Distribution Estimation Using Laplace Transforms

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Abstract
We propose two related methods for deriving probability distribution estimates using approximate rational Laplace transform representations. Whatever method is used, the result is a Coxian estimate for an arbitrary distribution form or plain sample data, with the algebra of the Coxian often simplifying to a generalized hyperexponential or phase-type. The transform (or, alternatively, the moment-generating function) is used to facilitate the computations and leads to an attractive algorithm. For method one, the first 2N − 1 derivatives of the transform are matched with those of an approximate rational function; for the second method, a like number of values of the transform are matched with those of the approximation. The numerical process in both cases begins with an empirical Laplace transform or truncation of the actual transform, and then requires only the solution of a relatively small system of linear equations, followed by root finding for a low-degree polynomial. Besides the computationally attractive features of the overall procedure, it addresses the question of the number of terms, or the order, involved in a generalized hyperexponential, phase-type, or Coxian distribution, a problem not adequately treated by existing methods. Coxian distributions are commonly used in the modeling of single-stage and network queueing problems, inventory theory, and reliability analyses. They are particularly handy in the development of large-scale model approximations.

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