Statistical inference for detrended point processes

We consider a multivariate point process with a parametric intensity process which splits into a stochastic factor bt and a trend function at of a squared polynomial form with exponents larger than . Such a process occurs in a situation where an underlying process with intensity bt can be observed on a transformed time scale only. On the basis of the maximum likelihood estimator for the unknown parameter a detrended (or residual) process is defined by transforming the occurrence times via integrated estimated trend function. It is shown that statistics (mean intensity, periodogram estimator) based on the detrended process exhibit the same asymptotic properties as they do in the case of the underlying process (without trend function). Thus trend removal in point processes turns out to be an appropriate method to reveal properties of the (unobservable) underlying process - a concept which is well established in time series. A numerical example of an earthquake aftershock sequence illustrates the performance of the method.