Kernel estimation in a nonparametric marker dependent hazard model
type: Item
instance of: scholarly article
title: Kernel estimation in a nonparametric marker dependent hazard model
zbMATH Open document ID: 0847.62023
published in: The Annals of Statistics
publication date: 1996-10-08T00:00:00Z
review text: Let \(Z(t)\) be a \(d\)-dimensional time dependent covariate or marker process, and let \(\lambda(t)\) be the stochastic hazard for an individual with history \(\{Z(s);\;s \leq t\}\). \textit{N. P. Jewell} and \textit{J. P. Nielsen} [Biometrika 80, No. 1, 153-164 (1993; Zbl 0772.62052)] discussed the distinction between covariate and marker. For our purposes this distinction is unimportant. We examine the following model: \[ \lambda(t) = a \{Z(t), t\} Y(t), \tag{1} \] where \(Y(t)\) is an indicator of survival at time \(t\). Submodels of (1) have a long tradition in survival analysis. An important special case is where the intensity depends only on the marker process, that is, \[ \lambda (t) = \alpha \{Z(t)\} Y(t).\tag{2} \] We call (2) the marker-only model. This occurs in medical applications where the exposure time is not known precisely. We make several contributions. First, we introduce an alternative kernel estimator of \(\alpha\) in (1) within the general counting process framework of \textit{I. W. McKeague} and \textit{K. J. Utikal} [Ann. Stat. 18, No. 3, 1172-1187 (1990; Zbl 0721.62087)]. Our procedure is analogous to the Nadaraya-Watson regression estimator in construction. It also works in the special case (2) where time is not observed. Second, we obtain expressions for the asymptotic bias of our estimator, which McKeague and Utikal did not do; thus we obtain the optimal rate of convergence. Third, our conditions for establishing global convergence are strictly weaker than those of McKeague and Utikal. This is a consequence of the difference in proof technique, rather than estimator.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62M09
Mathematics Subject Classification ID: 62G20
zbMATH DE Number: 885117
zbMATH Keywords: hazard function
zbMATH Keywords: survival analysis
zbMATH Keywords: covariate
zbMATH Keywords: marker process
zbMATH Keywords: marker-only model
zbMATH Keywords: kernel estimator
zbMATH Keywords: counting process
zbMATH Keywords: Nadaraya-Watson regression estimator
zbMATH Keywords: asymptotic bias
zbMATH Keywords: optimal rate of convergence
author: Jens Perch Nielsen
author: Oliver Linton
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176324321
OpenAlex ID: W1991454797
DOI: 10.1214/AOS/1176324321
description: scientific article; zbMATH DE number 885117