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Sparsity-regularized skewness estimation for the multivariate skew normal and multivariate skew t distributions

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  • Wang, Sheng
  • Zimmerman, Dale L.
  • Breheny, Patrick

Abstract

The multivariate skew normal (MSN) and multivariate skew t (MST) distributions have received considerable attention in the past two decades because of their appealing mathematical properties and their usefulness for modeling skewed data. We develop sparse regularization methodology for estimating the skewness parameters of these two distributions. This methodology facilitates skewness selection, i.e., the identification of those marginal indices of skewness, if any, that are equal to zero. Obstacles that render skewness selection infeasible for existing parameterizations of the two distributions are described, and a new parameterization that permits the circumvention of those obstacles is introduced. A penalized likelihood method for sparsity-based regularized skewness estimation using the new parameterization is proposed. Model selection consistency and the oracle property of the method are established. A simulation study demonstrates that the method is reasonably effective for skewness selection and, because of inclusion of a ridge penalty, is more effective at preventing the divergence of the shape parameter in small samples than the Q-penalty approach of Azzalini and Arellano-Valle (2013). The simulation study demonstrates further that our method may improve the estimation of all parameters of the MSN and MST distributions, not merely the skewnesses, when some of the skewnesses are zero.

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  • Wang, Sheng & Zimmerman, Dale L. & Breheny, Patrick, 2020. "Sparsity-regularized skewness estimation for the multivariate skew normal and multivariate skew t distributions," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    • Handle: RePEc:eee:jmvana:v:179:y:2020:i:c:s0047259x20302207
    DOI: 10.1016/j.jmva.2020.104639
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    References listed on IDEAS

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    1. Nash, John C., 2014. "On Best Practice Optimization Methods in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 60(i02).
    2. Christophe Ley & Davy Paindaveine, 2010. "On Fisher information matrices and profile log-likelihood functions in generalized skew-elliptical models," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 235-250.
    3. Hamparsum Bozdogan, 1987. "Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 345-370, September.
    4. 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.
    5. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    6. Branco, Márcia D. & Dey, Dipak K., 2001. "A General Class of Multivariate Skew-Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 99-113, October.
    7. 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.
    8. Arellano-Valle, Reinaldo B. & Azzalini, Adelchi, 2013. "The centred parameterization and related quantities of the skew-t distribution," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 73-90.
    9. Zhang, Yongli & Yang, Yuhong, 2015. "Cross-validation for selecting a model selection procedure," Journal of Econometrics, Elsevier, vol. 187(1), pages 95-112.
    10. Villa, Cristiano & Rubio, Francisco J., 2018. "Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula," Computational Statistics & Data Analysis, Elsevier, vol. 124(C), pages 197-219.
    11. Arellano-Valle, Reinaldo B. & Azzalini, Adelchi, 2008. "The centred parametrization for the multivariate skew-normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 99(7), pages 1362-1382, August.
    12. Reinaldo B. Arellano-Valle, 2010. "On the information matrix of the multivariate skew-t model," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 371-386.
    13. Shu-Ching Chang & Dale L. Zimmerman, 2016. "Skew-normal antedependence models for skewed longitudinal data," Biometrika, Biometrika Trust, vol. 103(2), pages 363-376.
    14. Hansheng Wang & Runze Li & Chih-Ling Tsai, 2007. "Tuning parameter selectors for the smoothly clipped absolute deviation method," Biometrika, Biometrika Trust, vol. 94(3), pages 553-568.
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