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

(z) = 
k∞
k=0 akzk 2 is positive, since (n − k)(akbn − anbk) > 0 for kNote. 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.