Smooth and non-smooth estimates of a monotone hazard

Abstract

We discuss a number of estimates of the hazard under the assumption that the hazard is monotone on an interval $[0,a]$. The usual isotonic least squares estimators of the hazard are inconsistent at the boundary points $0$ and $a$. We use penalization to obtain uniformly consistent estimators. Moreover, we determine the optimal penalization constants, extending related work in this direction by [ Statist. Sinica 3 (1993) 501–515; Ann. Statist. 27 (1999) 338–360]. Two methods of obtaining smooth monotone estimates based on a non-smooth monotone estimator are discussed. One is based on kernel smoothing, the other on penalization.