A General Class of Multivariate Skew-Elliptical Distributions

doi:10.1006/JMVA.2000.1960

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A General Class of Multivariate Skew-Elliptical Distributions

Journal Article•DOI•

A General Class of Multivariate Skew-Elliptical Distributions

Márcia D. Branco¹, Dipak K. Dey²•Institutions (2)

University of São Paulo¹, University of Connecticut²

01 Oct 2001-Journal of Multivariate Analysis (Academic Press, Inc.)-Vol. 79, Iss: 1, pp 99-113

TL;DR: In this article, a general class of multivariate skew-elliptical distributions is proposed, which contains the multivariate normal, Student's t, exponential power, and Pearson type II, with an extra parameter to regulate skewness.

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About: This article is published in Journal of Multivariate Analysis.The article was published on 2001-10-01 and is currently open access. It has received 629 citations till now. The article focuses on the topics: Elliptical distribution & Multivariate stable distribution.

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Journal Article•DOI•

Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution

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Adelchi Azzalini¹, Antonella Capitanio¹•Institutions (1)

University of Padua¹

01 May 2003-Journal of The Royal Statistical Society Series B-statistical Methodology

TL;DR: In this paper, a fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities.

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Abstract: Summary. A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew t-density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

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1,215 citations

Cites background or methods or result from "A General Class of Multivariate Ske..."

...These expressions are in agreement with those given by Branco and Dey (2001), but higher moments were not given there....
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...We shall now examine in detail two important cases of this form, namely when U* has either a PVIId+1 or a PIId+1 distribution, which are among those considered by Branco and Dey (2001). Proposition 4....
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...For the rest of this section,UÅ denotes a .d+1/-dimensional variate partitioned into a scalar component U0 and a d-dimensional vector U. Branco and Dey (2001) introduced a class of skew elliptical distributions generated by applying to a .d + 1/-dimensional elliptical variate the same conditioning…...
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...Moreover, for the reasons explained in Section 3.1, distribution (26) coincides in turn with the skew t-distribution of Branco and Dey (2001), although this equality is not visible from their derivation because they did not provide the above closed form expression of the density....
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...The first is to prove that the two forms of skew elliptical densities that were introduced by Azzalini and Capitanio (1999), page 599, and by Branco and Dey (2001) are closely connected....
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Journal Article•DOI•

Distributions generated by perturbation of symmetry with emphasis on a multivariate skew $t$ distribution

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Adelchi Azzalini¹, Antonella Capitanio¹•Institutions (1)

University of Padua¹

12 Nov 2009-arXiv: Methodology

TL;DR: In this paper, a general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities.

...read moreread less

Abstract: A fairly general procedure is studied to perturbate a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is general enough to encompass a number of recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew $t$ density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

...read moreread less

1,174 citations

Journal Article•DOI•

The Skew-normal Distribution and Related Multivariate Families.

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Adelchi Azzalini¹•Institutions (1)

University of Padua¹

01 Jun 2005-Scandinavian Journal of Statistics

TL;DR: In this paper, the authors provide an introductory overview of a portion of distribution theory which is currently under intense development and illustrate connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more.

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Abstract: . This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so-called skew-normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew-elliptical families, and some forms of semi-parametric formulations, extending the relevance of the field much beyond the original theme of ‘skewness’. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more.

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657 citations

Cites background from "A General Class of Multivariate Ske..."

...Proceeding along a different route, Branco & Dey (2001) start from a (d þ 1)-dimensional variable U ¼ ðU0; UT1 Þ T with location and scale parameters as in (21) but with assumption of normality replaced the one of elliptical form Ellð0; X ; ~f ðdþ1ÞÞ....
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...The above statements summarize results of Branco & Dey (2001), Gupta (2003) and Azzalini & Capitanio (2003)....
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...From the general expression (32), Branco & Dey (2001) formulate skew versions of various multivariate families: logistic, stable, Subbotin (or exponential power), t, Pearson type II, and more....
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...Since the Branco & Dey (2001) construction is not set in the form (12) with f0 of elliptical type as it had been considered by Azzalini & Capitanio (1999), the natural question arises of how the two approaches are related....
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Journal Article•DOI•

A new class of multivariate skew distributions with applications to Bayesian regression models

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Sujit K. Sahu¹, Dipak K. Dey², Márcia D. Branco•Institutions (2)

University of Southampton¹, University of Connecticut²

01 Jun 2003-Canadian Journal of Statistics-revue Canadienne De Statistique

TL;DR: In this paper, a new class of distributions by introducing skewness in multivariate ellip-tically symmetric distributions was developed, which is obtained by using transformation and conditioning.

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Abstract: The authors develop a new class of distributions by introducing skewness in multivariate ellip- tically symmetric distributions. The class, which is obtained by using transformation and conditioning, contains many standard families including the multivariate skew-normal and distributions. The authors obtain analytical forms of the densities and study distributional properties. They give practical applica- tions in Bayesian regression models and results on the existence of the posterior distributions and moments under improper priors for the regression coefficients. They illustrate their methods using practical examples.

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616 citations

Additional excerpts

...( b # , 0 + Y This density matches with the one obtained by Branco & Dey (2001) only in the univariate case....
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Book•

The Skew-Normal and Related Families

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Adelchi Azzalini¹, Antonella Capitanio²•Institutions (2)

University of Padua¹, University of Bologna²

01 Dec 2013

TL;DR: This comprehensive treatment, blending theory and practice, will be the standard resource for statisticians and applied researchers, and Assuming only basic knowledge of (non-measure-theoretic) probability and statistical inference, the book is accessible to the wide range of researchers who use statistical modelling techniques.

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Abstract: Preface 1. Modulation of symmetric densities 2. The skew-normal distribution: probability 3. The skew-normal distribution: statistics 4. Heavy and adaptive tails 5. The multivariate skew-normal distribution 6. Skew-elliptical distributions 7. Further extensions and other directions 8. Application-oriented work Appendices References.

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547 citations

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References

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Open Access

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Monograph•DOI•

Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance

[...]

Gennady Samorodnitsky, Murad S. Taqqu

01 Jun 1995-Journal of the American Statistical Association

TL;DR: In this paper, the authors introduce sample path properties such as boundedness, continuity, and oscillations, as well as integrability, and absolute continuity of the path in the real line.

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Abstract: Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable stochastic integrals and harmonizable processes Self-similar processes Chentsov random fields Introduction to sample path properties Boundedness, continuity and oscillations Measurability, integrability and absolute continuity Boundedness and continuity via metric entropy Integral representation Historical notes and extensions.

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2,611 citations

Book•

Symmetric Multivariate and Related Distributions

[...]

Kai-Tai Fang¹, Samuel Kotz, Kai Wang Ng•Institutions (1)

University of Wisconsin-Madison¹

01 Nov 1989

TL;DR: In this article, the authors define marginal distributions, moments and density marginal distributions moments density the relationship between (phi and f) conditional distributions properties of elliptically symmetric distributions mixtures of normal distributions robust statistics and regression model robust statistics regression model log-elliptical and additive logistic elliptical distributions multivariate log elliptical distribution additive logistics elliptical distribution complex elliptical symmetric distribution.

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Abstract: Part 1 Preliminaries: construction of symmetric multivariate distributions notation of algebraic entities and characteristics of random quantities the "d" operator groups and invariance dirichlet distribution problems 1. Part 2 Spherically and elliptically symmetric distributions: introduction and definition marginal distributions, moments and density marginal distributions moments density the relationship between (phi) and f conditional distributions properties of elliptically symmetric distributions mixtures of normal distributions robust statistics and regression model robust statistics regression model log-elliptical and additive logistic elliptical distributions multivariate log-elliptical distribution additive logistic elliptical distributions complex elliptically symmetric distributions. Part 3 Some subclasses of elliptical distributions: multiuniform distribution the characteristic function moments marginal distribution conditional distributions uniform distribution in the unit sphere discussion symmetric Kotz type distributions definition distribution of R(2) moments multivariate normal distributions the c.f. of Kotz type distributions symmetric multivariate Pearson type VII distributions definition marginal densities conditional distributions moments conditional distributions moments some examples extended Tn family relationships between Ln and Tn families of distributions order statistics mixtures of exponential distributions independence, robustness and characterizations problems V. Part 6 Multivariate Liouville distributions: definitions and properties examples marginal distributions conditional distribution characterizations scale-invariant statistics survival functions inequalities and applications.

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2,106 citations

"A General Class of Multivariate Ske..." refers background in this paper

...A comprehensive review of the properties and characterizations of elliptical distributions can be found in Fang et al. (1990)....
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...Skewed Pearson Type II Distribution The usual density function of the Pearson type II distribution (see Fang et al., 1990) has the following generator function g(k+1)(u)= 1[m+1+(k+1) 2] 1[m+1] ?...
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Journal Article•DOI•

The multivariate skew-normal distribution

[...]

Adelchi Azzalini, A. Dalla Valle

01 Dec 1996-Biometrika

TL;DR: In this article, a multivariate parametric family such that the marginal densities are scalar skew-normal is introduced, and its properties are studied with special emphasis on the bivariate case.

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Abstract: SUMMARY The paper extends earlier work on the so-called skew-normal distribution, a family of distributions including the normal, but with an extra parameter to regulate skewness. The present work introduces a multivariate parametric family such that the marginal densities are scalar skew-normal, and studies its properties, with special emphasis on the bivariate case.

...read moreread less

1,478 citations

Journal Article•DOI•

Statistical applications of the multivariate skew normal distribution

[...]

Adelchi Azzalini¹, Antonella Capitanio²•Institutions (2)

University of Padua¹, University of Bologna²

01 Jan 1999-Journal of The Royal Statistical Society Series B-statistical Methodology

TL;DR: Azzalini and Dalla Valle as discussed by the authors have discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter, and a further extension is described which introduces a skewing factor of an elliptical density.

...read moreread less

Abstract: Azzalini and Dalla Valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.

...read moreread less

1,130 citations

"A General Class of Multivariate Ske..." refers background in this paper

...It is to be noted that Azzalini and Capitanio (1999) also obtained a class of multivariate skew-elliptical distribution of the form similar to (2.1)....
[...]
...Interesting special cases are discussed in Kelker (1970), Fang and Zhang (1990), Fang, Kotz and Ng (1990), Azzalini and DallaValle (1996), and Azzalini and Capitanio (1999)....
[...]

Journal Article•DOI•

Statistical applications of the multivariate skew-normal distribution

[...]

Adelchi Azzalini¹, Antonella Capitanio²•Institutions (2)

University of Padua¹, University of Bologna²

11 Nov 2009-arXiv: Methodology

TL;DR: Azzalini and Dalla Valle as mentioned in this paper have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter.

...read moreread less

Abstract: Azzalini & Dalla Valle (1996) have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.

...read moreread less

1,046 citations