A general approach to Bahadur-Kiefer representations for M-estimators

this paper is to provide a very general approach to the determination of the exact rate at which the remainder R n in (1.1) goes to zero, both in terms of convergence in distribution and almost sure limiting behavior. The idea is to show, that after certain basic properties of an M--estimator have been established, such as existence and consistency, R n , when properly normed, has the same stochastic behavior as n !1 as a functional '(W n ) of a sequence of processes W n . The weak convergence of the W n yields the asymptotic distribution of the remainder term and the law of the iterated logarithm gives its exact almost sure rate of convergence to zero. We will show how this works explicitly in the proofs of Theorems 2.1, 2.4, 3.4 and in examples. The only previously known exact rate of almost sure convergence for R n was obtained by Kiefer (1967). We get his result as a special case of our Theorem 3.4. In Section 3, we use our methods to obtain exact almost sure and distributional rates for a diverse selection of M--estimators for location, including those studied by Carroll (1978), Jureckov'a (1980, 1985) and Jureckov'a and Sen (1987). In the multivariate case, we give particular attention to the example of the spatial medians and the k--means. We emphasize that all of these results are merely special cases of what can be derived from this very general approach to Bahadur--Kiefer representation for M--estimators. Our approach begins with the assumption that an M--estimator satisfies certain basic properties, such as existence and consistency. Existence is usually an elementary problem. Some good places to see how to establish consistency are Huber (1981) and Chapter 13 of Hoffmann-- Jorgensen (1994). We shall make frequent use of the methods and results from modern ..