Cornish-Fisher expansions for Poisson and negative binomial processes

Abstract

Negative binomial and Poisson processes fX(t);t 0g have cu- mulants proportional to the rate . So, formal Cornish-Fisher expansions in powers of 1=2 are available for their cumulative distribution function and quantiles. If T is an independent random variable with cumulants propor- tional to then the cumulants of X(T) are proportional to ; so, Cornish- Fisher expansions in powers of 1=2 are available for its cumulative distribu- tion function and quantiles. When T has a gamma distribution, alternative expansions in powers of 1 are available for the probability mass function of X(T). Let fX(t);t 0g be a point process and T be a random variable independent of it. The aim of this note is derive exact expressions for the cumulative distribution function, probability mass function and quantiles of X(T) for a class of point processes, a class that includes the Poisson and negative binomial processes. We believe that this is rst time exact expressions have been derived for the stated functions and for the stated class of point processes. These results are useful, for example, for modeling inventory demand when X(t) has independent increments and the distribution of X(t+ ) X( ) does not depend on . In inventory modeling X(T) is the lead-time demand corresponding to a lead-time T. Its distribution is used to determine when to order stock. The results will also be useful in other prominent application areas of Poisson processes, some of which are: electrical and electronic engineering, operations research, management science, computer science, physics, industrial engineering, economics, manufacturing engineering, neurosciences, medicine, civil engineering, environmental sciences, astronomy, mechanical engineering, geochemistry, optics, materials science, business, ecology, and educational research. The results will also be useful for negative binomial processes which are used for modeling events for each risk type in many chronic disease processes, vehicle