Analysis of Interval-Censored or Aggregated Point Process Data

The use of local likelihood methods (Tibshirani and Hastie, 1987; Loader, 1996) in the presence of interval-censored or aggregated data leads to a natural consideration of an EM-type strategy, or a local EM algorithm. We consider local EM to analyze point process data that are either interval-censored or aggregated into counts. We present the formulation of local EM algorithms to estimate density, intensity and risk and their implementations using piecewise constant functions. It is shown that the use of the piecewise constant function at the E-step explicitly results in an iteration that involves an expectation, maximization and smoothing step, or an EMS algorithm considered in Silverman et al. (1990). Consequently, we reveal a previously unknown connection between local EM and the EMS algorithm. From a theoretical perspective, local EM and the EMS algorithm complement each other. We demonstrate that an EMS algorithm rises naturally from a local likelihood consideration in the context of point processes while the EMS algorithm serves as a rapid implementation of local EM algorithms and provides theoretical tools to better understand the role of local EM.