Suppose X1, ...Xn are i.i.d. Poisson(). Consider testing the null hypothesis H0 : = 0 versus

Suppose X1, ...Xn are i.i.d. Poisson(λ). Consider testing the null hypothesis H0 : λ = λ0 versus the alternative, HA : λ>λ0.

(i) Consider the test φ1 n with rejection region n1/2[X¯ n − λ0] > z1−αλ1/2 0 , where

(zα) = α and is the cdf of a standard normal random variable. Find the limiting power of this test against λ0 + hn−1/2.

(ii) Alternatively, let g be a differentiable, monotone increasing function with g

(λ0) > 0, and consider the test φg n with rejection region n1/2

[g(X¯ n) − g(λ0)] > z1−α g

(λ0)λ1/2 0 .

Show that φ1 n and φg n are equivalent in the sense that, for any b > 0,

sup 0≤h≤b Eλ0+hn−1/2 |φ1 n − φg n | → 0 .
(iii) Can we replace b by ∞?