Quasi-Likelihood Regression with Multiple Indices and Smooth Link and Variance Functions

A flexible semi-parametric regression model is proposed for modeling the relationship between a response and multivariate predictor variables. The proposed multiple-index model with unknown link and variance functions is an extension of the single index model of Chiou & Müller (1998). The unknown functions are assumed to be smooth and are estimated nonparametrically. We propose data-adaptive methods for automatic smoothing parameter selection and for the choice of the number of indices M . This model is very flexible, easy to use and adapts to complex data structures. It provides efficient adaptive estimation through the variance function component in the sense that the asymptotic distribution is the same as if the nonparametric components are known. We develop iterative estimation schemes which include a constrained projection method for the case where the regression parameter vectors are mutually orthogonal. The proposed methods are illustrated with the analysis of data from a food folate experiment in a rat growth bioassay, and from a medfly reproduction experiment with various feeding schedules. Asymptotic properties of the estimated model components are also obtained.