Identification of power distribution mixtures through regression of exponentials

Given two independent non-degenerate positive random variables X and Y, Lukacs (Ann Math Stat 26:319–324, <CitationRef CitationID="CR14">1955</CitationRef>) proved that X/(X + Y) and X + Y are independent if and only if X and Y are gamma distributed with the same scale parameter. In this work, under the assumption X/U and U are independent, and X/U has a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${{\mathcal Be}(p,\,q)}$$</EquationSource> </InlineEquation> distribution, we characterize the distribution of (U, X) by the condition E(h(U − X)|X) = b, where h is allowed to be a linear combination of exponential functions. Since the assumption for X and U above is equivalent to X|U being <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\mathcal{B}e(p,\,1)}$$</EquationSource> </InlineEquation> scaled by U, hence our results can be viewed as identification of a power distribution mixture. Copyright Springer-Verlag 2013