Nonparametric estimation of a convex bathtub-shaped hazard function

TL;DR

This paper develops a nonparametric maximum likelihood estimator for convex hazard functions, proving its consistency, convergence rate, and asymptotic distribution without the need for tuning parameters.

Contribution

It introduces a new nonparametric MLE for convex hazard functions with proven statistical properties and no tuning parameter selection.

Findings

Estimator is consistent and converges at rate n^{2/5}

Asymptotic distribution theory established

No tuning parameter needed for the estimator

Abstract

In this paper, we study the nonparametric maximum likelihood estimator (MLE) of a convex hazard function. We show that the MLE is consistent and converges at a local rate of $n^{2/5}$ at points $x_0$ where the true hazard function is positive and strictly convex. Moreover, we establish the pointwise asymptotic distribution theory of our estimator under these same assumptions. One notable feature of the nonparametric MLE studied here is that no arbitrary choice of tuning parameter (or complicated data-adaptive selection of the tuning parameter) is required.