Modeling marked point processes via bivariate mixture transition distribution models

We propose new probability models for the analysis of marked point processes. These models deal with the type of data that arrive or are observed in possibly unequal time intervals, such as financial transactions and earthquakes, among others. The models treat both the time between event arrivals and the observed marks as stochastic processes. We adopt a class of bivariate distributions to form the bivariate mixture transition distribution. In these models the conditional bivariate distribution of the next observation given the past is a mixture of conditional distributions given each one of the last p observations or a selection of past p events. The identifiability of the model is investigated, and an EM algorithm is developed to obtain estimates of the model parameters. Simulation and real data examples are used to demonstrate the utility of these models.