Abstract: The distribution of the grid points at which a response function is observed in longitudinal or functional data applications is often informative and not independent of the response process. Here, we propose a covariation model for estimating and making inferences about this interrelation, where we treat the data as replicated realizations of a marked point process. We derive the maximum likelihood estimators and the asymptotic distribution of the estimators. The behavior of the estimators is examined using simulations. Lastly, we apply the model to an online auction data set, and show that there is a strong correlation between bidding patterns and price trajectories.

Key words and phrases: Doubly-stochastic process, Karhunen-Loève decomposition, latent-variable model, Poisson process.