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On some characterizations of the mixture of gamma distributions. (English) Zbl 1120.60004

Summary: Following A. K. Gupta and J. Wesolowski [Ann. Inst. Stat. Math. 49, 171–180 (1997; Zbl 0890.62007)], under the condition \(X/U\) and \(U\) are independent, \(X/U\) has a \({\mathcal B}e(p,q)\) distribution, and given \(X\) the conditional expectation of a certain function of \((U,X)\) is constant, we characterize the distribution of \((U,X)\). This problem is related to Lukacs type characterization, where both \(X\) and \(Y\) have to be gamma distributed with the same scale parameter, if both \(X\) and \(Y\), and \(X/(X+Y)\) and \(X+Y\) are independent. Among others, we prove if \(q=1\), and for some integer \(n\geq 1\), \(E(\sum^n_{i=1}a_i(U-X)^i\,|\,X)=b\), where \(a_1,\dots, a_n,b\), are real constants such that \(a_1^2+\cdots+a_n^2\neq 0\) and \(b\neq 0\), or for some real number \(n>0\), \(E((U-X)^n\,|\,X)=b\), where \(b>0\) is a constant, then the distribution of \((U,X)\) can be determined.

MSC:

60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions

Citations:

Zbl 0890.62007
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References:

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