Improving unbiasedness of the proportional hazards model estimator with Cox and Snell's bias approximation and jackknife resampling

Bias approximation has played an important role in the maximum likelihood estimation method, and numerous bias calculation techniques have been proposed under different contexts. For the semiparametric proportional hazards model, which is the standard regression method to study the time-to-event data, the existing work applied the bias formula and derived the approximate bias of Cox's estimator based on the partial likelihood function. In this work, we instead use the joint likelihood function and utilize the counting process approach to develop an approximate bias of Cox's estimator. Explicit expressions for the higher-order partial derivatives are derived, which facilitate the bias calculation techniques. We also incorporate the jackknife resampling method and propose a Jackknife-Cox-Snell method that processes the bias of Cox's estimator through two steps. The first step aims to remove the analytical terms derived from Cox and Snell's formula and the second step reduces the residual bias term. A comprehensive simulation study is performed to show the usefulness of the proposed bias-corrected method. We also apply the proposed method to two sets of survival data for comparison and illustration.