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Keywords = multivariate extended skew-Student distribution

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30 pages, 4253 KB  
Article
The Linear Skew-t Distribution and Its Properties
by C. J. Adcock
Stats 2023, 6(1), 381-410; https://doi.org/10.3390/stats6010024 - 23 Feb 2023
Cited by 1 | Viewed by 4353
Abstract
The aim of this expository paper is to present the properties of the linear skew-t distribution, which is a specific example of a symmetry modulated-distribution. The skewing function remains the distribution function of Student’s t, but its argument is simpler than that used [...] Read more.
The aim of this expository paper is to present the properties of the linear skew-t distribution, which is a specific example of a symmetry modulated-distribution. The skewing function remains the distribution function of Student’s t, but its argument is simpler than that used for the standard skew-t. The linear skew-t offers different insights, for example, different moments and tail behavior, and can be simpler to use for empirical work. It is shown that the distribution may be expressed as a hidden truncation model. The paper describes an extended version of the distribution that is analogous to the extended skew-t. For certain parameter values, the distribution is bimodal. The paper presents expressions for the moments of the distribution and shows that numerical integration methods are required. A multivariate version of the distribution is described. The bivariate version of the distribution may also be bimodal. The distribution is not closed under marginalization, and stochastic ordering is not satisfied. The properties of the distribution are illustrated with numerous examples of the density functions, table of moments and critical values. The results in this paper suggest that the linear skew-t may be useful for some applications, but that it should be used with care for methodological work. Full article
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42 pages, 6219 KB  
Article
Properties and Limiting Forms of the Multivariate Extended Skew-Normal and Skew-Student Distributions
by Christopher J. Adcock
Stats 2022, 5(1), 270-311; https://doi.org/10.3390/stats5010017 - 9 Mar 2022
Cited by 3 | Viewed by 3060
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Multivariate Statistics and Applications)
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13 pages, 712 KB  
Article
An Information-Theoretic Approach for Multivariate Skew-t Distributions and Applications
by Salah H. Abid, Uday J. Quaez and Javier E. Contreras-Reyes
Mathematics 2021, 9(2), 146; https://doi.org/10.3390/math9020146 - 11 Jan 2021
Cited by 23 | Viewed by 4040
Abstract
Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew- [...] Read more.
Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew-t (FMST) distributions. This class of flexible distributions allows handling asymmetry and tail weight behavior simultaneously. We find upper and lower bounds of Rényi entropy for these families. Numerical simulations illustrate the results for several scenarios: symmetry/asymmetry and light/heavy-tails. Finally, we present applications of our findings to a swordfish length-weight dataset to illustrate the behavior of entropies of the FMST distribution. Comparisons with the counterparts—the finite mixture of multivariate skew-normal and normal distributions—are also presented. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications)
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