A simple compound scan statistic useful for modeling insurance and risk management problems

Abstract

In the present work we study the distribution of a random sum of random variables which is related to a binary scan statistic for Markov dependent trials. The motivation of the model studied herein stems from several areas of applied science such as actuarial science, financial risk management, quality control and reliability, educational psychology, engineering, etc.Let us consider a sequence of binary success/failure trials and denote by Tk the waiting time for the first occurrence of two successes separated by at most k failures, where k≥0 is any integer. Let also Y1,Y2,… be a sequence of independent and identically distributed (i.i.d) discrete random variables, independent of Tk. In the present article we develop some results for the distribution of the compound random variable Sk=∑t=1TkYt and illustrate how these results can be profitably used to study models pertaining to actuarial science and financial risk management practice.