# 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}+O_{2}{texte{ et }}P={frac {O_{1}}{O_{1}+O_{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 {style d'affichage P_{je}={frac {O_{je}}{somme _{je=1}^{k}O_{je}}}} is independent of {displaystyle W=sum _{je=1}^{k}O_{je}} if and only if all the Y i have gamma distributions with the same scale parameter.[2] References ^ Lukacs, Eugène (1955). "A characterization of the gamma distribution". Annales de statistiques mathématiques. 26: 319–324. est ce que je:10.1214/aoms/1177728549. ^ Mosimann, Jacques E. (1962). "On the compound multinomial distribution, the multivariate {style d'affichage bêta } distribution, and correlation among proportions". Biometrika. 49 (1 et 2): 65–82. est ce que je:10.1093/biomet/49.1-2.65. JSTOR 2333468. Ng, O. 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. Catégories: Probability theoremsCharacterization of probability distributions

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