Isotonized smooth estimators of a monotone baseline hazard in the Cox model

TL;DR

This paper introduces two isotonic smooth estimators for a monotone baseline hazard in the Cox model, analyzing their asymptotic properties and comparing their performance through numerical simulations.

Contribution

It develops and compares two novel isotonic smooth estimators for the Cox model's baseline hazard, detailing their asymptotic behavior and practical performance.

Findings

Both estimators are asymptotically normal at rate n^{m/(2m+1)}.

The Grenander-type estimator is asymptotically equivalent to kernel smoothed isotonic estimators.

The maximum smoothed likelihood estimator has the same variance but different bias.

Abstract

We consider two isotonic smooth estimators for a monotone baseline hazard in the Cox model, a maximum smooth likelihood estimator and a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative baseline hazard. We show that they are both asymptotically normal at rate $n^{m/(2m+1)}$, where $m\geq 2$ denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuha\"a and Musta (2016). It turns out that the Grenander-type estimator is asymptotically equivalent to the kernel smoothed isotonic estimators, while the maximum smoothed likelihood estimator exhibits the same asymptotic variance but a different bias. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.