Moment-Based Approximation with Mixed Erlang Distributions

Moment-based approximations have been extensively analyzed over the years (see, e.g., Osogami and Harchol-Balter 2006 and references therein). A number of specific phase-type (and non phase-type) distributions have been considered to tackle the moment-matching problem (see, for instance, Johnson and Taaffe 1989). Motivated by the development of more flexible moment-based approximation methods, we develop and examine the use of finite mixture of Erlangs with a common rate parameter for the moment-matching problem. This is primarily motivated by Tijms (1994) who shows that this class of distributions can approximate any continuous positive distribution to an arbitrary level of accuracy, as well as the tractability of this class of distributions for various problems of interest in quantitative risk management. We consider separately situations where the rate parameter is either known or unknown. For the former case, a direct connection with a discrete moment-matching problem is established. A parallel to the s-convex stochastic order (e.g., Denuit et al. 1998) is also drawn. Numerical examples are considered throughout.