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

Abstract

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:1. • 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. 2. • The higher the number of events observed, the higher the estimate of the initial intensity of the event. 3. • Both the duration of observation and the timing of events contribute significantly to determining the rate at which the event rate changes over time.