Robust estimates in generalized partially linear single-index models

Abstract

A natural generalization of the well known generalized linear models is to allow only for some of the predictors to be modeled linearly while others are modeled nonparametrically. However, this model can face the so called “curse of dimensionality” problem that can be solved by imposing a nonparametric dependence on some unknown projection of the carriers. More precisely, we assume that the observations (y _i ,x _i ,t _i ), 1≤i≤n, are such that t _i ∈ℝ^q, x _i ∈ℝ^p and y _i |(x _i ,t _i )∼F(⋅,μ _i ) with $\mu_{i}=H (\eta(\boldsymbol{\alpha}^{\mathrm{T}}\mathbf{t}_{i})+\mathbf {x}_{i}^{\mathrm{T}}\boldsymbol{\beta} )$, for some known distribution function F and link function H. The function η:ℝ→ℝ and the parameters α and β are unknown and to be estimated. This model is known as the generalized partly linear single-index model.

In this paper, we introduce a family of robust estimates for the parametric and nonparametric components under a generalized partially linear single-index model. It is shown that the estimates of α and β are root-n consistent and asymptotically normally distributed. Through a Monte Carlo study, we compare the performance of the proposed estimators with that of the classical ones.