Let Xi,j , 1 i I, 1 j n be independent with Xi,j Poisson with mean i . The problem is

Let Xi,j , 1 ≤ i ≤ I, 1 ≤ j ≤ n be independent with Xi,j Poisson with mean λi . The problem is to test the null hypothesis that the λi are all the same versus they are not all the same. Consider the test that rejects the null hypothesis iff T ≡ n I

i=1(X¯i − X¯)2 X¯

is large, where X¯i =

j Xi,j /n and X¯ =

i X¯i/I.

(i) How large should the critical values be so that, if the null hypothesis is correct, the probability of rejecting the null hypothesis tends (as n → ∞ with I fixed) to the nominal level α.

(ii) Show that the test is pointwise consistent in power against any (λ1,..., λI), as long as the λi are not all equal.