Statistical inference for detrended point processes

Statistical inference for detrended point processes

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Article ID: iaor19951171
Country: Netherlands
Volume: 50
Issue: 2
Start Page Number: 331
End Page Number: 347
Publication Date: Apr 1994
Journal: Stochastic Processes and Their Applications
Authors:
Abstract:

The paper considers a multivariate point process with a parametric intensity process which splits into a stochastic factor equ1 and a trend function equ2 of a squared polynomial form with exponents larger than equ3. Such a process occurs in a situation where an underlying process with intensity equ4 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.

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