Local linear estimation and plug-in bandwidth selection for the ratio of two hazard rate functions

An estimator of the ratio of two hazard rate functions is derived, based on the local linear smoothing of a piecewise constant empirical estimator of the ratio. The asymptotic properties of the estimator are studied analytically and employed in developing a plug-in bandwidth selector which facilitates its optimal implementation. The asymptotic consistency and distribution of the practically useful (data-driven) version of the bandwidth rule are also studied herein. The potential of the method as a data-analytic tool is illustrated by its application in evaluating the effect on survival of a treatment response for primary biliary cirrhosis patients.

本文基于两类风险率函数（hazard rate functions）之比的分段常数经验估计量（piecewise constant empirical estimator）的局部线性平滑（local linear smoothing）方法，推导出该比值的估计量。对该估计量的渐近性质（asymptotic properties）开展解析研究，并基于此构建了可助力其最优实现的插件带宽选择器（plug-in bandwidth selector）。本文同时探究了该带宽准则中具备实际应用价值的数据驱动（data-driven）形式的渐近一致性（asymptotic consistency）与分布特征。通过将该方法应用于评估治疗应答对原发性胆汁性肝硬化（primary biliary cirrhosis）患者生存状况的影响，验证了其作为数据分析工具的应用潜力。