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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Official URL: http://sankhya.isical.ac.in/search/30a2/30a2018.ht...

## Abstract

Let X_{1}, X_{2}, ..., X_{n} are independent and positive random variates, Y_{i}=∑^{n}_{j=1}b_{ij} X_{j}, i=1, 2, ..., p, and Y_{t}=Π^{n}_{j=1} X_{j}^{btj}, 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 X_{1}, X_{2}, ..., X_{n} are examined. For X_{1}, ..., X_{n} independent variates with E(X^{-1}_{j}≠=0, j=1, 2, ..., n, Y_{i}=∑^{n}_{j=1}b_{ij}X^{-1}_{j}, i=1,2, ..., p and Y_{t}=∑^{n}_{j=1}b_{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 X_{1},…,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 X_{1},…,X_{n} under the conditions E(Y_{i}|Y_{p+1},...,Y_{n})= constant, i=1,2,…,p where Y_{i}=∑_{j=1}_{n}b_{ij}g(X_{j}), i=1,2,...,p, Y_{t}=∑^{n}_{j=1}b_{tj}X_{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.

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Deposited On: | 25 Nov 2011 12:42 |

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