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Abstract

By using empirical process theory we study a method addressed totesting for multimodality and estimating density contour clusters in higherdimensions. The method is based on the so-called excess mass. Given aprobability measure F and a class of sets in the d-dimensional Euclidean space, the excess mass is defined as the maximal difference between theF-measure and l times the Lebesgue measure of sets in the given class. The excess mass can be estimated by replacing F by the empirical measure. Thecorreponding maximizing sets can be used for estimating density contourclusters. Comparing excess masses over different classes yields informationabout the modality of the underlying probability measure. This can be usedto construct tests for multimodality. The asymptotic behaviour of theconsidered estimators and test statistics is studied for different classesof sets, including the classes of balls, ellipsoids and convex sets.