Average Derivatives for Hazard Functions

Abstract

This paper develops semiparametric kernel-based estimators of risk-specific hazard functions for competing risks data+ Both discrete and continuous failure times are considered+ The maintained assumption is that the hazard function depends on explanatory variables only through an index+ In contrast to existing semiparametric estimators, proportional hazards is not assumed+ The new estimators are asymptotically normally distributed+ The estimator of index coefficients is root-n consistent+ The estimator of hazard functions achieves the one-dimensional rate of convergence+ Thus the index assumption eliminates the "curse of dimensional-ity+" The estimators perform well in Monte Carlo experiments+ I thank Denise Doiron for stimulating my interest in this research project and Catherine de Fontenay, Hans Christian Kongsted, Lars Muus, seminar participants, and two anonymous referees for comments on an earlier version of the paper+ I gratefully acknowledge the hospitality of the University of Aarhus and the University of Copenhagen, where part of this research was

Asymptotic distributions of two “synthetic data” estimators for censored single-index models

“…Many estimation methods have been proposed for this model. Examples are, in the case of complete data, Powell et al [3] and Härdle and Stoker [2], Härdle et al [5], Ichimura [4], Xia et al [8] and Hristache et al [7]; in the case of censored data, GØrgens [22], Lu and Burke [27], and Wang et al [28]. Among them, the estimators of Härdle et al [5], Ichimura [4], and Wang et al [28] are obtained by minimizing a global cost function and need iterative algorithms.…”

“…These assumptions are feasible as well in the case of censored data, since such an initial estimate is acquirable from the existing methods, such as GØrgens [22] and Lu and Burke [27]. Let f θ (v) be the density function of θ τ X and D n = {x :…”

“…Under fixed censoring and truncation, Lewbel and Linton [21] proposed a novel technique of estimating a nonparametric regression function and its derivatives in two stages. For single-index models with censored data, GØrgens [22] developed semiparametric kernel-based estimators of risk-specific hazard functions for competing risk data. Other approaches to the single-index models include the spline smoothing method of Huang and Liu [23] and the local polynomial smoother of Nielsen et al [24] and Wang [25] for the proportional hazards models.…”

Randomly censored partially linear single-index models

“…Singh and Lu [30] studied censored nonparametric additive regression models based on some special data transformations. Recently, Lu and Burke [24] proposed a method called "censored average derivative estimation (CADE)'' for studying the estimation of the unknown multiple regression function; GØrgens [11] developed semiparametric kernel-based estimators of riskspecific hazard functions for competing risk data; Lu et al [25] investigated a class of partially linear single-index proportional hazards models for survival data. Under fixed censoring and truncation, Lewbel and Linton [21] proposed a novel technique of estimating nonparametric regression function and its derivatives in two stages.…”

Empirical likelihood for the class of single index hazard regression models

“…, functional ANOVA hazard model (Huang, Kooperberg, Stone, & Truong, 2000), partially linear additive hazard model (GØrgens, 2004;Huang, 1999), and single index hazard model (Huang & Liu, 2006;Lu, Chen, Song, & Singh, 2006;Wang, 2004). By extending the following single index hazard model (SIHM) in Huang and Liu (2006) λ(t|X) = λ 0 (t) exp{ψ (X ′ β)}.…”