Put
$$
a=\frac{p_0}{1-p_0},
\qquad
b=\frac{p_1}{1-p_1}.
$$
At a fixed alternative $\theta$, the approximate power of the <Wald test> increases as its variance
$$
V(n_0,n_1)=\frac a{n_0}+\frac b{n_1}
$$
decreases. For fixed total sample size $n=n_0+n_1$, the allocation problem is therefore
$$
\min_{n_0,n_1>0}\left\{\frac a{n_0}+\frac b{n_1}:n_0+n_1=n\right\},
$$
with integer rounding applied after solving the continuous problem. The second-order condition is positivity of the second derivative at the stationary point.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 2 / a / i / Solution

 Solution

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