On M-processes and M-estimation. (English) Zbl 0701.62074

Consider the usual linear regression model, \(Y_ j=x_ j'\theta +\sigma e_ j\), \(j=1,...,n\), where the dimension of the parameter vector \(\theta\) is p. This article is concerned with the asymptotic behaviour of M-estimators of \(\theta\) when p is allowed to increase with the sample size. This is done by using a stochastic equicontinuity argument. From this argument new results for one-step M-estimators, M-estimators and robust scale estimators are obtained. The objective of the article is to impose as weak conditions as possible on the criterion function and on the error distribution.
If for example \(\psi\), in usual M-estimation notations, has a bounded derivative, it is required that \(p^ 2(\log n)^{2+\gamma}/n\to 0,\gamma >0\), for contrast estimation. If \(\psi\) is discontinuous, a somewhat stronger condition on p is required: \(p^ 3(\log n)^ 2/n\to 0.\) To obtain the distribution of quadratic forms it is required that \(p^ 3(\log n)^{2+\gamma}/n\to 0,\gamma >0\), if \(\psi\) has a bounded derivative, and that \(p^ 4(\log n)^ 2/n\to 0,\) when \(\psi\) is discontinuous.
Reviewer: H.Nyquist


62J05 Linear regression; mixed models
62G05 Nonparametric estimation
62G35 Nonparametric robustness
60F05 Central limit and other weak theorems
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