Lukacs's proportion-sum independence theorem explained

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

W = Y _{1+Y} _{2andP} = (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

P _{i = (Y _{i})/(\sum ^{k} _{i =1} Y _{i})}

is independent of

W = \sum ^{k} _{i =1} Y _{i}

if and only if all the Y _i have gamma distributions with the same scale parameter.^[2]

References

• Book: Ng. W. N.. Tian. G-L. Tang. M-L. Dirichlet and Related Distributions. John Wiley & Sons, Ltd.. 2011. 978-0-470-68819-9. page 64. Lukacs's proportion-sum independence theorem and the corollary with a proof.

Notes and References

1. 10.1214/aoms/1177728549. Lukacs, Eugene. A characterization of the gamma distribution. Annals of Mathematical Statistics. 26. 1955. 2. 319–324. free.
2. Mosimann. James E.. On the compound multinomial distribution, the multivariate \beta distribution, and correlation among proportions. Biometrika. 1962. 49. 1 and 2. 65 - 82. 2333468. 10.1093/biomet/49.1-2.65.