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 for 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

1. Lukacs, Eugene (1955). "A characterization of the gamma distribution". Annals of Mathematical Statistics. 26: 319–324. doi:10.1214/aoms/1177728549.
2. Mosimann, James E. (1962). "On the compound multinomial distribution, the multivariate $\beta$ 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.