 Solution

Under the null, replace both parameters by a common $r$. The likelihood becomesThe pooled maximum-likelihood estimate isUnder an interior null $p=q=r$, part (ii) gives a centered asymptotic normal difference with variance $r(1-r)(1/(2M)+1/(2N))$. The law of large numbers makes $\hat{r}\to r$ in probability. By Slutsky's theorem, replacing $r$ by this consistent estimate givesFor the two-sided alternative, reject when $|Z|>z_{1-\alpha /2}$, the corresponding standard-normal quantile. This has asymptotic significance level $\alpha$.