A Generalization of the Gamma Distribution

Abstract

This paper concerns a generalization of the gamma distribution, the specific form being suggested by Liouville's extension to Dirichlet's integral formula [3]. In this form it also may be regarded as a special case of a function introduced by L. Amoroso [1] and R. d'Addario [2] in analyzing the distribution of economic income. (Also listed in [4] and [5].) In essence, the generalization (1) herein is accomplished by supplying a positive parameter, $p$, as an exponent in the exponential factor of the gamma distribution. The moment generating function is shown, and cumulative probabilities are related directly to the incomplete gamma function (tabulated in [6]). Distributions are given for various functions of independent "generalized gamma variates" thus defined, special attention being given to the sum of such variates. Convolution results occur in alternating series form, with coefficients whose evaluation may be tedious and lengthy. An upper bound is provided for the modulus of each term, and simplified computation methods are developed for some special cases. A corollary is derived showing that the researches of Robbins in [7] apply to a larger class of problems than was treated in [7]. Extensions of his methods lead to iterative formulae for the coefficients in series obtained for an even larger class of problems.