Higher Order Cumulants and Inference for a Class of Filtered Poisson Processes

R.J. Kulperger

doi:10.17713/ajs.v27i1&2.530

Authors

R.J. Kulperger Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, Canada

DOI:

https://doi.org/10.17713/ajs.v27i1&2.530

Abstract

Spectral methods are useful in the analysis of time series and point process data in Zd or Rd. Parameter estimates based on these often have a limiting Gaussian distribution, whose limiting variance depends upon integrals of the second, third and fourth order spectral densities. The effective evaluation of these spectral density integrals is needed. In image analysis (d = 2), these integrals are very time consuming to evaluate. This paper
considers a particular class of integrated or filtered Poisson processes. Using higher order cumulants, one can identify parameters up to a certain order. In some parametric cases these identify all the parameters of the process. In such cases it would then be possible to construct and hence simulate a filtered Poisson process for the estimator functional has the same asymptotics. Simulation or bootstrapping this process is a more efficient way of estimating or approximating the distribution of the parameter estimate. This paper
demonstrates the validity for such a method. Specific examples of scanning fluorescence correlation spectroscopy (S–FCS) and image correlation spectroscopy (ICS) are discussed and used as motivating examples. In these cases one can obtain estimates of an object that previously could only inferred indirectly.

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