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Addressing non-normality in multivariate analysis using the t-distribution

Author

Listed:
  • Felipe Osorio

    (Universidad Técnica Federico Santa María)

  • Manuel Galea

    (Pontificia Universidad Católica de Chile)

  • Claudio Henríquez

    (Pontificia Universidad Católica de Chile)

  • Reinaldo Arellano-Valle

    (Pontificia Universidad Católica de Chile)

Abstract

The main aim of this paper is to propose a set of tools for assessing non-normality taking into consideration the class of multivariate t-distributions. Assuming second moment existence, we consider a reparameterized version of the usual t distribution, so that the scale matrix coincides with covariance matrix of the distribution. We use the local influence procedure and the Kullback–Leibler divergence measure to propose quantitative methods to evaluate deviations from the normality assumption. In addition, the possible non-normality due to the presence of both skewness and heavy tails is also explored. Our findings based on two real datasets are complemented by a simulation study to evaluate the performance of the proposed methodology on finite samples.

Suggested Citation

  • Felipe Osorio & Manuel Galea & Claudio Henríquez & Reinaldo Arellano-Valle, 2023. "Addressing non-normality in multivariate analysis using the t-distribution," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 107(4), pages 785-813, December.
    • Handle: RePEc:spr:alstar:v:107:y:2023:i:4:d:10.1007_s10182-022-00468-2
    DOI: 10.1007/s10182-022-00468-2
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    References listed on IDEAS

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    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Bodnar, Taras & Gupta, Arjun K. & Parolya, Nestor, 2014. "On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 215-228.
    3. Sutradhar, B. C., 1993. "Score Test for the Covariance Matrix of the Elliptic t-Distribution," Journal of Multivariate Analysis, Elsevier, vol. 46(1), pages 1-12, July.
    4. Fiorentini, Gabriele & Sentana, Enrique & Calzolari, Giorgio, 2003. "Maximum Likelihood Estimation and Inference in Multivariate Conditionally Heteroscedastic Dynamic Regression Models with Student t Innovations," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(4), pages 532-546, October.
    5. Hong‐Tu Zhu & Sik‐Yum Lee, 2001. "Local influence for incomplete data models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(1), pages 111-126.
    6. Manuel Galea & David Cademartori & Roberto Curci & Alonso Molina, 2020. "Robust Inference in the Capital Asset Pricing Model Using the Multivariate t -distribution," JRFM, MDPI, vol. 13(6), pages 1-22, June.
    7. W.‐Y. Poon & Y. S. Poon, 1999. "Conformal normal curvature and assessment of local influence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(1), pages 51-61.
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