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Kernel estimation in a nonparametric marker dependent hazard model. (English) Zbl 0847.62023

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 \(\{Z(s);\;s \leq t\}\). N. P. Jewell and 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 I. W. McKeague and 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.

MSC:

62G05 Nonparametric estimation
62M09 Non-Markovian processes: estimation
62G20 Asymptotic properties of nonparametric inference
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