Quasi-Likelihood Regression with Unknown Link and Variance Functions
Journal
Journal of the American Statistical Association
Journal Volume
93
Journal Issue
444
Start Page
1376-1387
ISSN
0162-1459
1537-274X
Date Issued
1998-12
Author(s)
Hans-Georg Müller
Abstract
We consider the multiple regression model E(y) = μ, μ = g(xTβ), var(y) — [sgrave]2(μ) with predictors x, link function g, and variance function [sgrave]2(·). The aim is to reduce the assumptions in a fully parametric generalized linear model or a quasi-likelihood model by allowing the link and the variance functions to be unknown but smooth. These functions are then estimated nonparametrically, and the estimates are substituted into the quasi-likelihood function. We propose a three-stage approach to identify this semiparametric model by estimating the link function, the variance function, and the vector of regression coefficients in the linear predictor of the model. Consistency results for the link and the variance function estimators, as well as the asymptotic limiting distribution of the regression coefficients, are obtained. We show that the resulting parameter estimates are asymptotically efficient, as compared to the quasi-likelihood parameter estimates obtained for the case where link and variance functions are known. We suggest data-adaptive bandwidth choices based on deviance and Pearson's chi-squared statistic and show them to work well in a simulation study. We also discuss an application to quantal dose-response data that demonstrates the usefulness of the proposed method. © 1998 Taylor & Francis Group, LLC.
Subjects
Asymptotic efficiency
Curve estimation
Dose-response analysis
Estimating equation
Generalized linear model
Local polynomial fitting
Multiple regression
Semiparametric modeling
Smoothing
Publisher
Informa UK Limited
Type
journal article