On mode testing and empirical approximations to distributions

There are close connections between the theory of statistical inference under order restrictions and the theory of tests for unimodality. In particular, a result of Kiefer and Wolfowitz, on the error of convex approximations to empirical distribution functions, is basic to limit theory for the dip test for unimodality. We develop a version of Kiefer and Wolfowitz' result in the context of distributions that are strongly unimodal, and apply it and related limit theory to compare the powers, against local alternatives, of three different tests of unimodality. In this context it is shown that the dip, excess mass and bandwidth tests are all able to detect departures of size n^-35 (measured in terms of the distribution function) from the null hypothesis, where n denotes sample size; but are not able to detect departures of smaller order. Thus, they have similar powers.