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

doi:10.1111/1467-9868.00391

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

Journal Article•DOI•

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

Adelchi Azzalini¹, Antonella Capitanio¹•Institutions (1)

01 May 2003-Journal of The Royal Statistical Society Series B-statistical Methodology (Blackwell Publishing)-Vol. 65, Iss: 2, pp 367-389

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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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 or methods from "Distributions generated by perturba..."

...The above formulation is in the form presented by Azzalini & Capitanio (2003) and, in a slightly different way, by Genton & Loperfido (2005); despite the discrepancy in publication dates, these two papers have been developed independently and more or less simultaneously....
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...See Fang (2003) and Azzalini & Capitanio (2003) for a detailed analysis of these aspects....
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...More details and some numerical illustrations are given by Azzalini & Capitanio (2003)....
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...The problem has been tackled by Azzalini & Capitanio (2003) where, although a general coincidence could not be established, it has been shown that this is however valid at least for various important cases, notably the multivariate Pearson types II and VII families; the latter family is of special…...
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...The above statements summarize results of Branco & Dey (2001), Gupta (2003) and Azzalini & Capitanio (2003)....
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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

Book Chapter•DOI•

Automated high-dimensional flow cytometric data analysis

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Saumyadipta Pyne¹, Xinli Hu¹, Kui Wang², Elizabeth J. Rossin¹, Tsung-I Lin³, Lisa M. Maier¹, Clare Baecher-Allan⁴, Geoffrey J. McLachlan², Pablo Tamayo¹, David A. Hafler¹, Philip L. De Jager¹, Jill P. Mesirov¹ - Show less +8 more•Institutions (4)

Broad Institute¹, University of Queensland², National Chung Hsing University³, Brigham and Women's Hospital⁴

25 Apr 2010

TL;DR: This work presents a direct multivariate finite mixture modeling approach, using skew and heavy-tailed distributions, to address the complexities of flow cytometric analysis and to deal with high-dimensional cytometric data without the need for projection or transformation.

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Abstract: Flow cytometry is widely used for single cell interrogation of surface and intracellular protein expression by measuring fluorescence intensity of fluorophore-conjugated reagents We focus on the recently developed procedure of Pyne et al (2009, Proceedings of the National Academy of Sciences USA 106, 8519-8524) for automated high- dimensional flow cytometric analysis called FLAME (FLow analysis with Automated Multivariate Estimation) It introduced novel finite mixture models of heavy-tailed and asymmetric distributions to identify and model cell populations in a flow cytometric sample This approach robustly addresses the complexities of flow data without the need for transformation or projection to lower dimensions It also addresses the critical task of matching cell populations across samples that enables downstream analysis It thus facilitates application of flow cytometry to new biological and clinical problems To facilitate pipelining with standard bioinformatic applications such as high-dimensional visualization, subject classification or outcome prediction, FLAME has been incorporated with the GenePattern package of the Broad Institute Thereby analysis of flow data can be approached similarly as other genomic platforms We also consider some new work that proposes a rigorous and robust solution to the registration problem by a multi-level approach that allows us to model and register cell populations simultaneously across a cohort of high-dimensional flow samples This new approach is called JCM (Joint Clustering and Matching) It enables direct and rigorous comparisons across different time points or phenotypes in a complex biological study as well as for classification of new patient samples in a more clinical setting.

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354 citations

Cites background from "Distributions generated by perturba..."

...The multivariate skew t distribution is defined by introducing skewness in a multivariate elliptically symmetric t distribution (26)...
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Journal Article•DOI•

The Generalized Hyperbolic Skew Student’s t-Distribution

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Kjersti Aas¹, Ingrid Hobæk Haff¹•Institutions (1)

Norwegian Computing Center¹

01 Mar 2006-Journal of Financial Econometrics

TL;DR: In this article, a special case of the generalized hyperbolic (GH) family, called the GH skew Student's t-distribution, has been proposed, which has the important property that one tail has polynomial and the other exponential behavior.

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Abstract: In this article we argue for a special case of the generalized hyperbolic (GH) family that we denote as the GH skew Student’s t-distribution. This distribution has the important property that one tail has polynomial and the other exponential behavior. Further, it is the only subclass of the GH family of distributions having this property. Although the GH skew Student’s t-distribution has been previously proposed in the literature, it is not well known, and specifically, its special tail behavior has not been addressed. This article presents empirical evidence of exponential/polynomial tail behavior in skew financial data, and demonstrates the superiority of the GH skew Student’s t-distribution with respect to data fit compared with some of its competitors. Through VaR and expected shortfall calculations we show why the exponential/polynomial tail behavior is important in practice.

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351 citations

Cites background or methods or result from "Distributions generated by perturba..."

...A third alternative is the skew Student’s t-distribution proposed by Azzalini and Capitanio (2003) (which coincides with the skew t-distribution of Branco and Dey (2001)), having a density on the form...
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...Backtesting shows that the GH skew Student’s t-distribution outperforms the NIG and skew Student’s t-distribution provided by Azzalini and Capitanio (2003) when expected shortfall is used as a risk measure, and is also slightly better at predicting VaR....
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...For Azzalini’s skew Student’s t-distribution, we used the numerical maximum likelihood estimation scheme given in Azzalini and Capitanio (2003) , which has been implemented in the sn-package for R. The CPU time pr. iteration was approximately 0.01 for NIG, approximately 0.02 for GH skew Student’s t and approximately 0.04 for Azzalini’s skew Student’s t, whereas the number of iterations needed until convergence was slightly larger for the GH ......
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...If the data in addition are very skewed, it is also superior to the skew Student’s t-distribution provided by Azzalini and Capitanio (2003) ....
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...For Azzalini’s skew Student’s t-distribution, we used the numerical maximum likelihood estimation scheme given in Azzalini and Capitanio (2003), which has been implemented in the sn-package for R....
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Posted Content•

The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map

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William T. Shaw¹, I. R. C. Buckley•Institutions (1)

King's College London¹

05 Jan 2009-arXiv: Statistical Finance

TL;DR: In this article, a technique for introducing skewness or kurtosis into a symmetric or other distribution is proposed, based on a "transmutation map" which is the functional composition of the cumulative distribution function of one distribution with the inverse cumulative distribution (quantile) function of another.

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Abstract: Motivated by the need for parametric families of rich and yet tractable distributions in financial mathematics, both in pricing and risk management settings, but also considering wider statistical applications, we investigate a novel technique for introducing skewness or kurtosis into a symmetric or other distribution. We use a "transmutation" map, which is the functional composition of the cumulative distribution function of one distribution with the inverse cumulative distribution (quantile) function of another. In contrast to the Gram-Charlier approach, this is done without resorting to an asymptotic expansion, and so avoids the pathologies that are often associated with it. Examples of parametric distributions that we can generate in this way include the skew-uniform, skew-exponential, skew-normal, and skew-kurtotic-normal.

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348 citations

Additional excerpts

...In Section 4 we conclude....
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References

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Journal Article•

A class of distributions which includes the normal ones

[...]

Adelchi Azzalini

01 Jan 1985-Scandinavian Journal of Statistics

TL;DR: In this paper, a nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard.

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Abstract: On introduit une nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard

...read moreread less

2,470 citations

"Distributions generated by perturba..." refers background or methods in this paper

...A reviewer of this paper remarked that, if we set d = 1, density (26) does not reduce to the form 2 t1.y; ν/T1.αy; ν/, which seems to be the ‘most natural’ univariate form of skew t-density generated by lemma 1 of Azzalini (1985), a forerunner of proposition 1....
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...In connection with the SN distribution, Azzalini (1985) and Azzalini and Capitanio (1999) have highlighted some problematic aspects of the likelihood function....
[...]
...In fact, one could reverse the reasoning and claim that lemma 1 of Azzalini (1985) ‘should’ have been stated in the form of proposition 1 for d = 1; in other words, there is no reason to restrict w.y/ to the linear form αy, especially outside the normal case....
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...The multivariate distribution sketched by Azzalini (1985), section 4, and the multiple-constraint model outlined by Arnold and Beaver (2000a), section 6, has a G which is the product of m (m 1) terms of type Φ.αiyi/ or Φ.αTi y + bi/ respectively....
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Book•

Symmetric Multivariate and Related Distributions

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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

"Distributions generated by perturba..." refers background in this paper

...U|U0 = z/ does not depend on Qz if and only if UÅ is Gaussian; see theorem 4.12 of Fang et al. (1990)....
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...Using results in Fang et al. (1990), pages 82–83, we have c1 f̃ Qz .y2/ = Γ....
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...The proof is identical with that of proposition 4, considering the densities of marginal and conditional distributions of PII, as defined in Fang et al. (1990), pages 89–91....
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...12 of Fang et al. (1990). In this case, the integral in (16) becomes , so that , ! ....
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...For a full treatment of this topic, we refer the reader to Fang et al. (1990)....
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Journal Article•DOI•

The multivariate skew-normal distribution

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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.

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1,478 citations

Journal Article•DOI•

Statistical applications of the multivariate skew normal distribution

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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.

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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

"Distributions generated by perturba..." refers background or methods or result in this paper

...Other properties of quadratic forms of SN variables are given by Azzalini & Capitanio (1999) , Genton et al. (2001) and Loperfido (2001)....
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...As a further level of generalisation of the normal distribution, Azzalini & Capitanio (1999, p. 599) have presented a lemma which leads to the construction of a ‘skew elliptical’ density, which is an elliptical density multiplied by a suitable skewing factor, in such a way that the product is still a proper density....
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...while, for the expressions of , we use results given by Azzalini & Capitanio (1999) and by Genton et al. (2001)....
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...The first is to prove that the two forms of skew elliptical densities introduced by Azzalini & Capitanio (1999, p. 599) and by Branco & Dey (2001) are closely connected....
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...Notice that Propositions 8 and 9 of Azzalini & Capitanio (1999) have added conditions on the parameter, but these are not necessary....
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Journal Article•DOI•

On Bayesian modeling of fat tails and skewness

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Carmen Fernandez, Mark F. J. Steel

01 Mar 1998-Journal of the American Statistical Association

TL;DR: In this paper, a Bayesian analysis of linear regression models that can account for skewed error distributions with fat tails is presented. But the authors do not consider whether the tail behavior is affected by skewness.

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Abstract: We consider a Bayesian analysis of linear regression models that can account for skewed error distributions with fat tails The latter two features are often observed characteristics of empirical datasets, and we formally incorporate them in the inferential process A general procedure for introducing skewness into symmetric distributions is first proposed Even though this allows for a great deal of flexibility in distributional shape, tail behavior is not affected Applying this skewness procedure to a Student t distribution, we generate a “skewed Student” distribution, which displays both flexible tails and possible skewness, each entirely controlled by a separate scalar parameter The linear regression model with a skewed Student error term is the main focus of the article We first characterize existence of the posterior distribution and its moments, using standard improper priors and allowing for inference on skewness and tail parameters For posterior inference with this model, we suggest

...read moreread less

829 citations

"Distributions generated by perturba..." refers methods in this paper

...Alternative proposals of univariate skew t-distributions have been made by Fernández and Steel (1998), constructed similarly to the so-called two-piece normal density, and by Jones (2001), developed by Jones and Faddy (2003), which is based on a suitable transformation of a beta density....
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