Lukacs's proportion-sum independence theorem

Lukacs's proportion-sum independence theorem In statistics, Lukacs's proportion-sum independence theorem is a result that is used when studying proportions, in particular the Dirichlet distribution. It is named after Eugene Lukacs.[1] The theorem If Y1 and Y2 are non-degenerate, independent random variables, then the random variables {displaystyle W=Y_{1}+Y_{2}{text{ and }}P={frac {Y_{1}}{Y_{1}+Y_{2}}}} are independently distributed if and only if both Y1 and Y2 have gamma distributions with the same scale parameter.

Corollary Suppose Y i, i = 1, ..., k be non-degenerate, independent, positive random variables. Then each of k − 1 random variables {displaystyle P_{i}={frac {Y_{i}}{sum _{i=1}^{k}Y_{i}}}} is independent of {displaystyle W=sum _{i=1}^{k}Y_{i}} if and only if all the Y i have gamma distributions with the same scale parameter.[2] References ^ Lukacs, Eugene (1955). "A characterization of the gamma distribution". Annals of Mathematical Statistics. 26: 319–324. doi:10.1214/aoms/1177728549. ^ Mosimann, James E. (1962). "On the compound multinomial distribution, the multivariate {displaystyle beta } distribution, and correlation among proportions". Biometrika. 49 (1 and 2): 65–82. doi:10.1093/biomet/49.1-2.65. JSTOR 2333468. Ng, W. N.; Tian, G-L; Tang, M-L (2011). Dirichlet and Related Distributions. John Wiley & Sons, Ltd. ISBN 978-0-470-68819-9. page 64. Lukacs's proportion-sum independence theorem and the corollary with a proof. Categories: Probability theoremsCharacterization of probability distributions