For a randomized test $\varphi(X_1,\ldots,X_n)\in[0,1]$, level $\alpha$ over the null neighborhood means
$$
\sup_{P\in\mathcal P_\epsilon(P_0)}\mathbb E_{P^n}\varphi\leq\alpha.
$$
A <maximin test> maximizes its worst-case power
$$
\inf_{P\in\mathcal P_\epsilon(P_1)}\mathbb E_{P^n}\varphi
$$
among all tests satisfying that robust level constraint.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 3 / c / Solution

A maximin test maximizes the smallest power over a composite alternative subject to controlling the largest rejection probability over a composite null.


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