Quasi-Likelihood Regression with Unknown Link and Variance Functions

Abstract

We consider the multiple regression model E ( y ) = μ, μ = g (x T β), 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.