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On the exact distribution of the maximum of absolutely continuous dependent random variables

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

Abstract

We derive the exact probability density function of the maximum of arbitrary absolutely continuous dependent random variables and of absolutely continuous exchangeable random variables. We show this density is related to the family of fundamental skew distributions. In particular, we examine the case where the random variables have an elliptically contoured distribution. We study some particular examples based on the multivariate normal and multivariate Student t distributions, and discuss numerical computation issues. We illustrate our results on a genetic selection problem and on an autoregressive time series model of order one.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:stapro:v:78:y:2008:i:1:p:27-35
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    References listed on IDEAS

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    1. 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.
    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. 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.
    4. Loperfido, Nicola, 2002. "Statistical implications of selectively reported inferential results," Statistics & Probability Letters, Elsevier, vol. 56(1), pages 13-22, January.
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    Citations

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    Cited by:

    1. Karavias, Yiannis & Tzavalis, Elias, 2012. "Generalized �Fixed-T Panel Unit Root Tests Allowing for Structural Breaks," MPRA Paper 43128, University Library of Munich, Germany.
    2. 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.
    3. Müller K. & Richter W.-D., 2016. "Extreme value distributions for dependent jointly ln,p-symmetrically distributed random variables," Dependence Modeling, Sciendo, vol. 4(1), pages 1-33, February.
    4. Klaus Müller & Wolf-Dieter Richter, 2015. "Exact extreme value, product, and ratio distributions under non-standard assumptions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 99(1), pages 1-30, January.
    5. Loperfido, Nicola, 2008. "A note on skew-elliptical distributions and linear functions of order statistics," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3184-3186, December.
    6. 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.
    7. 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.
    8. 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.
    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. Ayyub Sheikhi & Ahad Jamalizadeh, 2011. "Regression analysis using order statistics," Statistical Papers, Springer, vol. 52(4), pages 885-892, November.
    12. 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, Sciendo, vol. 4(1), pages 1-29, February.
    13. Reinaldo Arellano-Valle & Luis Castro & Graciela González-Farías & Karla Muñoz-Gajardo, 2012. "Student-t censored regression model: properties and inference," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(4), pages 453-473, November.
    14. Yiannis Karavias & Elias Tzavalis, "undated". "Testing for unit roots in panels with structural changes, spatial and temporal dependence when the time dimension is finite," Discussion Papers 14/03, University of Nottingham, Granger Centre for Time Series Econometrics.
    15. Yiannis Karavias & Elias Tzavalis, "undated". "Generalized fixed-T panel unit root tests allowing for structural breaks," Discussion Papers 12/02, University of Nottingham, Granger Centre for Time Series Econometrics.

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