Minimum volume sets and generalized quantile processes

Bahadur-Kiefer approximations for generalized quantile processes as defined in Einmahl and Mason (1992) are given which generalize results for the classical one-dimensional quantile processes. An as application we consider the special case of the volume process of minimum volume sets in classes of subsets of the d-dimensional Euclidean space. Minimum volume sets can be used as estimators of level sets of a density and might be useful in cluster analysis. The volume of minimum volume sets itself can be used for robust estimation of scale. Consistency results and rates of convergence for minimum volume sets are given. Rates of convergence of minimum volume sets can be used to obtain Bahadur-Kiefer approximations for the corresponding volume process and vice versa. A generalization of the minimum volume approach to non-i.i.d. problems like regression and spectral analysis of time series is discussed.