Functional Data Analysis for Point Processes with Rare Events

In various applications, one encounters samples of objects, where each object consists of a small number of repeated event times observed over a fixed time interval. For such rare events data there are no flexible methods available that can be applied when the shapes of the intensity functions that generate the observed event times are not known or vary substantially between objects. We model the underlying intensity functions as nonparametric object-specific random functions. Applying a novel functional method to obtain the covariance structure of the associated random densities, we reconstruct object-specific density functions that reflect the distribution of event times. We demonstrate in simulations that the proposed functional approach is superior to conventional nonparametric methods, as it borrows strength from the entire sample of objects, rather than aiming at the estimation of each object’s density separately. Our method is based on a key relationship that allows to reduce the covariance estimation problem for random densities to the simpler problem of estimating a non-random joint density from pooled event times. We describe an application to model bid arrivals for a sample of online auctions and also include asymptotic justifications of the methodology.