STATISTICAL-INFERENCE FOR DETRENDED POINT-PROCESSES

We consider a multivariate point process with a parametric intensity p rocess which splits into a stochastic factor b(t) and a trend function a(t) of a squared polynomial form with exponents larger than - 1/2. S uch a process occurs in a situation where an underlying process with i ntensity b(t) 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 occurr ence times via integrated estimated trend function. It is shown that s tatistics (mean intensity, periodogram estimator) based on the detrend ed process exhibit the same asymptotic properties as they do in the ca se of the underlying process (without trend function). Thus trend remo val 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 earthqu ake aftershock sequence illustrates the performance of the method.