Skew-elliptical Distributions Research Articles

Bivariate Tail Conditional Co-Expectation for elliptical distributions

In this paper, we consider a random vector X=(X1,X2) following a multivariate Elliptical distribution and we provide an explicit formula for E(X|X≤X˜), i.e., the expected value of the bivariate random variable X conditioned to the event X≤X˜, with X˜∈R2. Such a conditional expectation has an intuitive interpretation in the context of risk measures. Specifically, E(X|X≤X˜) can be interpreted as the Tail Conditional Co-Expectation of X (TCoES). Our main result analytically proves that for a large number of Elliptical distributions, the TCoES can be written as a function of the probability density function of the Skew Elliptical distributions introduced in the literature by the pioneering work of Azzalini (1985). Some numerical experiments based on empirical data show the usefulness of the obtained results for real-world applications.

Multivariate doubly truncated moments for generalized skew-elliptical distributions with applications

ABSTRACT In this paper, we focus on multivariate doubly truncated first two moments of generalized skew-elliptical distributions. This class of distributions includes many useful distributions, such as skew-normal, skew Student- t , skew-logistic and skew-Laplace-normal distributions, as special cases. The formulas of multivariate doubly truncated covariance (MDTCov) for generalized skew-elliptical distributions are also given. Further, we compute multivariate doubly truncated expectations (MDTEs) and MDTCovs for 2 -variate skew-normal, skew-Student- t , skew-logistic and skew-Laplace-normal distributions, and use Monte-Carlo method to simulate and compare with the above results. As applications, the results of multivariate tail conditional expectation (MTCE) and multivariate tail covariance (MTCov) for generalized skew-elliptical distributions are derived. In addition, an optimal problem involving MDTE and MDTCov risk measures is proposed. Finally, we use real data to fit skew-normal distribution and to discuss MTCEs and MTCovs of logarithm of adjusted prices for two portfolios consisting of three companies from S&P (Standard & Poor’s) sectors.

The Tail Mean–Variance optimal capital allocation under the extended skew-elliptical distribution

In this paper we introduce a novel Tail Mean-Variance (TMV) model intended to optimize capital allocation in financial decision-making. The TMV model has many intriguing properties, such as considering the variability and tail risk of the loss function simultaneously. We specifically consider the multivariate extended skew-elliptical (ESE) distributions, which are widely applicable in financial and insurance data modeling. Additionally, we explore the probabilistic properties of the multivariate extended skew-elliptical (ESE) distributions and present explicit formulas for the TMV model within this distribution. Finally, a numerical example is provided to illustrate the results.

Stochastic representations and probabilistic characteristics of multivariate skew-elliptical distributions

The family of multivariate skew-normal distributions has many interesting properties. It is shown here that these hold for a general class of skew-elliptical distributions. For this class, several stochastic representations are established and then their probabilistic properties, such as characteristic function, moments, quadratic forms as well as transformation properties, are investigated.

Doubly truncated expectation and variance of univariate generalized skew-elliptical distributions with applications

In this article, doubly truncated expectation (DTE) and variance (DTV) of univariate generalized skew-elliptical (GSE) distributions are investigated. In addition, we present an alternative form of DTE and DTV for this class of distributions in terms of the hazard function. This class of distributions includes many skewing distributions, for instance, generalized skew-normal, skew-Student-t, skew-logistic, skew-Laplace, and skew-Pearson type VII distributions. Also, we define truncated generalized skew-elliptical distributions and give the relations between moments of truncated distributions and truncated moments of distributions. Specially, we use the EM algorithm to give maximum likelihood estimation of parameters for generalized skew-elliptical distributions. Further, we apply our results to present tail conditional expectation (TCE) and tail variance (TV) for GSE distributions. We also structure an optimal portfolio selection involving DTE and DTV, and give its optimal solution. As an illustrative example, DTE and DTV of a skew-normal random variable are estimated by the Monte Carlo method. Finally, we use real data to fit and select the best distributions, and analysis TCE and TV for the logarithm of adjusted price of three companies (stocks) from S & P (Standard & Poor’s) sectors.

Elliptical and Skew-Elliptical Regression Models and Their Applications to Financial Data Analytics

Various statistical distributions have played significant roles in financial data analytics in recent decades. Among these, elliptical modeling has gained popularity, while the study and application of skew-elliptical modeling have garnered increased attention in various domains. This paper begins by acknowledging the notable accomplishments and contributions of Professor Chris Heyde in the field of financial data modeling. We provide a comprehensive review of elliptical and skew-elliptical modeling, summarizing the latest advancements. In particular, we focus on the characteristics, estimation methods, and diagnostics of elliptical and skew-elliptical distributions in regression and time series models, as well as copula modeling. Furthermore, we discuss several related applications in regression and time series models, including estimation and diagnostic methods. The main objective of this paper is to address the critical need for accurately identifying the underlying elliptical distribution, whether it is elliptical or skew-elliptical. This identification is essential for conducting local influence diagnostics and employing appropriate regression methods using suitable elliptical modeling techniques. To illustrate this process, we present examples that demonstrate the identification of the elliptical distribution, starting with the Box–Jenkins methodology and progressing to copula modeling. The inclusion of copula modeling is motivated by its effectiveness in conjunction with elliptical and skew-elliptical distributions, as it aids in distinguishing between the two. Ultimately, the findings of this paper offer valuable insights, as correctly determining the elliptical and skew-elliptical distribution enables the application of suitable local influence and regression methods, thereby contributing to financial portfolio management, business analytics, and insurance analytics, ensuring the accurate specification of models.

Some results on multivariate measures of elliptical and skew-elliptical distributions: higher-order moments, skewness and kurtosis

<abstract><p>The kurtosis and skewness of distributions are important measures that can describe the shape of a distribution, and there have been many results for symmetric distributions, but there are still many difficulties and challenges in the characterization of skew distributions. Based on the results of Mardia's and Song's kurtosis measures of elliptical distributions obtained by Zografos <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, we generalize the results and study some measures for elliptical and skew-elliptical distributions. We also derive the expressions of moments of skew-elliptical distributions in terms of the ones of skew-normals and take skew-$ t $, skew-Pearson type Ⅶ and skew-Pearson type Ⅱ distributions as examples.</p></abstract>

A spatial heterogeneity mixed model with skew-elliptical distributions

A spatial heterogeneity mixed model with skew-elliptical distributions

Multivariate tail covariance risk measure for generalized skew-elliptical distributions

In this paper, the multivariate tail covariance (MTCov) for generalized skew-elliptical distributions is considered. Some special cases for this distribution, such as generalized skew-normal, generalized skew student-t, generalized skew-logistic and generalized skew-Laplace distributions, are also considered. In order to test the theoretical feasibility of our results, the MTCov for skewed and non skewed normal distributions is computed and compared. Finally, we give a special formula of the MTCov for generalized skew-elliptical distributions.

Optimal tests for elliptical symmetry: Specified and unspecified location

Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most papers in the literature indeed are dealing with the null hypothesis of elliptical symmetry with specified location and actually address location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam’s asymptotic theory of statistical experiments, and optimality is to be understood locally and asymptotically within the family of generalized skew-elliptical distributions. The tests we are proposing are enjoying all the desirable properties of a “good” test of elliptical symmetry: they have simple asymptotic distributions under the entire null hypothesis of elliptical symmetry with unspecified radial density and shape parameter; they are affine-invariant, computationally fast, intuitively understandable, and not too demanding in terms of moments. While achieving optimality against generalized skew-elliptical alternatives, they remain quite powerful under a much broader class of non-elliptical distributions and significantly outperform the available competitors.

A New Robust Class of Skew Elliptical Distributions

A new robust class of multivariate skew distributions is introduced. Practical aspects such as parameter estimation method of the proposed class are discussed, we show that the proposed class can be fitted under a reasonable time frame. Our study shows that the class of distributions is capable to model multivariate skewness structure and does not suffer from the curse of dimensionality as heavily as other distributions of similar complexity do, such as the class of canonical skew distributions. We also derive a nested form of the proposed class which appears to be the most flexible class of multivariate skew distributions in literature that has a closed-form density function. Numerical examples on two data sets, i) a data set containing daily river flow data recorded in the UK; and ii) a data set containing biomedical variables of athletes collected by the Australian Institute of Sports, are demonstrated. These examples further support the practicality of the proposed class on moderate dimensional data sets.

Tail conditional moment for generalized skew-elliptical distributions

ABSTRACT Substantial changes in the financial markets and insurance companies have needed the development of the structure of the risk benchmark, which is the challenge addressed in this paper. We propose a theorem that expands the tail conditional moment (TCM) measure from elliptical distributions to wider classes of skew-elliptical distributions. This family of distributions is suitable for modeling asymmetric phenomena. We obtain the analytical formula for the TCM for skew-elliptical distributions to help well to figure out the risk behavior along the tail of loss distributions. We derive four significant results and generalize the tail conditional skewness (TCS) and the tail conditional kurtosis (TCK) measures for generalized skew-elliptical distributions, which are used to determine the skewness and the kurtosis in the tail of loss distributions. The proposed TCM measure has been applied to well-known families of generalized skew-elliptical distributions. We also provide a practical example of a portfolio problem by calculating the proposed TCM measure for the weighted sum of generalized skew-elliptical distributions.

The tail mean–variance optimal portfolio selection under generalized skew-elliptical distribution

In the insurance and financial markets, events of extreme losses happen in the tail of return distributions, and investors are sensitive to these losses. The Tail Mean–Variance (TMV) criterion focuses on the rare risk but large losses, and it has recently been used in financial management for portfolio selection. In this paper, the proposed TMV criterion is based on the two measures of risk, i.e., the Tail Conditional Expectation (TCE) and Tail Variance (TV) under Generalized Skew-Elliptical (GSE) distribution. We obtain an explicit solution with simple implementation and use a convex optimization approach for the TMV optimization problem under the GSE distribution. We also provide a practical example of a portfolio optimization problem using the proposed TMV criterion. The empirical results show that the optimal portfolio performance can be improved by controlling the tail variability of returns distribution.

Justifying Mean-Variance Portfolio Selection when Asset Returns Are Skewed

We show that, in the presence of a risk-free asset, the return distribution of every portfolio is determined by its mean and variance if and only if asset returns follow a specific skew-elliptical distribution. Thus, contrary to common belief among academics and practitioners, skewed returns do not allow a rejection of mean-variance analysis. Our work differs from Chamberlain's [Chamberlain G (1983) A characterization of the distributions that imply mean-variance utility functions. J. Econom. Theory 29(1):185–201.] by focusing on the returns of portfolios, where the weights over the risk-free asset and the risky assets sum to unity. Furthermore, it extends Meyer's [Meyer J, Rasche RH (1992) Sufficient conditions for expected utility to imply mean-standard deviation rankings: Empirical evidence concerning the location and scale condition. Econom. J. (London) 102(410):91–106.] by introducing elliptical noise into their generalized location-scale framework. To emphasize the relevance of our skew-elliptical model, we additionally provide empirical evidence that it cannot be rejected for the returns of typical portfolios of common stocks or popular alternative investments.This paper was accepted by Kay Giesecke, finance.

TAIL CONDITIONAL EXPECTATIONS FOR GENERALIZED SKEW-ELLIPTICAL DISTRIBUTIONS

This paper deals with the multivariate tail conditional expectation (MTCE) for generalized skew-elliptical distributions. We present tail conditional expectation for univariate generalized skew-elliptical distributions and MTCE for generalized skew-elliptical distributions. There are many special cases for generalized skew-elliptical distributions, such as generalized skew-normal, generalized skew Student-t, generalized skew-logistic and generalized skew-Laplace distributions.

Generalized kernel-based inverse regression methods for sufficient dimension reduction

The linearity condition and the constant conditional variance assumption popularly used in sufficient dimension reduction are respectively close to elliptical symmetry and normality. However, it is always the concern about their restrictiveness. In this article, we give systematic studies to provide insight into the reasons why the popularly used sliced inverse regression and sliced average variance estimation need these conditions. Then we propose a new framework to relax these conditions and suggest generalized kernel-based inverse regression methods to handle a class of mixture multivariate unified skew-elliptical distributions.

Tail variance for Generalized Skew-Elliptical distributions

Notable changes in financial markets have required the development of a standard structure for risk measurement, and obtaining an appropriate risk measurement from historical data is the challenge that is addressed in this article. In recent years, insurance and investment experts are interested in focusing on the use of the tail conditional expectation (TCE) because it has usable and desirable features in different situations. It is well-known that the tail conditional expectation as a risk measurement provides information about the mean of the tail of the loss distribution, while the tail variance (TV) measures the deviation of the loss from this mean along the tail of the distribution. In this paper, we present a theorem that extends the tail variance formula from the elliptical distributions to a rich class of Generalized Skew-Elliptical (GSE) distributions. We develop this theory for the four main classes of skew-elliptical distributions, including the Skew-Normal, Skew-Student-t, Skew-Logistic and Skew-Laplace distributions and obtain the proposed TV measure for them.

Tail densities of skew-elliptical distributions

Skew-elliptical distributions constitute a large class of multivariate distributions that account for both skewness and a variety of tail properties. This class has simpler representations in terms of densities rather than cumulative distribution functions, and the tail density approach has previously been developed to study tail properties when multivariate densities have more tractable forms. The special skew-elliptical structure allows for derivations of specific forms for the tail densities for those skew-elliptical copulas that admit probability density functions, under heavy and light tail conditions on density generators. The tail densities of skew-elliptical copulas are explicit and depend only on tail properties of the underlying density generator and conditions on the skewness parameters. In the heavy-tail case skewness parameters affect tail densities of the skew-elliptical copulas more profoundly than that in the light tail case, whereas in the latter case the tail densities of skew-elliptical copulas are only proportional to the tail densities of symmetrical elliptical copulas. Various examples, including tail densities of skew-normal and skew-t distributions, are given.

Testing a Multivariate Distribution for Generalized Skew Ellipticity

We consider the problem of testing whether a sample comes from a family of the multivariate generalized skew-elliptical distributions with an unknown location parameter, an unknown scaling matrix, ...

A proof for the existence of multivariate singular generalized skew-elliptical density functions

We derive the density functions of multivariate singular generalized skew-elliptical distributions, present their characteristic function, and derive explicit formulas for the expectation and the covariance matrix. This letter generalizes results given in Díaz-García et al. (2002) and Young et al. (2016) about the existence of multivariate singular elliptical and multivariate singular skew-normal density functions, respectively.