Optimal hypothesis testing uses sufficient statistics?

Question

Cross post Optimal hypothesis testing uses sufficient statistics?.

In statistical estimation with any convex risk for a model with a sufficient statistic, in seeking optimal estimators it suffices to consider functions of the sufficient statistic. This follows from (conditional) Jensen's inequality: if $g$ is an estimator for $\theta$ and $T=T(X)$ is a sufficient statistic, then $h(X)=E[g(X)|T]$ is an estimator of $\theta$ satisfying $R(h,\theta)\leq R(g,\theta)$ for all $\theta$, where $R$ denotes the risk.

My question is, does a version of this result hold in hypothesis testing? That is, the goal is to define a test (i.e. a function $\phi$ of the data with $\phi(X)\in\{0,1\}$ with optimality properties, say a good sum of type I and type II error, and the conclusion would be that it suffices to consider tests depending only on the sufficient statistic $T(X)$. Note that directly applying the same argument doesn't give anything useful: $E[\phi(X)\mid T]$ is no longer a hypothesis test since it takes values in $[0,1]$.

References of any kind welcome: I suspect there may be lecture notes with some result of this kind, I just haven't been able to find any.

Answer 1

$\newcommand\th\theta\newcommand\Th\Theta$It is natural, standard, and convenient to allow randomized tests, with values in $[0,1]$, and then your objection will be removed: $E(\phi(X)|T)$ is a randomized test.

Recall that a Neyman--Pearson test for a simple null hypothesis $H_0\colon\th=\th_0$ vs. a simple alternative $H_1\colon\th=\th_1$, which is optimal at the level equal to its size, is based on the ratio $$\frac{f_{\th_1}(x)}{f_{\th_0}(x)}=\frac{g(\th_1,T(x))}{g(\th_0,T(x))},\tag{1}\label{1}$$ which is a function of $T(x)$, where $(f_\th)$ is the statistical model (that is, a family of probability densities with respect to a measure), $T$ is a sufficient statistic for this model, and $g$ is the function that enters the Fisher–Neyman factorization $$f_\th(x)=g(\th,T(x))h(x).$$ So, the (optimal) Neyman--Pearson test is a function of any sufficient statistic. Note also that, if the model is discrete and the prescribed level of significance $\alpha$ of the test is an arbitrary number in $(0,1)$, then the (optimal) Neyman--Pearson test will usually have to be randomized -- see e.g. this.

Similarly to \eqref{1}, the likelihood ratio test for composite hypotheses $H_0\colon\th\in\Th_0$ vs. $H_1\colon\th\not\in\Th_0$ is based on the ratio $$\frac{\sup_{\th\in\Th_0}f_\th(x)}{\sup_{\th\in\Th}f_\th(x)} =\frac{\sup_{\th\in\Th_0}g(\th,T(x))}{\sup_{\th\in\Th}g(\th,T(x))},\tag{2}\label{2}$$ which is a function of $T(x)$, for any sufficient statistic $T$.