Statistica Sinica

Peter Hall and Matthew York

Abstract:It is known that Silverman's bootstrap test for multimodality tends towards conservatism, even in large samples, in the sense that the actual level tends to be less than the nominal one. In this paper, and in the context of testing for a single mode, we propose a means of calibrating Silverman's test so as to improve its level accuracy. The calibration takes two forms -- first, an asymptotic approach, which involves identifying the limiting distribution of the test statistic and adjusting for its departure from a hypothetical Uniform distribution; and second, a Monte Carlo technique, which enables a degree of correction for second-order effects to be incorporated. As an aid to applying Silverman's test to contexts rather different from those he envisaged, for example to the case of testing for the number of modes of a density in a compact interval, we show that the modes of the density estimator remain separated as the bandwidth is decreased, at least until a point is reached where the first three derivatives of the estimator vanish simultaneously. Theoretical and numerical properties of alternative forms of Silverman's test are addressed.

Key words and phrases:Bandwidth, bootstrap, density estimation, kernel methods, mode, Monte Carlo methods, smoothing parameter.