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Percy Brill

University of Windsor, Natural Sciences and Engineering Research Council of Canada



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Mei Ling Huang

University of Windsor, Natural Sciences and Engineering Research Council of Canada



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475 – Statistical Computing and Inference

Approximating the Finite-Time t Probability Distributions

Sponsor: Section on Statistical Computing
Keywords: renewal process, no-mean inter-arrivals, excess, age, total life, regenerative process, integral equations, L1 distance for pdfs

Percy Brill

University of Windsor, Natural Sciences and Engineering Research Council of Canada

Mei Ling Huang

University of Windsor, Natural Sciences and Engineering Research Council of Canada

We study an extreme renewal process with no-mean Pareto(II) inter-arrival times and shape parameter in (0,1]. We give integral equations for the probability density functions (pdfs) of the finite-time t excess, age and total life; the limiting pdfs as t tends to infinity do not exist because the Pareto(II) distribution has no mean. We approximate the time-t pdfs for large t using the corresponding limiting pdfs of a renewal process having right-truncated Pareto(II) inter-arrival times, the same shape parameter, and truncation point K > 0, which have very simple formulas. We apply an L1 metric with values in (0, 1), to measure the distance between the approximating pdfs and time-t pdfs.

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