IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v78y2008i1p27-35.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(07)00167-8
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Verena Monschang & Mark Trede & Bernd Wilfling, 2023. "Multi-horizon uniform superior predictive ability revisited: A size-exploiting and consistent test," CQE Working Papers 10623, Center for Quantitative Economics (CQE), University of Muenster.
    7. 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.
    8. Yiannis Karavias & Elias Tzavalis, 2014. "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.
    9. Tae Yoon Lee & Paul Gustafson & Mohsen Sadatsafavi, 2023. "Closed-Form Solution of the Unit Normal Loss Integral in 2 Dimensions, with Application in Value-of-Information Analysis," Medical Decision Making, , vol. 43(5), pages 621-626, July.
    10. Karavias, Yiannis & Tzavalis, Elias, 2012. "Generalized �Fixed-T Panel Unit Root Tests Allowing for Structural Breaks," MPRA Paper 43128, University Library of Munich, Germany.
    11. 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.
    12. 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.
    13. 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.
    14. Giovanni Fonseca & Federica Giummolè & Paolo Vidoni, 2021. "A note on simultaneous calibrated prediction intervals for time series," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 317-330, March.
    15. 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.
    16. Ayyub Sheikhi & Ahad Jamalizadeh, 2011. "Regression analysis using order statistics," Statistical Papers, Springer, vol. 52(4), pages 885-892, November.
    17. Kim Christensen & Roel Oomen & Roberto Renò, 2016. "The Drift Burst Hypothesis," CREATES Research Papers 2016-28, Department of Economics and Business Economics, Aarhus University.
    18. 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.
    19. Yiannis Karavias & Elias Tzavalis, 2012. "Generalized fixed-T panel unit root tests allowing for structural breaks," Discussion Papers 12/02, University of Nottingham, Granger Centre for Time Series Econometrics.
    20. Mohsen Sadatsafavi & Tae Yoon Lee & Laure Wynants & Andrew J Vickers & Paul Gustafson, 2023. "Value-of-Information Analysis for External Validation of Risk Prediction Models," Medical Decision Making, , vol. 43(5), pages 564-575, July.
    21. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. 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.
    3. 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.
    4. Barry C. Arnold & Robert J. Beaver, 2007. "Skewing Around: Relationships Among Classes of Skewed Distributions," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 153-162, June.
    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. 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. 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.
    8. 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.
    9. 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.
    10. 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.
    11. Roohollah Roozegar & Ahad Jamalizadeh & Mehdi Amiri & Tsung-I Lin, 2018. "On the exact distribution of order statistics arising from a doubly truncated bivariate elliptical distribution," METRON, Springer;Sapienza Università di Roma, vol. 76(1), pages 99-114, April.
    12. 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.
    13. Reinaldo B. Arellano-Valle & Marc G. Genton, 2010. "Multivariate extended skew-t distributions and related families," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 201-234.
    14. Reinaldo B. Arellano-Valle & Adelchi Azzalini, 2022. "Some properties of the unified skew-normal distribution," Statistical Papers, Springer, vol. 63(2), pages 461-487, April.
    15. Arellano-Valle, Reinaldo B. & Genton, Marc G. & Loschi, Rosangela H., 2009. "Shape mixtures of multivariate skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 91-101, January.
    16. Sharon Lee & Geoffrey McLachlan, 2013. "On mixtures of skew normal and skew $$t$$ -distributions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 7(3), pages 241-266, September.
    17. Mohsen Maleki & Darren Wraith & Reinaldo B. Arellano-Valle, 2019. "A flexible class of parametric distributions for Bayesian linear mixed models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 543-564, June.
    18. Song, Pingfan & Tan, Changchun & Wang, Shaochen, 2019. "On the moment generating function for random vectors via inverse survival function," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 345-350.
    19. Kheradmandi, Ameneh & Rasekh, Abdolrahman, 2015. "Estimation in skew-normal linear mixed measurement error models," Journal of Multivariate Analysis, Elsevier, vol. 136(C), pages 1-11.
    20. Kim, Hyoung-Moon & Genton, Marc G., 2011. "Characteristic functions of scale mixtures of multivariate skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 102(7), pages 1105-1117, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:78:y:2008:i:1:p:27-35. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.