Solution

Under the null, replace both parameters by a common

r

. The likelihood becomes

L_{0} (r) = (n 2 N) (m 2 M) r^{n + m} (1 - r)^{2 (N + M) - n - m} .

The pooled maximum-likelihood estimate is

r = \frac{n + m}{2 ( N + M )} .

Under an interior null

p = q = r

, part (ii) gives a centered asymptotic normal difference with variance

r (1 - r) (1/ (2 M) + 1/ (2 N))

r \to r

r

by this consistent estimate gives

Z = \frac{p - q}{( 1/ ( 2 M ) + 1/ ( 2 N )) r ( 1 - r )} d N (0, 1) .

For the two-sided alternative, reject when

∣ Z ∣ > z_{1 - α /2}

, the corresponding standard-normal quantile. This has asymptotic significance level

α

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