Lukacs's proportion-sum independence theorem
{{Short description|Theorem about independent random variables}}
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.{{Cite journal|doi= 10.1214/aoms/1177728549|author= Lukacs, Eugene|title=A characterization of the gamma distribution|journal=Annals of Mathematical Statistics|volume=26|year=1955|issue= 2|pages=319–324|doi-access=free}}
The theorem
If Y1 and Y2 are non-degenerate, independent random variables, then the random variables
:
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
:
if and only if all the Y i have gamma distributions with the same scale parameter.{{cite journal| last=Mosimann| first=James E.| title=On the compound multinomial distribution, the multivariate distribution, and correlation among proportions| journal=Biometrika| year=1962| volume=49| issue=1 and 2|pages=65–82|jstor=2333468| doi=10.1093/biomet/49.1-2.65}}
References
{{reflist}}
- {{cite book|last1=Ng| first1=W. N.|last2=Tian| first2=G-L| last3=Tang| first3=M-L| title=Dirichlet and Related Distributions| publisher=John Wiley & Sons, Ltd.| year=2011|isbn=978-0-470-68819-9}} page 64. [https://books.google.com/books?id=k8GS868oyo4C&pg=PT81 Lukacs's proportion-sum independence theorem and the corollary] with a proof.