(i) The noncentral 2 and F distributions have strictly monotone likelihood ratio. (ii) Under the assumptions of...

(i) The noncentral χ2 and F distributions have strictly monotone likelihood ratio.

(ii) Under the assumptions of Section 7.1, the hypothesis H : ψ2 ≤ ψ2 0 (ψ0 > 0 given) remains invariant under the transformations Gi(i = 1, 2, 3) that were used to reduce H : ψ = 0, and there exists a UMP invariant test with rejection region W > C

. The constant C is determined by Pψ0 {W > C

} = α, with the density of W given by (7.6).

[(i): Let f (z) = ∞

k=0 bk zk/

∞

k=0 ak zk



where the constants ak , bk are > 0 and ak zk and bk zk converge for all z > 0, and suppose that bk/ak < bk+1/ak+1 for all k. Then f

(z) =



k

(n − k)(akbn − anbk )zk+n−1

!∞

k=0 ak zk

"2 is positive, since (n − k)(akbn − anbk ) > 0 for k < n, and hence f is increasing.]

Note. The noncentral χ2 and F-distributions are in fact STP∞ [see, for example, Marshall and Olkin (1979) and Brown, Johnstone, and MacGibbon (1981)], and there thus exists a test of H : ψ = ψ0 against ψ = ψ0 which is UMP among all tests that are both invariant and unbiased.