Asymptotic properties of the GMLE in the case 1 interval‐censorship model with discrete inspection times

Abstract

AbstractWe consider the case 1 interval censorship model in which the survival time has an arbitrary distribution function F0 and the inspection time has a discrete distribution function G. In such a model one is only able to observe the inspection time and whether the value of the survival time lies before or after the inspection time. We prove the strong consistency of the generalized maximum‐likelihood estimate (GMLE) of the distribution function F0 at the support points of G and its asymptotic normality and efficiency at what we call regular points. We also present a consistent estimate of the asymptotic variance at these points. The first result implies uniform strong consistency on [0, ∞) if F0 is continuous and the support of G is dense in [0, ∞). For arbitrary F0 and G, Peto (1973) and Tumbull (1976) conjectured that the convergence for the GMLE is at the usual parametric rate n½ Our asymptotic normality result supports their conjecture under our assumptions. But their conjecture was disproved by Groeneboom and Wellner (1992), who obtained the nonparametric rate ni under smoothness assumptions on the F0 and G.