Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 3 / b

Writing $v_H=p_H(1-p_H)$ and $v_L=p_L(1-p_L)$, Neyman allocation minimizes the variance $v_H/n_H+v_L/n_L$ at fixed total sample size and gives

Minimizing the expected failures $n_H(1-p_H)+n_L(1-p_L)$ at a fixed variance gives the ethically optimal ratioIt allocates more patients to the dose with the larger response probability.

Neyman allocation optimizes information and may allocate more patients to the arm whose Bernoulli variance is nearer its maximum at $1/2$. The ethical allocation instead favors the arm with higher response probability. They coincide only for special parameter values; in general ethical gain is purchased with some loss of precision relative to Neyman allocation.