(i) For testing H0 : = 0 against H1 : = 1 when X is...

(i) For testing H0 : θ = 0 against H1 : θ = θ1 when X is N(θ, 1), given any 0 <α< 1 and any 0 <π< 1 (in the notation of the preceding problem), there exists θ1 and x such that

(a) H0 is rejected when X = x but

(b) P(H0 | x) is arbitrarily close to 1.

(ii) The paradox of part (i) is due to the fact that α is held constant while the power against θ1 is permitted to get arbitrarily close to 1. The paradox disappears if α is determined so that the probabilities of type I and type II error are equal [but see Berger and Sellke (1987)].

[For a discussion of such paradoxes, see Lindley (1957), Bartlett (1957), Schafer

(1982, 1988) and Robert (1993).]