Approximating Probability Distributions in an Extreme Renewal Process

• Activity Number: 475 - Statistical Computing and Inference
• Type: Contributed
• Date/Time: Wednesday, August 2, 2017 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Computing
• Author(s): Percy Brill* and Mei Ling Huang
• Companies: University of Windsor and Brock University
• Keywords: renewal process ; no-mean inter-arrivals ; excess, age, total life ; regenerative process ; integral equations ; L1 distance for pdfs

Abstract

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.