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The distribution of the product of powers of independent uniform random variables — A simple but useful tool to address and better understand the structure of some distributions

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  • Arnold, Barry C.
  • Coelho, Carlos A.
  • Marques, Filipe J.

Abstract

What is the distribution of the product of given powers of independent uniform (0, 1) random variables? Is this distribution useful? Is this distribution commonly used in some contexts? Is this distribution somehow related to the distribution of the product of other random variables? Are there some test statistics with this distribution? This paper will give the answers to the above questions. It will be seen that the answer to the last four questions above is: yes! We will show how particular choices of the numbers of variables involved and their powers will result in interesting and useful distributions and how these distributions may help us to shed some new light on some well-known distributions and also how it may help us to address, in a much simpler way, some distributions usually considered to be rather complicated as is the case with the exact distribution of a number of statistics used in Multivariate Analysis, including some whose exact distribution up until now is not available in a concise and manageable form.

Suggested Citation

  • Arnold, Barry C. & Coelho, Carlos A. & Marques, Filipe J., 2013. "The distribution of the product of powers of independent uniform random variables — A simple but useful tool to address and better understand the structure of some distributions," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 19-36.
    • Handle: RePEc:eee:jmvana:v:113:y:2013:i:c:p:19-36
    DOI: 10.1016/j.jmva.2011.04.006
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    References listed on IDEAS

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    1. Coelho, Carlos A., 1998. "The Generalized Integer Gamma Distribution--A Basis for Distributions in Multivariate Statistics," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 86-102, January.
    2. Nagar, Daya K. & Chen, Jie & Gupta, Arjun K., 2004. "Distribution and percentage points of the likelihood ratio statistic for testing circular symmetry," Computational Statistics & Data Analysis, Elsevier, vol. 47(1), pages 79-89, August.
    3. Fang, C. & Krishnaiah, P. R. & Nagarsenker, B. N., 1982. "Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 12(4), pages 597-611, December.
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    Cited by:

    1. Filipe Marques & Carlos Coelho, 2013. "Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions," Computational Statistics, Springer, vol. 28(5), pages 2091-2115, October.
    2. Carlos A. Coelho & Anuradha Roy, 2013. "Testing of hypothesis of a block compound symmetric covariance matrix," Working Papers 0179mss, College of Business, University of Texas at San Antonio.
    3. Marcel Nutz & Jaime San Martin & Xiaowei Tan, 2018. "Convergence to the Mean Field Game Limit: A Case Study," Papers 1806.00817, arXiv.org, revised May 2019.
    4. Carlos A. Coelho & Anuradha Roy, 2017. "Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(2), pages 308-330, June.

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