Abstract
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 \({{\mathcal Be}(p,\,q)}\) 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 \({\mathcal{B}e(p,\,1)}\) scaled by U, hence our results can be viewed as identification of a power distribution mixture.
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Huang, WJ., Su, NC. Identification of power distribution mixtures through regression of exponentials. Stat Papers 54, 227–241 (2013). https://doi.org/10.1007/s00362-011-0421-2
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DOI: https://doi.org/10.1007/s00362-011-0421-2
Keywords
- Beta distribution
- Characterization
- Constant regression
- Conditional expectation
- Gamma distribution
- Mixture distributions