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On the exact distribution of linear combinations of order statistics from dependent random variables

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  • Arellano-Valle, Reinaldo B.
  • Genton, Marc G.

Abstract

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.

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  • Arellano-Valle, Reinaldo B. & Genton, Marc G., 2007. "On the exact distribution of linear combinations of order statistics from dependent random variables," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 1876-1894, November.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:10:p:1876-1894
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    1. Loperfido, Nicola, 2002. "Statistical implications of selectively reported inferential results," Statistics & Probability Letters, Elsevier, vol. 56(1), pages 13-22, January.
    2. Arellano-Valle, Reinaldo B. & Genton, Marc G., 2005. "On fundamental skew distributions," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 93-116, September.
    3. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    4. Corrado Crocetta & Nicola Loperfido, 2005. "The exact sampling distribution of L-statistics," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 213-223.
    5. Chunsheng Ma & John Robinson, 1999. "Saddlepoint Approximations for the Difference of Order Statistics and Studentized Sample Quantiles," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 563-577.
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    1. Jamalizadeh, A. & Balakrishnan, N., 2010. "Distributions of order statistics and linear combinations of order statistics from an elliptical distribution as mixtures of unified skew-elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1412-1427, July.
    2. Müller K. & Richter W.-D., 2016. "Extreme value distributions for dependent jointly ln,p-symmetrically distributed random variables," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-33, February.
    3. Müller K. & Richter W.-D., 2017. "Exact distributions of order statistics from ln,p-symmetric sample distributions," Dependence Modeling, De Gruyter, vol. 5(1), pages 221-245, August.
    4. Jamalizadeh, A. & Balakrishnan, N. & Salehi, Mehdi, 2010. "Order statistics and linear combination of order statistics arising from a bivariate selection normal distribution," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 445-451, March.
    5. Monica Billio & Ludovic Calès & Dominique Guegan, 2010. "A Cross-Sectional Performance Measure for Portfolio Management," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00523466, HAL.
    6. Lui, Kung-Jong & Chang, Kuang-Chao, 2009. "Corrigendum to: "Testing homogeneity of risk difference in stratified randomized trials with noncompliance" [Comput. Statist. Data Anal. 53 (2008) 209-221]," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1529-1529, February.
    7. Madadi, Mohsen & Khalilpoor, Parisa & Jamalizadeh, Ahad, 2015. "Regression mean residual life of a system with three dependent components with normal lifetimes," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 182-191.
    8. Monica Billio & Ludovic Calès & Dominique Guegan, 2010. "A Performance Measure of Zero-Dollar Long/Short Equally Weighted Portfolios," Post-Print halshs-00476038, HAL.
    9. Jamalizadeh, A. & Mehrali, Y. & Balakrishnan, N., 2009. "Recurrence relations for bivariate t and extended skew-t distributions and an application to order statistics from bivariate t," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 4018-4027, October.
    10. Jamalizadeh, A. & Balakrishnan, N., 2009. "Prediction in a trivariate normal distribution via a linear combination of order statistics," Statistics & Probability Letters, Elsevier, vol. 79(21), pages 2289-2296, November.
    11. Arellano-Valle, Reinaldo B. & Genton, Marc G., 2008. "On the exact distribution of the maximum of absolutely continuous dependent random variables," Statistics & Probability Letters, Elsevier, vol. 78(1), pages 27-35, January.
    12. Mehdi Amiri & Yaser Mehrali & Narayanaswamy Balakrishnan & Ahad Jamalizadeh, 2022. "Efficient recursive computational algorithms for multivariate t and multivariate unified skew-t distributions with applications to inference," Computational Statistics, Springer, vol. 37(1), pages 125-158, March.
    13. Müller K. & Richter W.-D., 2016. "Exact distributions of order statistics of dependent random variables from ln,p-symmetric sample distributions, n ∈ {3,4}," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-29, February.
    14. Ayyub Sheikhi & Yaser Mehrali & Mahbanoo Tata, 2013. "On the exact joint distribution of a linear combination of order statistics and their concomitants in an exchangeable multivariate normal distribution," Statistical Papers, Springer, vol. 54(2), pages 325-332, May.
    15. Ali Genç, 2012. "Distribution of linear functions from ordered bivariate log-normal distribution," Statistical Papers, Springer, vol. 53(4), pages 865-874, November.
    16. Napat Rujeerapaiboon & Daniel Kuhn & Wolfram Wiesemann, 2018. "Chebyshev Inequalities for Products of Random Variables," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 887-918, August.
    17. R. Arellano-Valle & Ahad Jamalizadeh & H. Mahmoodian & N. Balakrishnan, 2014. "$$L$$ L -statistics from multivariate unified skew-elliptical distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(4), pages 559-583, May.

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