Symmetry

Research

30 pages, 480 KiB

Open AccessFeature PaperArticle

Robust and Nonrobust Linking of Two Groups for the Rasch Model with Balanced and Unbalanced Random DIF: A Comparative Simulation Study and the Simultaneous Assessment of Standard Errors and Linking Errors with Resampling Techniques

by Alexander Robitzsch

Symmetry 2021, 13(11), 2198; https://doi.org/10.3390/sym13112198 - 18 Nov 2021

Cited by 14 | Viewed by 1719

Abstract

In this article, the Rasch model is used for assessing a mean difference between two groups for a test of dichotomous items. It is assumed that random differential item functioning (DIF) exists that can bias group differences. The case of balanced DIF is [...] Read more.

In this article, the Rasch model is used for assessing a mean difference between two groups for a test of dichotomous items. It is assumed that random differential item functioning (DIF) exists that can bias group differences. The case of balanced DIF is distinguished from the case of unbalanced DIF. In balanced DIF, DIF effects on average cancel out. In contrast, in unbalanced DIF, the expected value of DIF effects can differ from zero and on average favor a particular group. Robust linking methods (e.g., invariance alignment) aim at determining group mean differences that are robust to the presence of DIF. In contrast, group differences obtained from nonrobust linking methods (e.g., Haebara linking) can be affected by the presence of a few DIF effects. Alternative robust and nonrobust linking methods are compared in a simulation study under various simulation conditions. It turned out that robust linking methods are preferred over nonrobust alternatives in the case of unbalanced DIF effects. Moreover, the theory of M-estimation, as an important approach to robust statistical estimation suitable for data with asymmetric errors, is used to study the asymptotic behavior of linking estimators if the number of items tends to infinity. These results give insights into the asymptotic bias and the estimation of linking errors that represent the variability in estimates due to selecting items in a test. Moreover, M-estimation is also used in an analytical treatment to assess standard errors and linking errors simultaneously. Finally, double jackknife and double half sampling methods are introduced and evaluated in a simulation study to assess standard errors and linking errors simultaneously. Half sampling outperformed jackknife estimators for the assessment of variability of estimates from robust linking methods. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

18 pages, 1505 KiB

Open AccessArticle

On a Vector-Valued Measure of Multivariate Skewness

by Nicola Loperfido

Symmetry 2021, 13(10), 1817; https://doi.org/10.3390/sym13101817 - 29 Sep 2021

Cited by 1 | Viewed by 1092

Abstract

The canonical skewness vector is an analytically simple function of the third-order, standardized moments of a random vector. Statistical applications of this skewness measure include semiparametric modeling, independent component analysis, model-based clustering, and multivariate normality testing. This paper investigates some properties of the [...] Read more.

The canonical skewness vector is an analytically simple function of the third-order, standardized moments of a random vector. Statistical applications of this skewness measure include semiparametric modeling, independent component analysis, model-based clustering, and multivariate normality testing. This paper investigates some properties of the canonical skewness vector with respect to representations, transformations, and norm. In particular, the paper shows its connections with tensor contraction, scalar measures of multivariate kurtosis and Mardia’s skewness, the best-known scalar measure of multivariate skewness. A simulation study empirically compares the powers of tests for multivariate normality based on the squared norm of the canonical skewness vector and on Mardia’s skewness. An example with financial data illustrates the statistical applications of the canonical skewness vector. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

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20 pages, 983 KiB

Open AccessFeature PaperArticle

Cumulants of Multivariate Symmetric and Skew Symmetric Distributions

by Sreenivasa Rao Jammalamadaka, Emanuele Taufer and Gyorgy H. Terdik

Symmetry 2021, 13(8), 1383; https://doi.org/10.3390/sym13081383 - 29 Jul 2021

Cited by 1 | Viewed by 1806

Abstract

This paper provides a systematic and comprehensive treatment for obtaining general expressions of any order, for the moments and cumulants of spherically and elliptically symmetric multivariate distributions; results for the case of multivariate t-distribution and related skew-t distribution are discussed in [...] Read more.

This paper provides a systematic and comprehensive treatment for obtaining general expressions of any order, for the moments and cumulants of spherically and elliptically symmetric multivariate distributions; results for the case of multivariate t-distribution and related skew-t distribution are discussed in some detail. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

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21 pages, 4037 KiB

Open AccessArticle

Canonical Correlations and Nonlinear Dependencies

by Nicola Maria Rinaldo Loperfido

Symmetry 2021, 13(7), 1308; https://doi.org/10.3390/sym13071308 - 20 Jul 2021

Cited by 1 | Viewed by 1897

Abstract

Canonical correlation analysis (CCA) is the default method for investigating the linear dependence structure between two random vectors, but it might not detect nonlinear dependencies. This paper models the nonlinear dependencies between two random vectors by the perturbed independence distribution, a multivariate semiparametric [...] Read more.

Canonical correlation analysis (CCA) is the default method for investigating the linear dependence structure between two random vectors, but it might not detect nonlinear dependencies. This paper models the nonlinear dependencies between two random vectors by the perturbed independence distribution, a multivariate semiparametric model where CCA provides an insight into their nonlinear dependence structure. The paper also investigates some of its probabilistic and inferential properties, including marginal and conditional distributions, nonlinear transformations, maximum likelihood estimation and independence testing. Perturbed independence distributions are closely related to skew-symmetric ones. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

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22 pages, 1193 KiB

Open AccessArticle

Multivariate Skew t-Distribution: Asymptotics for Parameter Estimators and Extension to Skew t-Copula

by Tõnu Kollo, Meelis Käärik and Anne Selart

Symmetry 2021, 13(6), 1059; https://doi.org/10.3390/sym13061059 - 11 Jun 2021

Cited by 5 | Viewed by 2345

Abstract

Symmetric elliptical distributions have been intensively used in data modeling and robustness studies. The area of applications was considerably widened after transforming elliptical distributions into the skew elliptical ones that preserve several good properties of the corresponding symmetric distributions and increase possibilities of [...] Read more.

Symmetric elliptical distributions have been intensively used in data modeling and robustness studies. The area of applications was considerably widened after transforming elliptical distributions into the skew elliptical ones that preserve several good properties of the corresponding symmetric distributions and increase possibilities of data modeling. We consider three-parameter p-variate skew t-distribution where p-vector

μ

is the location parameter,

Σ : p \times p

is the positive definite scale parameter, p-vector

α

is the skewness or shape parameter, and the number of degrees of freedom

ν

is fixed. Special attention is paid to the two-parameter distribution when

μ = 0

that is useful for construction of the skew t-copula. Expressions of the parameters are presented through the moments and parameter estimates are found by the method of moments. Asymptotic normality is established for the estimators of

Σ

and

α

. Convergence to the asymptotic distributions is examined in simulation experiments. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

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16 pages, 7841 KiB

Open AccessArticle

Skewness-Based Projection Pursuit as an Eigenvector Problem in Scale Mixtures of Skew-Normal Distributions

by Jorge M. Arevalillo and Hilario Navarro

Symmetry 2021, 13(6), 1056; https://doi.org/10.3390/sym13061056 - 11 Jun 2021

Cited by 2 | Viewed by 1802

Abstract

This paper addresses the projection pursuit problem assuming that the distribution of the input vector belongs to the flexible and wide family of multivariate scale mixtures of skew normal distributions. Under this assumption, skewness-based projection pursuit is set out as an eigenvector problem, [...] Read more.

This paper addresses the projection pursuit problem assuming that the distribution of the input vector belongs to the flexible and wide family of multivariate scale mixtures of skew normal distributions. Under this assumption, skewness-based projection pursuit is set out as an eigenvector problem, described in terms of the third order cumulant matrix, as well as an eigenvector problem that involves the simultaneous diagonalization of the scatter matrices of the model. Both approaches lead to dominant eigenvectors proportional to the shape parametric vector, which accounts for the multivariate asymmetry of the model; they also shed light on the parametric interpretability of the invariant coordinate selection method and point out some alternatives for estimating the projection pursuit direction. The theoretical findings are further investigated through a simulation study whose results provide insights about the usefulness of skewness model-based projection pursuit in the statistical practice. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

27 pages, 435 KiB

Open AccessArticle

Distribution-Based Entropy Weighting Clustering of Skewed and Heavy Tailed Time Series

by Raffaele Mattera, Massimiliano Giacalone and Karina Gibert

Symmetry 2021, 13(6), 959; https://doi.org/10.3390/sym13060959 - 28 May 2021

Cited by 9 | Viewed by 3186

Abstract

The goal of clustering is to identify common structures in a data set by forming groups of homogeneous objects. The observed characteristics of many economic time series motivated the development of classes of distributions that can accommodate properties, such as heavy tails and [...] Read more.

The goal of clustering is to identify common structures in a data set by forming groups of homogeneous objects. The observed characteristics of many economic time series motivated the development of classes of distributions that can accommodate properties, such as heavy tails and skewness. Thanks to its flexibility, the skewed exponential power distribution (also called skewed generalized error distribution) ensures a unified and general framework for clustering possibly skewed and heavy tailed time series. This paper develops a clustering procedure of model-based type, assuming that the time series are generated by the same underlying probability distribution but with different parameters. Moreover, we propose to optimally combine the estimated parameters to form the clusters with an entropy weighing k-means approach. The usefulness of the proposal is shown by means of application to financial time series, demonstrating also how the obtained clusters can be used to form portfolio of stocks. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

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32 pages, 2412 KiB

Open AccessEditor’s ChoiceArticle

Copulaesque Versions of the Skew-Normal and Skew-Student Distributions

by Christopher Adcock

Symmetry 2021, 13(5), 815; https://doi.org/10.3390/sym13050815 - 6 May 2021

Cited by 4 | Viewed by 2027

Abstract

A recent paper presents an extension of the skew-normal distribution which is a copula. Under this model, the standardized marginal distributions are standard normal. The copula itself depends on the familiar skewing construction based on the normal distribution function. This paper is concerned [...] Read more.

A recent paper presents an extension of the skew-normal distribution which is a copula. Under this model, the standardized marginal distributions are standard normal. The copula itself depends on the familiar skewing construction based on the normal distribution function. This paper is concerned with two topics. First, the paper presents a number of extensions of the skew-normal copula. Notably these include a case in which the standardized marginal distributions are Student’s t, with different degrees of freedom allowed for each margin. In this case the skewing function need not be the distribution function for Student’s t, but can depend on certain of the special functions. Secondly, several multivariate versions of the skew-normal copula model are presented. The paper contains several illustrative examples. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

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11 pages, 325 KiB

Open AccessArticle

Multivariate Tail Moments for Log-Elliptical Dependence Structures as Measures of Risks

by Zinoviy Landsman and Tomer Shushi

Symmetry 2021, 13(4), 559; https://doi.org/10.3390/sym13040559 - 28 Mar 2021

Cited by 2 | Viewed by 1543

Abstract

The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, [...] Read more.

The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, risk managers confront a system of mutually dependent risks, not only one risk. Thus, it is important to measure risks while capturing their dependence structure. In this short paper, we compute the multivariate risk measures, multivariate tail conditional expectation, and multivariate tail covariance measure for the family of log-elliptical distributions, which captures the dependence structure of the risks while focusing on the tail of their distributions, i.e., on extreme loss events. We then study our result and examine special cases, as well as the optimal portfolio selection using such measures. Finally, we show how the given multivariate tail moments can also be computed for log-skew elliptical models based on similar approaches given for the log-elliptical case. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

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9 pages, 269 KiB

Open AccessArticle

Simple New Proofs of the Characteristic Functions of the F and Skew-Normal Distributions

by Jun Zhao, Sung-Bum Kim, Seog-Jin Kim and Hyoung-Moon Kim

Symmetry 2020, 12(12), 2041; https://doi.org/10.3390/sym12122041 - 10 Dec 2020

Cited by 3 | Viewed by 1933

Abstract

For a statistical distribution, the characteristic function (CF) is crucial because of the one-to-one correspondence between a distribution function and its CF and other properties. In order to avoid the calculation of contour integrals, the CFs of two popular distributions, the F and [...] Read more.

For a statistical distribution, the characteristic function (CF) is crucial because of the one-to-one correspondence between a distribution function and its CF and other properties. In order to avoid the calculation of contour integrals, the CFs of two popular distributions, the F and the skew-normal distributions, are derived by solving two ordinary differential equations (ODEs). The results suggest that the approach of deriving CFs by the ODEs is effective for asymmetric distributions. A much simpler approach is proposed to derive the CF of the multivariate F distribution in terms of a stochastic representation without using contour integration. For a special case of the multivariate F distribution where the variable dimension is one, its CF is consistent with that of the former (univariate) F distribution. This further confirms that the derivations are reasonable. The derivation is quite simple, and is suitable for presentation in statistics theory courses. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Multivariate Statistics and Data Science)

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