Smooth estimation of a monotone hazard and a monotone density under random censoring

TL;DR

This paper introduces kernel smoothed Grenander-type estimators for monotone hazard and density functions under right censoring, demonstrating their convergence rates and Gaussian limit distributions, with boundary correction considerations.

Contribution

It develops boundary-corrected kernel estimators for monotone hazard and density functions under censoring, establishing their convergence rates and asymptotic distributions.

Findings

Estimators converge at rate n^{2/5}.

Limit distribution at fixed points is Gaussian.

Boundary correction allows uniform consistency away from endpoints.

Abstract

We consider kernel smoothed Grenander-type estimators for a monotone hazard rate and a monotone density in the presence of randomly right censored data. We show that they converge at rate $n^{2/5}$ and that the limit distribution at a fixed point is Gaussian with explicitly given mean and variance. It is well-known that standard kernel smoothing leads to inconsistency problems at the boundary points. It turns out that, also by using a boundary correction, we can only establish uniform consistency on intervals that stay away from the end point of the support (though we can go arbitrarily close to the right boundary).