A Semiparametric Regression Model for Panel Count Data: When Do Pseudo-likelihood Estimators Become Badly Inefficient?

We consider estimation in a particular semiparametric regression model for the mean of a counting process under the assumption of “panel count” data. The basic model assumption is that the conditional mean function of the counting process is of the form E{N(t)|Z} = exp(θ′Z)Λ(t) where Z is a vector of covariates and Λ is the baseline mean function. The “panel count” observation scheme involves observation of the counting process N for an individual at a random number K of random time points; both the number and the locations of these time points may differ across individuals. We study maximum pseudo-likelihood and maximum likelihood estimators θ̂ n and θ̂n of the regression parameter θ. The pseudo-likelihood estimators are fairly easy to compute, while the full maximum likelihood estimators pose more challenges from the computational perspective. We derive expressions for the asymptotic variances of both estimators under the proportional mean model. Our primary aim is to understand when the pseudo-likelihood estimators have very low efficiency relative to the full maximum likelihood estimators. The upshot is that the pseudo-likelihood estimators can have arbitrarily small efficiency relative to the full maximum likelihood estimators when the distribution of K, the number of observation time points per individual, is very heavy-tailed. 2 Jon A. Wellner, Ying Zhang, and Hao Liu