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

- The Cox regression model with a monotone baseline hazard is considered.
- Two isotonized smooth estimators of the baseline hazard are proposed.
- These estimators are shown to be asymptotically normal.
- They exhibit the same asymptotic variance but different biases.
- Numerical results on pointwise confidence intervals are provided.

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 nm∕(2m+1), where m≥2 denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuhaä 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.