Asymptotics for trimmed k-means and associated tolerance zones

Impartial trimming procedures with respect to general ‘penalty’ functions, Φ, have been recently introduced in Cuesta-Albertos et al. (1997. Ann. 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 (Φ(x)=x²) and of the impartial trimmed k-median estimator (Φ(x)=x).In spite of the additional complexity coming from the several groups setting, the empirical quantile methodology used in Butler (1982. Ann. 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).