A notes bivariate power law processes: conditional intensity and parameter estimation techniques

The point process model effectively represents the number of random events occurring over time through its intensity function. When events are of two types, a bivariate point process allows simultaneous analysis of each event's intensity. This study develops a conditional intensity model for a non-homogeneous bivariate point process over time with an event rate approach that follows a certain pattern over time in the form of a time-dependent power law intensity function with two parameters, an initial intensity parameter and a control parameter governing the change in the event rate over time. Parameter estimation is performed using the maximum likelihood method derived from the probability of one event occurring in a very short interval and the non-occurrence at other times. The results of the analysis show that:•The effect of observation duration on model parameters is not linear but depends on its interaction with the pattern of changes in the event rate over time.•The higher the number of events observed, the higher the estimate of the initial intensity of the event.•Both the duration of observation and the timing of events contribute significantly to determining the rate at which the event rate changes over time.