On the asymptotic properties of a flexible hazard estimator

• 成果类型: Article
• 署名作者: Strawderman, RL; Tsiatis, AA
• 署名单位: Harvard University; Harvard T.H. Chan School of Public Health
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1996
• 页码: 41-63
• 关键词: Nonparametric regression; counting-processes; LARGE-SAMPLE; models; CONVERGENCE; inference; rates

摘要

Suppose one has a stochastic time-dependent covariate Z(t), and is interested in estimating the hazard relationship lambda(t\(Z) over bar(t)) = omega(Z(t)), where (Z) over bar(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(.), we demonstrate that the difference between the estimator exp(s(n)(.)) and omega(.) goes to 0 in probability in sup-norm for any k(n) = n(phi), phi is an element of (0, 1). In addition, if phi > 1/5, then exp((s) over cap(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)(.) is a linear spline, and exposes some rather interesting behavior.