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

$W=Y_1+Y_2\text{ and }P = \frac{Y_1}{Y_1+Y_2}$

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

$P_i=\frac{Y_i}{\sum_{i=1}^k Y_i}$

is independent of

$W=\sum_{i=1}^k Y_i$

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.