 and Eakzk andEbkz k converge for all z > O,andsupposethat bk/ak < bk+l/ak+1 for all k. Then (:E akzk)2 k-O LL(n - k)(akb" - anbk)zk+ll-I k 0 for k < n, and hence j is increasing.] t For literature on noncentral F. see Johnson and Kotz (1970. Vol. 2) and Tiku (1985b).

Note. The noncentral X2 • and F-distributions are in fact STPoo [see for example Marshall and Olkin (1979) and Brown, Johnstone and MacGibbon (1981»), and there thus exists a test of H: I/J = I/Jo against I/J * I/Jo which is UMP among all tests that are both invariant and unbiased.