On semiparametric -estimation in single-index regression

Abstract

In this paper we analyze a large class of semiparametric M −estimators for singleindex models, including semiparametric quasi-likelihood and semiparametric maximum likelihood estimators. Some possible applications to robustness are also mentioned. The definition of these estimators involves a kernel regression estimator for which a bandwidth rule is necessary. Given the semiparametric M −estimation problem, we propose a natural bandwidth choice by joint maximization of the M −estimation criterion with respect to the parameter of interest and the bandwidth. In this way we extend a methodology first introduced by Härdle, Hall and Ichimura (1993) for semiparametric least-squares. We prove asymptotic normality for our semiparametric estimator. We derive the asymptotic equivalence between our bandwidth and the optimal bandwidth obtained through weighted cross-validation. Empirical evidence obtained from simulations suggests that our bandwidth improves the higher order asymptotics of the semiparametric M −estimator when it replaces the usual bandwidth chosen by cross-validation.

Semiparametric estimation of binary response models with endogenous regressors

“…. , h dv+1 ), for our simulation study we follow Härdle, Hall, and Ichimura (1993) and Delecroix, Hristache, and Patilea (2005), and consider the following pragmatic approach: we treat the bandwidths as additional parameters of the estimated likelihood and perform the maximization with respect to both β and h. That is, we use the first component of…”

Rank-based estimating equation with non-ignorable missing responses via empirical likelihood

“…On the other hand, as the estimation of both π(x, y) and m 0 (x, γ) involve selecting the bandwidth, similar to Delecroix et al (2006), we consider a joint minimization of D h ∈ {h : c 1 n −α 1 < h < c 2 n −α 2 }, for some c 1 , c 2 > 0, 1/8 < α 1 < α 2 < 1/4. The score function ϕ that appears in the theoretical development of the paper is taken to be the Wilcoxon score function; that is ϕ(u) = √ 12(u − 1/2).…”

Variable bandwidth local maximum likelihood type estimation for diffusion processes

“…Reference [20] defined maximum likelihood type robust estimates of regression and investigated the asymptotic properties. From then on, the maximum likelihood type robust estimation (M-estimation) has been discussed by many authors, for example, [21,22] and the references therein. Meanwhile, some modified maximum likelihood type estimators were developed, such as the local maximum likelihood type estimator (local M-estimators), which is a combination of the local linear regression and the M-estimation regression, so the nice properties of local linear estimator and M-estimator persist.…”