Semiparametric fiducial inference for Cox models
Yifan Cui, Jan Hannig, Paul Edlefsen
TL;DR
This work introduces a semiparametric fiducial inference framework for Cox models, using a data-generating algorithm to construct a generalized fiducial distribution for the log hazard ratio $\beta$ while treating the baseline hazard nonparametrically. A Gibbs sampler built on conic-optimization principles samples fiducial points for $\beta$ (and the baseline hazard) from inequalities derived by inverting the data-generating mechanism, with the Cox partial likelihood providing a link to classical inference. The authors prove consistency of the fiducial estimator and establish a Bernstein–von Mises result, ensuring asymptotically normal behavior and valid confidence sets; simulations show improved finite-sample performance relative to maximum partial likelihood, especially when MLE struggles, and the method extends to a broad class of semiparametric models. A real-data HIV trial application demonstrates practical utility in small subgroups, and the framework offers extensions to constrained models, additive hazards, and time-varying effects, highlighting fiducial inference as a viable, principled alternative in semiparametric survival analysis.
Abstract
R. A. Fisher introduced the concept of fiducial as a potential replacement for the Bayesian posterior distribution in the 1930s. During the past century, fiducial approaches have been explored in various parametric and nonparametric settings. However, to the best of our knowledge, no fiducial inference has been developed in the realm of semiparametric statistics. In this paper, we propose a novel fiducial approach for semiparametric models. To streamline our presentation, we use the Cox proportional hazards model, which is the most popular model for the analysis of survival data, as a running example. Other models and extensions are also discussed. In our experiments, we find our method to perform well especially in situations when the maximum likelihood estimator fails.