Local estimators in multivariate generalized linear models with varying coefficients
type: Item
instance of: scholarly article
title: Local estimators in multivariate generalized linear models with varying coefficients
zbMATH Open document ID: 0936.62085
author: Gerhard Tutz
author: Göran Kauermann
published in: Computational Statistics
publication date: 2000-03-02T00:00:00Z
review text: In varying-coefficients models there are two types of regressors, the usual covariate \(x\) and the effect modifier \(u\). Let the data be given by \((y_i,x_i,u_i),\;i=1,\dots,n\), with the \(q\)-dimensional response \(y_i\) observed at covariates \((x_i,u_i)\). Let the varying-coefficient model be of the form \[ \mu_i=E(y_i |x_i,u_i)=h(Z_i\beta_i),\;\beta_i=\beta(u_i),\;Z_i=Z(x_i). \] For the estimation of \(\beta(u)\) the local likelihood function \[ l_{\gamma}^{(u)}(\beta(u))=\sum_{i=1}^n \omega_{\gamma}(u,u_i)[y'_i\theta_i(\beta(u))-k(\theta_i(\beta(u)))]/ \phi, \] is considered where \(\theta_i(\beta(u))\) is the natural parameter corresponding to the expectation \(\mu_i=h(Z_i\beta(u))\). Analogous to local likelihood estimates the locally weighted least squares estimate of \(\beta_i\) is based on the criterion \[ \sum_{s=1}^m \omega_{\gamma}(u_i,u_s) \sum_{j=1}^{m_s} [g(y_{js}-Z_{sj}\beta_i]' C_{sj}(\mu_{sj})^{-1}[g(y_{js}-Z_{sj}\beta_i]. \] Maximization of this expression yields the locally weighted least squares estimate. Extensions to the case of multivariate smoothing are considered. An example illustrates the usefulness of this approach.
Mathematics Subject Classification ID: 62J12
Mathematics Subject Classification ID: 62H12
Mathematics Subject Classification ID: 62-08
zbMATH DE Number: 1409727
zbMATH Keywords: varying coefficients
zbMATH Keywords: local likelihood smoothing
zbMATH Keywords: locally weighted least squares
reviewed by: M. P. Moklyachuk
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1409727