Open Access
February 1996 On the asymptotic properties of a flexible hazard estimator
Robert L. Strawderman, Anastasios A. Tsiatis
Ann. Statist. 24(1): 41-63 (February 1996). DOI: 10.1214/aos/1033066198


Suppose one has a stochastic time-dependent covariate $Z(t)$, and is interested in estimating the hazard relationship $\lambda(t|\overline{Z}(t)) = \omega(Z(t))$, where $\overline{Z}(t)$ denotes the history of $Z(t)$ up to and including time t. In this paper, we consider a model of the form $\exp(s_n(Z(t)))$, where $s_n(Z(t))$ is a spline of finite but arbitrary order, and investigate the behavior of the maximum likelihood estimator of the hazard as the number of knots in the spline function increases with the sample size at some rate $k_n = o(n)$. For twice continuously differentiable $\omega(\cdot)$, we demonstrate that the difference between the estimator $\exp(s_n(\cdot))$ and $\omega(\cdot)$ goes to 0 in probability in sup-norm for any $k_n = n^{\phi}, \phi \epsilon (0, 1)$. In addition, if $\phi > 1/5$, then $\exp(\hat{s}_n(Z(t))) - \omega(Z(t))$, properly normalized, is asymptotically standard normal. A large-sample approximation to the variance is derived in the case where $s_n(\cdot)$ is a linear spline, and exposes some rather interesting behavior.


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Robert L. Strawderman. Anastasios A. Tsiatis. "On the asymptotic properties of a flexible hazard estimator." Ann. Statist. 24 (1) 41 - 63, February 1996.


Published: February 1996
First available in Project Euclid: 26 September 2002

zbMATH: 0853.62030
MathSciNet: MR1389879
Digital Object Identifier: 10.1214/aos/1033066198

Primary: 62F12 , 62G05
Secondary: 60G44 , 62M99 , 62P10

Keywords: Censoring , hazard function , martingale , Spline , Survival analysis , time-dependent covariate

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 1 • February 1996
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