Two-stage local M-estimation of additive models

Abstract

This paper studies local M-estimation of the nonparametric components of additive models. A two-stage local M-estimation procedure is proposed for estimating the additive components and their derivatives. Under very mild conditions, the proposed estimators of each additive component and its derivative are jointly asymptotically normal and share the same asymptotic distributions as they would be if the other components were known. The established asymptotic results also hold for two particular local M-estimations: the local least squares and least absolute deviation estimations. However, for general two-stage local M-estimation with continuous and nonlinear ψ-functions, its implementation is time-consuming. To reduce the computational burden, one-step approximations to the two-stage local M-estimators are developed. The one-step estimators are shown to achieve the same efficiency as the fully iterative two-stage local M-estimators, which makes the two-stage local M-estimation more feasible in practice. The proposed estimators inherit the advantages and at the same time overcome the disadvantages of the local least-squares based smoothers. In addition, the practical implementation of the proposed estimation is considered in details. Simulations demonstrate the merits of the two-stage local M-estimation, and a real example illustrates the performance of the methodology.