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EM algorithm using overparameterization for the multivariate skew-normal distribution

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  • Abe, Toshihiro
  • Fujisawa, Hironori
  • Kawashima, Takayuki
  • Ley, Christophe

Abstract

A stochastic representation with a latent variable often enables us to make an EM algorithm to obtain the maximum likelihood estimate. The skew-normal distribution has such a simple stochastic representation with a latent variable, and consequently one expects to have a convenient EM algorithm. However, even for the univariate skew-normal distribution, existing EM algorithms constructed using a stochastic representation require a solution of a complicated estimating equation for the skewness parameter, making it difficult to extend such an idea to the multivariate skew-normal distribution. A stochastic representation with overparameterization is proposed, which has not been discussed yet. The approach allows the construction of an efficient EM algorithm in a closed form, which can be extended to a mixture of multivariate skew-normal distributions. The proposed EM algorithm can be regarded as an accelerated version with momentum (which is known as an acceleration technique of the algorithm in optimization) of a recently proposed EM algorithm. The novel EM algorithm is applied to real data and compared with the command msn.mle from the R package sn.

Suggested Citation

  • Abe, Toshihiro & Fujisawa, Hironori & Kawashima, Takayuki & Ley, Christophe, 2021. "EM algorithm using overparameterization for the multivariate skew-normal distribution," Econometrics and Statistics, Elsevier, vol. 19(C), pages 151-168.
    • Handle: RePEc:eee:ecosta:v:19:y:2021:i:c:p:151-168
    DOI: 10.1016/j.ecosta.2021.03.003
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    References listed on IDEAS

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    1. Clécio S. Ferreira & Víctor H. Lachos & Heleno Bolfarine, 2016. "Likelihood-based inference for multivariate skew scale mixtures of normal distributions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 100(4), pages 421-441, October.
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    4. 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.
    5. 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.
    6. Adelchi Azzalini, 2005. "The Skew‐normal Distribution and Related Multivariate Families," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 159-188, June.
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    1. Azzalini, Adelchi, 2022. "An overview on the progeny of the skew-normal family— A personal perspective," Journal of Multivariate Analysis, Elsevier, vol. 188(C).

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