Measuring Mass Concentrations and Estimating Density Contour Clusters-An Excess Mass Approach

Abstract

By using empirical process theory, the so-called excess mass approach is studied. It can be applied to various statistical problems, especially in higher dimensions, such as testing for multimodality, estimating density contour clusters, estimating nonlinear functionals of a density, density estimation, regression problems and spectral analysis. We mainly consider the problems of testing for multimodality and estimating density contour clusters, but the other problems also are discussed. The excess mass (over $\mathbb{C})$ is defined as a supremum of a certain functional defined on $\mathbb{C}$, where $\mathbb{C}$ is a class of subsets of the $d$-dimensional Euclidean space. Comparing excess masses over different classes $\mathbb{C}$ yields information about the modality of the underlying probability measure $F$. This can be used to construct tests for multimodality. If $F$ has a density $f$, the maximizing sets of the excess mass are level sets or density contour clusters of $f$, provided they lie in $\mathbb{C}$. The excess mass and the density contour clusters can be estimated from the data. Asymptotic properties of these estimators and of the test statistics are studied for general classes $\mathbb{C}$, including the classes of balls, ellipsoids and convex sets.