On the asymptotic properties of a flexible hazard estimator

Abstract

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.