World Library  
Flag as Inappropriate
Email this Article

Lukacs's proportion-sum independence theorem

Article Id: WHEBN0024187952
Reproduction Date:

Title: Lukacs's proportion-sum independence theorem  
Author: World Heritage Encyclopedia
Language: English
Subject: Generalized Dirichlet distribution, List of statistics articles
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

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 ii = 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. ^
  2. ^
  • page 64. Lukacs's proportion-sum independence theorem and the corollary with a proof.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.