Some characterizations of the gamma distribution

Abstract

Let X₁, X₂, ..., X_n are independent and positive random variates, Y_i=∑ⁿ_j=1b_ij X_j, i=1, 2, ..., p, and Y_t=Πⁿ_j=1 X_j^b_tj, t=p+1, ...,n. In this paper, the conditions under which the equations E(Y_i|Y_p+1...,Y_n)=constant, i=1, 2,..., p, imply gamma distributions for X₁, X₂, ..., X_n are examined. For X₁, ..., X_n independent variates with E(X^-1_j≠=0, j=1, 2, ..., n, Y_i=∑ⁿ_j=1b_ijX^-1_j, i=1,2, ..., p and Y_t=∑ⁿ_j=1b_tj X_j, t=p+1,..., n, the equations E(Y_i|Y_p+1,...,Y_n)= constant, i=1, 2, ..., p, imply γor conjugate γ distributions for X₁,…,X_n under the same conditions on the coefficients b_ij's as before. The proofs require the solution of the two types of new functional equations. We have also investigated the form of distribution functions for the random variates X₁,…,X_n under the conditions E(Y_i|Y_p+1,...,Y_n)= constant, i=1,2,…,p where Y_i=∑_j=1nb_ijg(X_j), i=1,2,...,p, Y_t=∑ⁿ_j=1b_tjX_j, t=p+1,...,n and g(x) is a continuous function of x. The genaral form of the density function under some conditions is shown to be f(x)=exp[-αƒ^α_a(g(y)-β)dy], where α and β are parameters and a is a constant of integration.