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Point process diagnostics based on weighted second-order statistics and their asymptotic properties. (English) Zbl 1332.60070

Summary: A new approach for point process diagnostics is presented. The method is based on extending second-order statistics for point processes by weighting each point by the inverse of the conditional intensity function at the point’s location. The result is generalized versions of the spectral density, \(R/S\) statistic, correlation integral and \(K\)-function, which can be used to test the fit of a complex point process model with an arbitrary conditional intensity function, rather than a stationary Poisson model. Asymptotic properties of these generalized second-order statistics are derived, using an approach based on martingale theory.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62M99 Inference from stochastic processes
62M02 Markov processes: hypothesis testing
62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems

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