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04/24/01 Colloquium
DR. JIAN-JIAN REN
DEPARTMENT OF MATHEMATICS
TULANE UNIVERSITY
DEPARTMENT OF MATHEMATICS
TULANE UNIVERSITY
Regression M-Estimators With Non-I.I.D. Doubly Censored Data
Abstract: Motivated by problems arisen in breast cancer research (Peer et. el. 1993; Ren and Peer, 2000), this talk discusses the M-estimator for the linear regression model with fixed design when the response variables are subject to doubly censoring. This is a difficult problem because the available data set is a non-i.i.d. doubly censored random sample.First, for this study we find a way to express the usual M-estimator for the linear model with fixed design and complete response observations, a non-i.i.d. complete random sample, as a functional of a Generalized Weighted Bivariate Empirical Process, then we derive its asymptoticnormality directly through the Hadamard differentiability property of this functional and the weak convergence of this weighted empirical process. The implication of this result is twofold: (i) It reveals the direct relation between the M-estimator and the distribution function of the error variables in the linear model with fixed design, which leads to the construction of the M-estimator when the response variables are subject to doubly censoring; (ii) In literature, the Hadamard differentiability approach has been successfully used to study the asymptotic properties of various important statistics based on i.i.d. random samples by Bickel and Freedman (1981), Fernholz (1983), Sen (1988), Gill (1989), Ren and Sen (1995, 2000), Van der Vaart and Wellner (1996), Ren and Gu (1997), among others. The current result shows that this attractive formulation can also be used to deal with problems based on non-i.i.d. random samples.
