Approximations to distributions of statistics used for testing hypotheses about the number of modes of a population

A commonly considered method for testing the null hypothesis that a population has a single mode is based on a nonparametric density estimator. It is due to B.W. Silverman, and involves calculating the smallest bandwidth, (h) over tilde say, such that the curve produced by the estimator has precisely one mode. The null hypothesis is rejected if the value of (h) over tilde is ''too large''. In this paper we develop analytical and numerical approximations to the asymptotic distribution of this test statistic. We argue that an asymptotic test based on such approximations is more accurate than bootstrap tests based on either the same or smaller resample sizes, and suggest that even more accurate tests may be derived by combining our asymptotic test with a bootstrap one in an appropriate way. Related problems, of testing for the number of shoulder points of a density or for the number of points of infection, are also considered.