Computation of nonparametric convex hazard estimators via profile methods

Nonparametric estimation of a convex bathtub-shaped hazard function

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.

Computation of nonparametric convex hazard estimators via profile methods

This paper proposes a profile likelihood algorithm to compute the nonparametric maximum likelihood estimator of a convex hazard function. The maximisation is performed in two steps: First the support reduction algorithm is used to maximise the likelihood over all hazard functions with a given point of minimum (or antimode). Then it is shown that the profile (or partially maximised) likelihood is quasi-concave as a function of the antimode, so that a bisection algorithm can be applied to find the maximum of the profile likelihood, and hence also the global maximum. The new algorithm is illustrated using both artificial and real data, including lifetime data for Canadian males and females.