Smooth Plug‐in Inverse Estimators in the Current Status Continuous Mark Model

Abstract

ABSTRACT. We consider the problem of estimating the joint distribution function of the event time and a continuous mark variable when the event time is subject to interval censoring case 1 and the continuous mark variable is only observed in case the event occurred before the time of inspection. The nonparametric maximum likelihood estimator in this model is known to be inconsistent. We study two alternative smooth estimators, based on the explicit (inverse) expression of the distribution function of interest in terms of the density of the observable vector. We derive the pointwise asymptotic distribution of both estimators.

A maximum smoothed likelihood estimator in the current status continuous mark model

“…On the second coordinate we took 5 cells for the MSLE, which means that the bias on the second coordinate does not play a role, since 0.6 is then a point of the grid for the second coordinate, and on the first coordinate we took the number of cells between 4 (for n = 500) and 7 (for n = 10 000). The results were obtained by generating 10 000 samples for each value of (t, z), considered in the table, and each sample size n. We compared the results with the MSE's of the plug-in estimator F (1) n , studied in Groeneboom et al (2011), and defined by…”

“…The choice of δ n of order n −1/5 is probably optimal. One can also choose ε n of this order, but it is probably better to choose ε n δ n , see for a discussion on this matter Groeneboom et al (2011). Further remarks on the problem of binwidth choice can be found in section 4.…”

“…Asymptotic results for K → ∞ as n → ∞ are not yet known. Another approach to obtain a consistent estimator is adopted in Groeneboom et al (2011). There a plug-in inverse estimator for the bivariate distribution function F 0 , using kernel estimators, is studied and its asymptotic distribution is derived.…”

“…We prove that for a histogramtype smoothing of the observation distribution the resulting MSLE is consistent under certain conditions. Contrary to the plug-in inverse estimator studied by Groeneboom et al (2011), the MSLE is a real distribution function.…”

Bayesian nonparametric estimation in the current status continuous mark model

“…This is in agreement with the results in . Groeneboom, Jongbloed & Witte (2011) and Groeneboom, Jongbloed & Witte (2012) consider the exact setting of this paper using frequentist estimation methods. In Groeneboom, Jongbloed & Witte (2011) two plug-in inverse estimators are proposed.…”