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Properties and Limiting Forms of the Multivariate Extended Skew-Normal and Skew-Student Distributions

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  • Christopher J. Adcock

    (Sheffield University Management School, University of Sheffield, Sheffield S10 1FL, UK
    UCD Michael Smurfit Graduate Business School, University College Dublin, Carysfort Avenue, Blackrock, D04 V1W8 Dublin, Ireland)

Abstract

This paper is concerned with the multivariate extended skew-normal [MESN] and multivariate extended skew-Student [MEST] distributions, that is, distributions in which the location parameters of the underlying truncated distributions are not zero. The extra parameter leads to greater variability in the moments and critical values, thus providing greater flexibility for empirical work. It is reported in this paper that various theoretical properties of the extended distributions, notably the limiting forms as the magnitude of the extension parameter, denoted τ in this paper, increases without limit. In particular, it is shown that as τ → − ∞ , the limiting forms of the MESN and MEST distributions are different. The effect of the difference is exemplified by a study of stockmarket crashes. A second example is a short study of the extent to which the extended skew-normal distribution can be approximated by the skew-Student.

Suggested Citation

  • Christopher J. Adcock, 2022. "Properties and Limiting Forms of the Multivariate Extended Skew-Normal and Skew-Student Distributions," Stats, MDPI, vol. 5(1), pages 1-42, March.
    • Handle: RePEc:gam:jstats:v:5:y:2022:i:1:p:17-311:d:767501
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    References listed on IDEAS

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    1. Saralees Nadarajah & Samuel Kotz, 2008. "Moments of truncated t and F distributions," Portuguese Economic Journal, Springer;Instituto Superior de Economia e Gestao, vol. 7(1), pages 63-73, April.
    2. 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.
    3. Antonella Capitanio, 2020. "On The Canonical Form Of Scale Mixtures Of Skew-Normal Distributions," Statistica, Department of Statistics, University of Bologna, vol. 80(2), pages 145-160.
    4. Barry Arnold & Robert Beaver & Richard Groeneveld & William Meeker, 1993. "The nontruncated marginal of a truncated bivariate normal distribution," Psychometrika, Springer;The Psychometric Society, vol. 58(3), pages 471-488, September.
    5. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    6. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
    7. C. Adcock, 2010. "Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution," Annals of Operations Research, Springer, vol. 176(1), pages 221-234, April.
    8. Travis Loux & Orlando Davy, 2021. "Adjusting Published Estimates for Exploratory Biases Using the Truncated Normal Distribution," The American Statistician, Taylor & Francis Journals, vol. 75(3), pages 294-299, July.
    9. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    10. A. Capitanio & A. Azzalini & E. Stanghellini, 2003. "Graphical models for skew‐normal variates," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 129-144, March.
    11. Adcock, C.J., 2014. "Mean–variance–skewness efficient surfaces, Stein’s lemma and the multivariate extended skew-Student distribution," European Journal of Operational Research, Elsevier, vol. 234(2), pages 392-401.
    12. Aigner, Dennis & Lovell, C. A. Knox & Schmidt, Peter, 1977. "Formulation and estimation of stochastic frontier production function models," Journal of Econometrics, Elsevier, vol. 6(1), pages 21-37, July.
    13. William Horrace, 2015. "Moments of the truncated normal distribution," Journal of Productivity Analysis, Springer, vol. 43(2), pages 133-138, April.
    14. Ogasawara, Haruhiko, 2021. "A non-recursive formula for various moments of the multivariate normal distribution with sectional truncation," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
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