Moment distributions of phase-type (abstract only)

Abstract

Both matrix-exponential and phase-type distributions have a number of important closure properties. Among those are the distributions of the age and residual life-time of a stationary renewal process with inter-arrivals of either type. In this talk we show that the spread, which is the sum of the age an residual life-time, is also phase-type distributed. Moreover, we give some explicit representations. The spread is known to have a first order moment distribution. If X is a positive random variable and ?i is its i'th moment, then the function fi(x) = xif(x)/?i is a density function, and the corresponding distribution is called the i'th order moment distribution. We prove that the classes of matrix-exponential or phase-type distributions are closed under the formation of moment distributions of any order. Other distributions which are closed under the formation of moment distributions are e.g. log-normal, Pareto and gamma distributions. We provide explicit representations for both the matrix-exponential class and for the phase-type distributions, where the latter class may also use the former representations, but for various reasons it is desirable to establish a phase-type representation when dealing with phase-type distributions. For the first order distribution we present an explicit formula for the related Lorenz curve and Gini index. Moment distributions of orders one, two and three have been extensively used in areas such as economy, physics, demography and civil engineering.