Asymptotics for trimmed k-means and associated tolerance zones

Impartial trimming procedures with respect to general 'penalty' functions, di, have been recently introduced in Cuesta-Albertos et al. (1997. Arm. Statist. 25, 553-576) in the (generalized) k-means framework. Under regularity assumptions, for real-valued samples, we obtain the asymptotic normality both of the impartial trimmed k-mean estimator (Phi(x) = x(2)) and of the impartial trimmed k-median estimator (Phi(x) = x). In spite of the additional complexity coming from the several groups setting, the empirical quantile methodology used in Butler (1982. Arm. Statist. 10, 197-204) for the LTS estimator (and subsequently in Tableman (1994. Statist. Probab. Lett. 19, 387-398) for the LTAD estimator) also works in our framework. Besides their relevance for the robust estimation of quantizers, our results open the possibility of considering asymptotic distribution-free tolerance regions, constituted by unions of intervals, for predicting a future observation, generalizing the idea in Butler (1982). (C) 1999 Elsevier Science B.V. All rights reserved. AMS classifications: Primary 62G20; 62G15; secondary 62G35.