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, 1955 ) 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 ℬe(p, q) 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 ℬe(p, 1) scaled by U , hence our results can be viewed as identification of a power distribution mixture.