## The Annals of Statistics

### Two estimators of the mean of a counting process with panel count data

#### Abstract

We study two estimators of the mean function of a countingprocess based on “panel count data.” The setting for “panel count data” is one in which $n$ independent subjects, each with a counting process with common mean function, are observed at several possibly different times duringa study. Following a model proposed by Schick and Yu, we allow the number of observation times, and the observation times themselves, to be random variables. Our goal is to estimate the mean function of the counting process. We show that the estimator of the mean function proposed by Sun and Kalbfleisch can be viewed as a pseudo-maximum likelihood estimator when a non-homogeneous Poisson process model is assumed for the counting process. We establish consistency of both the nonparametric pseudo maximum likelihood estimator of Sun and Kalbfleisch and the full maximum likeli- hood estimator, even if the underlying counting process is not a Poisson process.We also derive the asymptotic distribution of both estimators at a fixed time $t$, and compare the resulting theoretical relative efficiency with finite sample relative efficiency by way of a limited Monte-Carlo study.

#### Article information

Source
Ann. Statist., Volume 28, Number 3 (2000), 779-814.

Dates
First available in Project Euclid: 12 March 2002

https://projecteuclid.org/euclid.aos/1015951998

Digital Object Identifier
doi:10.1214/aos/1015951998

Mathematical Reviews number (MathSciNet)
MR1792787

Zentralblatt MATH identifier
1105.62372

Subjects
Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties 62N01: Censored data models

#### Citation

Wellner, Jon A.; Zhang, Ying. Two estimators of the mean of a counting process with panel count data. Ann. Statist. 28 (2000), no. 3, 779--814. doi:10.1214/aos/1015951998. https://projecteuclid.org/euclid.aos/1015951998

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