Local partial likelihood estimation in proportional hazards regression

TL;DR

This paper introduces a novel local partial likelihood method for directly estimating the relative risk function in proportional hazards models, improving upon previous approaches by considering neighborhoods around two points simultaneously.

Contribution

It proposes a new approach that estimates the difference in risk functions by selecting observations in shrinking neighborhoods around two points, with rigorous asymptotic analysis.

Findings

Asymptotic properties of the estimator are established.

Variance of the estimator can be easily estimated.

Simulation study confirms the effectiveness of the method.

Abstract

Fan, Gijbels and King [Ann. Statist. 25 (1997) 1661--1690] considered the estimation of the risk function $\psi (x)$ in the proportional hazards model. Their proposed estimator is based on integrating the estimated derivative function obtained through a local version of the partial likelihood. They proved the large sample properties of the derivative function, but the large sample properties of the estimator for the risk function itself were not established. In this paper, we consider direct estimation of the relative risk function $\psi (x_2)-\psi (x_1)$ for any location normalization point $x_1$. The main novelty in our approach is that we select observations in shrinking neighborhoods of both $x_1$ and $x_2$ when constructing a local version of the partial likelihood, whereas Fan, Gijbels and King [Ann. Statist. 25 (1997) 1661--1690] only concentrated on a single neighborhood, resulting in the cancellation of the risk function in the local likelihood function. The asymptotic properties of our estimator are rigorously established and the variance of the estimator is easily estimated. The idea behind our approach is extended to estimate the differences between groups. A simulation study is carried out.