Two Estimators of the Mean of a Counting Process with Panel Count Data

We study two estimators of the mean function of a counting process 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 di erent times during a 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 Kalb eisch 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 Kalb eisch and the full maximum likelihood estimator even if the underlying counting process is not a Poisson process We also derive the asymptotic distribution of both estimators at a xed time t and compare the resulting theoretical relative e ciency with nite sample relative e ciency by way of a limited monte carlo study Research supported in part by National Science Foundation grant DMS and NIAID grant R AI Research supported in part by National Science Foundation grant DMS AMS subject classi cations Primary F F secondary J J