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.
Contents
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
is independent of
if and only if all the Y i have gamma distributions with the same scale parameter.
References
Lukacs's proportion-sum independence theorem Wikipedia(Text) CC BY-SA