Some characterizations of the gamma distribution

Khatri, C. G. ; Radhakrishna Rao, C. (1968) Some characterizations of the gamma distribution Sankhya: The Indian Journal of Statistics, 30 (2). pp. 157-166. ISSN 0972-7671

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Let X1, X2, ..., Xn are independent and positive random variates, Yi=∑nj=1bij Xj, i=1, 2, ..., p, and Ytnj=1 Xjbtj, t=p+1, ...,n. In this paper, the conditions under which the equations E(Yi|Yp+1...,Yn)=constant, i=1, 2,..., p, imply gamma distributions for X1, X2, ..., Xn are examined. For X1, ..., Xn independent variates with E(X-1j≠=0, j=1, 2, ..., n, Yi=∑nj=1bijX-1j, i=1,2, ..., p and Yt=∑nj=1btj Xj, t=p+1,..., n, the equations E(Yi|Yp+1,...,Yn)= constant, i=1, 2, ..., p, imply γor conjugate γ distributions for X1,…,Xn under the same conditions on the coefficients bij'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,…,Xn under the conditions E(Yi|Yp+1,...,Yn)= constant, i=1,2,…,p where Yi=∑j=1nbijg(Xj), i=1,2,...,p, Yt=∑nj=1btjXj, 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[-αƒαa(g(y)-β)dy], where α and β are parameters and a is a constant of integration.

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