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

Hendrik Paul Lopuhaä, Eni Musta*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

5 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)43-67
Number of pages25
JournalJournal of Statistical Planning and Inference
Publication statusPublished - 2017


  • Asymptotic normality
  • Cox regression model
  • Hazard rate
  • Isotonic estimation
  • Isotonized smoothed Breslow estimator
  • Kernel smoothing
  • Maximum smoothed likelihood estimator


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