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On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach

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  • Soltani, Ahmad Reza
  • Roozegar, Rasool

Abstract

In this article we employ certain techniques in divided differences to relate the generalized Stieltjes transform of the distribution of a randomly weighted average of independent random variables X1,…,Xm to the generalized Stieltjes transforms of the distribution functions F1,…,Fm; Xi∼Fi,i=1,…,m. The random weights are assumed to be cuts of [0,1] by m−1 ordered statistics of independent and identically uniformly distributed random variables U1,…,Un on [0,1]; m≤n. Soltani and Homei (2009) treated the case m=n using the Schwartz distribution theory. We identified fairly large classes of randomly weighted average distributions by their generalized Stieltjes transforms; in particular including the uniform, Wigner and certain power semicircle distributions.

Suggested Citation

  • Soltani, Ahmad Reza & Roozegar, Rasool, 2012. "On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 1012-1020.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:5:p:1012-1020
    DOI: 10.1016/j.spl.2012.02.007
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    References listed on IDEAS

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    1. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1683, August.
    2. Morganna Carmem Diniz & Edmundo de Souza e Silva & H. Richard Gail, 2002. "Calculating the Distribution of a Linear Combination of Uniform Order Statistics," INFORMS Journal on Computing, INFORMS, vol. 14(2), pages 124-131, May.
    3. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    4. Karlin, S. & Micchelli, C. A. & Rinott, Y., 1986. "Multivariate splines: A probabilistic perspective," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 69-90, October.
    5. Soltani, A.R. & Homei, H., 2009. "Weighted averages with random proportions that are jointly uniformly distributed over the unit simplex," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1215-1218, May.
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    Cited by:

    1. Soltani, A.R., 2022. "Recursive integral equations for random weights averages: Exponential functions and Cauchy distribution," Statistics & Probability Letters, Elsevier, vol. 190(C).
    2. Hazhir Homei, 2015. "A novel extension of randomly weighted averages," Statistical Papers, Springer, vol. 56(4), pages 933-946, November.
    3. Julian Górny & Erhard Cramer, 2019. "From B-spline representations to gamma representations in hybrid censoring," Statistical Papers, Springer, vol. 60(4), pages 1119-1135, August.
    4. Roozegar, Rasool & zarch, Hamid Reza Taherizadeh, 2021. "On the asymptotic distribution of randomly weighted averages of random vectors," Statistics & Probability Letters, Elsevier, vol. 179(C).
    5. Roozegar, Rasool & Soltani, A.R., 2015. "On the asymptotic behavior of randomly weighted averages," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 269-272.

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