Matching moments to phase distributions: Mixtures of erlang distributions of common order

Abstract

One approach to the moment-matching problem for phase distributions to to restrict selection to an appropriate subset of phase distributions. We investigate the use of mixtures of Erlang distributions of common order to match moments feasible for distributions with support on . We show that, except for special cases, the first κ (finite) moments of any nondegenerate distribution with support on can be matched by a mixture of Erlang distributions of (sufficiently high) common order. Moreover, we show that any κ-tuple of first κ moments feasible for a mixture of n-stage Erlang distributions feasible for a mixture of or fewer . The three-moment-matching problem is considered in detail. The set of pairs of second and third standardized moments feasible for mixtures of is characterized. An analytic expression is derived for the minimum order, n, such that a given set of first three moments is feasible for a mixture of .Expressions are also given for the parameters of the unique mixture of two that matches a feasible set of first three moments. Methods for implementation of these results are suggested and evaluated. In our evaluation, we consider distributional properties such as dimension, numerical stability, and density-function shape