M-estimators of structural parameters in pseudolinear models.
type: Item
instance of: scholarly article
title: M-estimators of structural parameters in pseudolinear models.
zbMATH Open document ID: 1060.62029
DOI: 10.1023/A:1023027929079
author: Friedrich Liese
author: Igor Vajda
published in: Applications of Mathematics
publication date: 2005-05-04T00:00:00Z
full work available at URL: https://eudml.org/doc/33033
review text: The authors consider the consistency of M-estimators of the structural parameter \(\beta _0\) in a pseudo-linear model, based on independent observations \(Y_1,\ldots ,Y_n,\) with \(Y_i\) distributed according to the distribution function \(F(y-\theta _i)\), \(\theta _i=u(z_i^{\prime }\beta _0)\), \(i=1,\ldots ,n.\) The M-estimator is defined as \(\arg \min _{\beta \in B}\sum _{i=1}^n \rho (Y_i-\tau \circ u(z_i^{\prime }\beta ))\) with given functions \(\tau \) and \(u.\) It explicitly depends on \(Y_n\) and on the \((p\times n)\) matrix \(z_n.\) Under a series of regularity conditions on \(\rho\), \(F,\) for monotone and continuous \(\tau \) and \(u,\) and bounded \(z\) satisfying some concentration condition, the authors derive two identifiability conditions on \(M_n(\theta )=n^{-1} \sum _{i=1}^n\rho (Y_i-\tau (\theta ))\) and on its expectation in a neighborhood of the true \(\theta _0,\) sufficient for the consistency of the M-estimator of \(\beta _0.\) Analogous sufficient conditions are derived for the consistency of M-estimators of \(\beta _0\) in the model with random \(z_n.\) The results are obtained with the aid of consistency considerations for the parent parameter \(\theta ,\) that can be of interest of their own. The results apply also to regression quantiles and similar M-estimators generated by a not very smooth \(\rho\). It is shown that the pseudolinear model is more general than the generalized linear model, with whom it coincides only in some special cases.
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62J12
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62F35
zbMATH DE Number: 2165522
zbMATH Keywords: M-estimator
zbMATH Keywords: pseudo-linear models
MaRDI profile type: Publication
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OpenAlex ID: W121223805
description: scientific article; zbMATH DE number 2165522