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Symmetry
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Recent Advances on Symmetry in Mathematical Statistics
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Recent Advances on Symmetry in Mathematical Statistics

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (29 February 2020) | Viewed by 22999

Special Issue Editor

Prof. Dr. Christophe Ley

E-Mail Website
Guest Editor
Department of Applied Mathematics, Computer Science and Statistics, Ghent University, 9000 Gent, Belgium
Interests: mathematical statistics; machine learning; data science; applied probability and interdisciplinary research

Special Issue Information

Dear colleagues,

In this Special Issue of Symmetry we want to present research articles and review articles on the fundamental concept of Symmetry in Mathematical Statistics.

Symmetry is one of the most important structural assumptions and simplifying hypotheses in statistics, playing a major role, for instance, in the identifiability of location or intercept under nonparametric conditions. It is therefore no wonder that there exists a plethora of tests for symmetry and, in case of rejection, various skew alternatives to model the data.

We are therefore especially interested in contributions regarding tests for symmetry on the real line, in higher-dimensional spaces or on non-linear manifolds (e.g., the sphere), as well as in the construction of flexible skew alternatives to the hypothesis of symmetry, in cases where the latter fails to hold. Comparisons between existing tests resp. existing skew distributions are especially welcome, as well as new results/findings on existing tests and distributions.

Prof. Christophe Ley
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Directional statistics
  • Elliptical Symmetry
  • Flexible modelling
  • High-dimensional Data
  • Hypothesis testing
  • Rotational Symmetry
  • Skew distributions
  • Tests for symmetry

Published Papers (6 papers)

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Research

14 pages, 323 KiB  
Article
Bayesian Inference for Skew-Symmetric Distributions
by Fatemeh Ghaderinezhad, Christophe Ley and Nicola Loperfido
Symmetry 2020, 12(4), 491; https://doi.org/10.3390/sym12040491 - 25 Mar 2020
Cited by 5 | Viewed by 3745
Abstract
Skew-symmetric distributions are a popular family of flexible distributions that conveniently model non-normal features such as skewness, kurtosis and multimodality. Unfortunately, their frequentist inference poses several difficulties, which may be adequately addressed by means of a Bayesian approach. This paper reviews the main [...] Read more.
Skew-symmetric distributions are a popular family of flexible distributions that conveniently model non-normal features such as skewness, kurtosis and multimodality. Unfortunately, their frequentist inference poses several difficulties, which may be adequately addressed by means of a Bayesian approach. This paper reviews the main prior distributions proposed for the parameters of skew-symmetric distributions, with special emphasis on the skew-normal and the skew-t distributions which are the most prominent skew-symmetric models. The paper focuses on the univariate case in the absence of covariates, but more general models are also discussed. Full article
(This article belongs to the Special Issue Recent Advances on Symmetry in Mathematical Statistics)
37 pages, 848 KiB  
Article
A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications
by Chris Adcock and Adelchi Azzalini
Symmetry 2020, 12(1), 118; https://doi.org/10.3390/sym12010118 - 7 Jan 2020
Cited by 30 | Viewed by 5170
Abstract
Within the context of flexible parametric families of distributions, much work has been dedicated in recent years to the theme of skew-symmetric distributions, or symmetry-modulated distributions, as we prefer to call them. The present contribution constitutes a review of this area, with special [...] Read more.
Within the context of flexible parametric families of distributions, much work has been dedicated in recent years to the theme of skew-symmetric distributions, or symmetry-modulated distributions, as we prefer to call them. The present contribution constitutes a review of this area, with special emphasis on multivariate skew-elliptical families, which represent the subset with more immediate impact on applications. After providing background information of the distribution theory aspects, we focus on the aspects more relevant for applied work. The exposition is targeted to non-specialists in this domain, although some general knowledge of probability and multivariate statistics is assumed. Given this aim, the mathematical profile is kept to the minimum required. Full article
(This article belongs to the Special Issue Recent Advances on Symmetry in Mathematical Statistics)
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8 pages, 259 KiB  
Article
A Note on Distributions in the Second Chaos
by Pauliina Ilmonen and Lauri Viitasaari
Symmetry 2019, 11(12), 1487; https://doi.org/10.3390/sym11121487 - 6 Dec 2019
Viewed by 1778
Abstract
In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X = ∑ k ≥ 1 λ k ( ξ k 2 − 1 ) , where [...] Read more.
In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X = ∑ k ≥ 1 λ k ( ξ k 2 − 1 ) , where ξ k ∼ N ( 0 , 1 ) are independent. We compute the Lévy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever λ k ≠ 0 infinitely often. Our work generalises some known results presented in the literature. Full article
(This article belongs to the Special Issue Recent Advances on Symmetry in Mathematical Statistics)
22 pages, 641 KiB  
Article
Comparison and Classification of Flexible Distributions for Multivariate Skew and Heavy-Tailed Data
by Slađana Babić, Christophe Ley and David Veredas
Symmetry 2019, 11(10), 1216; https://doi.org/10.3390/sym11101216 - 30 Sep 2019
Cited by 10 | Viewed by 3197
Abstract
We present, compare and classify popular families of flexible multivariate distributions. Our classification is based on the type of symmetry (spherical, elliptical, central symmetry or asymmetry) and the tail behaviour (a single tail weight parameter or multiple tail weight parameters). We compare the [...] Read more.
We present, compare and classify popular families of flexible multivariate distributions. Our classification is based on the type of symmetry (spherical, elliptical, central symmetry or asymmetry) and the tail behaviour (a single tail weight parameter or multiple tail weight parameters). We compare the families both theoretically (relevant properties and distinctive features) and with a Monte Carlo study (comparing the fitting abilities in finite samples). Full article
(This article belongs to the Special Issue Recent Advances on Symmetry in Mathematical Statistics)
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21 pages, 4475 KiB  
Article
From Symmetry to Asymmetry on the Disc Manifold: Modeling of Marion Island Data
by Andriette Bekker, Priyanka Nagar, Mohammad Arashi and Hannes Rautenbach
Symmetry 2019, 11(8), 1030; https://doi.org/10.3390/sym11081030 - 9 Aug 2019
Cited by 1 | Viewed by 3197
Abstract
The joint modeling of angular and linear observations is crucial as data of this nature are prevalent in multiple disciplines, for example the joint modeling of wind direction and another climatological variable such as wind speed or air temperature, the direction an animal [...] Read more.
The joint modeling of angular and linear observations is crucial as data of this nature are prevalent in multiple disciplines, for example the joint modeling of wind direction and another climatological variable such as wind speed or air temperature, the direction an animal moves and the distance moved, or wave direction and wave height. Hence, there is a need for developing flexible distributions on the hyper-disc, which has support of the interior of the hyper-sphere, as it allows for modeling the combination of angular and linear observations. This paper addresses this need by developing flexible distributions for the disc that have the ability to capture any inherent bimodality present in the data. A new class of bivariate distributions is proposed which has support on the unit disc in two dimensions that includes, as a special case, the existing Möbius distribution on the disc. This class is obtained by expressing the density function in a general form using a measurable function termed as generator. Special cases of this generator are considered to demonstrate the flexibility. By applying a conformal mapping to the generator function a new Möbius distribution class emanates. This class of bivariate distributions on the disc is the first to account for bimodality and skewness present in the data. The flexible behavior of the proposed models in terms of bimodality and skewness is graphically demonstrated. Preliminary evidential analysis of the wind data observed at Marion Island reveals the absence of unimodality in the data. The fit of the proposed models, which account for bimodality, to the Marion Island wind data were evaluated analytically and visually. Full article
(This article belongs to the Special Issue Recent Advances on Symmetry in Mathematical Statistics)
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18 pages, 524 KiB  
Article
MaxSkew and MultiSkew: Two R Packages for Detecting, Measuring and Removing Multivariate Skewness
by Cinzia Franceschini and Nicola Loperfido
Symmetry 2019, 11(8), 970; https://doi.org/10.3390/sym11080970 - 1 Aug 2019
Cited by 6 | Viewed by 5431
Abstract
The R packages MaxSkew and MultiSkew measure, test and remove skewness from multivariate data using their third-order standardized moments. Skewness is measured by scalar functions of the third standardized moment matrix. Skewness is tested with either the bootstrap or under normality. Skewness is [...] Read more.
The R packages MaxSkew and MultiSkew measure, test and remove skewness from multivariate data using their third-order standardized moments. Skewness is measured by scalar functions of the third standardized moment matrix. Skewness is tested with either the bootstrap or under normality. Skewness is removed by appropriate linear projections. The packages might be used to recover data features, as for example clusters and outliers. They are also helpful in improving the performances of statistical methods, as for example the Hotelling’s one-sample test. The Iris dataset illustrates the usages of MaxSkew and MultiSkew. Full article
(This article belongs to the Special Issue Recent Advances on Symmetry in Mathematical Statistics)
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