Composite quasi-likelihood for single-index models with massive datasets

Abstract

The single-index models (SIMs) provide an efficient way of coping with high-dimensional nonparametric estimation problems and avoid the “curse of dimensionality.” Many existing estimation procedures for SIMs were built on least square loss, which is popular for its mathematical beauty but is non-robust to non-normal errors and outliers. This article addressed the question of both robustness and efficiency of estimation methods based on a new data-driven weighted linear combination of convex loss functions instead of only quadratic loss for SIMs. The optimal weights can be chosen to provide maximum efficiency and these optimal weights can be estimated from data. As a specific example, we introduce a robust method of composite least square and least absolute deviation methods. Moreover, we extend the proposed method to the analysis of massive datasets via a divide-and-conquer strategy. The proposed approach significantly reduces the required primary memory and the resulting estimate is as efficient as if the entire dataset was analyzed simultaneously. The asymptotic normality of the proposed estimators is established. The simulation studies and real data applications are conducted to illustrate the finite sample performance of the proposed methods.