## 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:

 6e+06 Probability distributions: general theory 6.2e+11 Characterization and structure theory of statistical distributions

Zbl 0890.62007
Full Text:

### References:

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