Lukacs's proportion-sum independence theorem
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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
- [math]\displaystyle{ W=Y_1+Y_2\text{ and }P = \frac{Y_1}{Y_1+Y_2} }[/math]
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
- [math]\displaystyle{ P_i=\frac{Y_i}{\sum_{i=1}^k Y_i} }[/math]
is independent of
- [math]\displaystyle{ W=\sum_{i=1}^k Y_i }[/math]
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 (2): 319–324. doi:10.1214/aoms/1177728549.
- ↑ Mosimann, James E. (1962). "On the compound multinomial distribution, the multivariate [math]\displaystyle{ \beta }[/math] distribution, and correlation among proportions". Biometrika 49 (1 and 2): 65–82. doi:10.1093/biomet/49.1-2.65.
- 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.
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