Statistica Sinica 17(2007), 1091-1114

THE MAXIMUM LIKELIHOOD METHOD WITH

ESTIMATED NUISANCE PARAMETERS IN HAZARD

RATE MODELS WITH DISCONTINUITIES

Dietmar Ferger

Dresden University of Technology

Abstract: Let $X_1, \ldots, X_n$ be i.i.d. with common hazard function, a step function with exactly one jump. The location of the jump is the parameter of interest and is to be estimated based on our sample. We prove consistency and convergence in law of our estimators with rate $n$ and non-normal limit distribution. There is also $L_p$ -convergence with exact rate $n^{-1}$. This statistical experiment is non-regular in the sense of Ibragimov and Has'minskii (1981). Our approach is extended to general hazard functions with one jump-point. The basic idea can also be used in a complete nonparametric framework.

Key words and phrases: Argmax of Poisson-process, change-point, maximum-likelihood, nonparametric hazard.