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On the asymptotic properties of a flexible hazard estimator. (English) Zbl 0853.62030

Summary: Suppose one has a stochastic time-dependent covariate \(Z(t)\), and is interested in estimating the hazard relationship \(\lambda (t \mid \overline Z(t)) = \omega (Z(t))\), where \(\overline Z(t)\) denotes the history of \(Z(t)\) up to and including time \(t\). 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 \(k_n = n^\varphi\), \(\varphi \in (0,1)\). In addition, if \(\varphi > 1/5\), then \(\exp (\widehat s_n (Z(t))) - \omega (Z(t))\), 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.

MSC:

62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62F12 Asymptotic properties of parametric estimators
60G44 Martingales with continuous parameter
62M99 Inference from stochastic processes
62P10 Applications of statistics to biology and medical sciences; meta analysis
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