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.