On Non‐parametric Testing, the Uniform Behaviour of the t‐test, and Related Problems

. In this article, we revisit some problems in non‐parametric hypothesis testing. First, we extend the classical result of Bahadur & Savage [Ann. Math. Statist. 25 (1956) 1115] to other testing problems, and we answer a conjecture of theirs. Other examples considered are testing whether or not the mean is rational, testing goodness‐of‐fit, and equivalence testing. Next, we discuss the uniform behaviour of the classical t‐test. For most non‐parametric models, the Bahadur–Savage result yields that the size of the t‐test is one for every sample size. Even if we restrict attention to the family of symmetric distributions supported on a fixed compact set, the t‐test is not even uniformly asymptotically level α. However, the convergence of the rejection probability is established uniformly over a large family with a very weak uniform integrability type of condition. Furthermore, under such a restriction, the t‐test possesses an asymptotic maximin optimality property.