Multivariate Stable Distributions Research Articles

Application of Stable Random Vector with Gaussian Copula

More and more real-world datasets have heavy-tailed distribution, while the calculations for these distributions in multi-dimensional cases are complex. This work shows a method to investigate data of multivariate heavy-tailed distributions. The sufficient condition for every a-stable random vector is that it has α-stable marginals and Gaussian copula. From that results, we have a procedure testing stable distribution of multi-dimensional data and a formula representing density functions of multivariate stable distribution. Adopted a new tool, datasets about daily returns of 4 stocks on HoSE and 3 grains were analyzed.

Multivariate Subgaussian Stable Distributions in R

We introduce and showcase [mvpd](https://CRAN.R-project.org/package=mvpd) (an acronym for *m*ulti*v*ariate *p*roduct *d*istributions), a package that uses a product distribution approach to calculating multivariate subgaussian stable distribution functions. The family of multivariate subgaussian stable distributions are elliptically contoured multivariate stable distributions that contain the multivariate Cauchy and the multivariate normal distribution. These distributions can be useful in modeling data and phenomena that have heavier tails than the normal distribution (more frequent occurrence of extreme values). Application areas include log returns for stocks, signal processing for radar and sonar data, astronomy, and hunting patterns of sharks.

Multivariate Regression with Stable Errors Using Order Statistics

In this paper, we investigate a multivariate regression model with multivariate heavy-tailed stable errors. Since in heavy-tailed data, especially in multivariate stable distributions, some moments do not exist; classical multivariate regression methods do not perform well. We suggest using an effective property of the existence of some moments of order statistics stable distribution. We propose a method for trimming the data set using this property. Then, we estimate the regression coefficients based on the rest of the ordered data. We calculate the trimmed data set based on the error’s tail index and skewness parameters. Also, we analytically compute the bias and variance of the introduced estimators of the regression parameters. Finally, we study the performance of the proposed methods with ordinary least squares via a simulation study and a real data set.

Some analytical results on bivariate stable distributions with an application in operational risk

The multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails. Although one-dimensional stable distributions are well known, there are many open questions in the multivariate regime, since the tractability of the multivariate Gaussian universe, does not extend to non-Gaussian multivariate stable distributions. In this work, we provide the Laplace transform of bivariate stable distributions and its certain cut in the first quadrant. Given the lack of a closed-form likelihood function, we propose to estimate the parameters by means of Approximate Maximum Likelihood, a simulation-based method with desirable asymptotic properties. Simulation experiments and an application to truncated operational losses illustrate the applicability of the model.

Another look at contagion across United States and European financial markets: Evidence from the credit default swaps markets

AbstractThe paper looks at the results of Apergis, Christou and Kynigakis (2019) and proposes a novel model that allows time variation in volatility, skewness and kurtosis, based on multivariate stable distributions. The analysis also looks at bank sector CDS, insurance sector CDS, sovereign bonds, equity and volatility indices. The findings corroborate their results and indicate significant evidence of contagion, especially through the channels of co‐skewness and co‐kurtosis. In addition, it establishes a higher order channel of causality between co‐skewness and co‐kurtosis.

Estimation of the parameters of multivariate stable distributions

In this paper, we first discuss some of the well-known methods available in the literature for the estimation of the parameters of a univariate/multivariate stable distribution. Based on the available methods, a new hybrid method is proposed for the estimation of the parameters of a univariate stable distribution. The proposed method is further used for the estimation of the parameters of a strictly multivariate stable distribution. The efficiency, accuracy and simplicity of the new method is shown through Monte-Carlo simulation. Finally, we apply the proposed method to the univariate and bivariate financial data.

Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

Gaussian copula of stable random vectors and application

In this paper, we present a new method to investigate data of multivariate heavy-tailed distributions. We show that for any given number $\alpha \in (0;2]$, each Gaussian copula is also the copula of an $\alpha$-stable random vector. Simultaneously, every random vector is $\alpha$-stable if its marginals are $\alpha$-stable and its copula is a Gaussian copula. The result is used to build up a formula representing density functions of $\alpha$-stable random vectors with Gaussian copula. Adopting a new tool, the paper points out that pairs of GPS signals recording latitude and longitude of a fixed point have two-dimensional stable distribution, and in the most of cases, vectors of daily returns in stock market data have multivariate stable distributions with Gaussian copulas.

Interpretation of the measure of dependence for multivariate stable random variables using factor model

The problem of interpretation the measures of dependence of multivariate stable random variables in the space of these variables was considered. It is noted that both the examined distribution laws and the existing measures of dependence are parameterized using characteristic functions, i.e. in the frequency domain, not in the space of random variables. This is what causes the relevance of the studied problem. The paper presents a model of symmetric mixing of hidden factors that allows to consider the dependence between random variables within the factor model. Under this approach, a set of identically distributed observed random variables is seen as the result of a linear transformation of the same number of independent random variables (hidden factors). It is shown that for the multivariate normal distribution, the correlation coefficient can be interpreted as a measure of the differences between the linear transformation of factors and identical transformation. The analysis of parameterization forms of multivariate stable distribution laws was performed. The subclass of such distributions, for which it is possible to represent the observed random variables by a linear combination of the independent variables was singled out. It is important to note that this subclass is parameterized unambiguously, in contrast to the general case of multivariate stable laws. It is shown that within the selected subclass of multivariate stable distributions, the measure of dependence has the same meaning as the correlation coefficient for the multivariate normal law.

Bayesian analysis of multivariate stable distributions using one-dimensional projections

In this paper we take up Bayesian inference in general multivariate stable distributions. We exploit the representation of Matsui and Takemura (2009) for univariate projections, and the representation of the distributions in terms of their spectral measure. We present efficient MCMC schemes to perform the computations when the spectral measure is approximated discretely or, as we propose, by a normal distribution. Appropriate latent variables are introduced to implement MCMC. In relation to the discrete approximation, we propose efficient computational schemes based on the characteristic function.

Goodness-of-fit tests for multivariate stable distributions based on the empirical characteristic function

We consider goodness-of-fit testing for multivariate stable distributions. The proposed test statistics exploit a characterizing property of the characteristic function of these distributions and are consistent under some conditions. The asymptotic distribution is derived under the null hypothesis as well as under local alternatives. Conditions for an asymptotic null distribution free of parameters and for affine invariance are provided. Computational issues are discussed in detail and simulations show that with proper choice of the user parameters involved, the new tests lead to powerful omnibus procedures for the problem at hand.

Analytical-Numeric Formulas for the Probability Density Function of Multivariate Stable and Geo-Stable Distributions

The main problem in applying the multivariate stable and geo-stable distributions is the lack of a closed-form formula for the probability density function. In this article, we provide a solution by using the connection between these two distributions and fractional calculus. The analytical-numeric formulas for the probability density function for these two probability distributions are obtained via three analytic approximation methods—homotopy perturbation method, Adomian decomposition method, and variational iteration method. Finally, the convergent analysis of the applied methods is discussed.

Bayesian Inference in Multivariate Stable Distributions

In this paper we take up Bayesian inference in general, multivariate stable distributions. We use approximate Bayesian computation (ABC) along with carefully crafted proposal distributions for the implementation of MCMC. The problem of selecting summary statistics in ABC is resolved through the use of the characteristic function. Two important problems in multivariate stable distributions are: (i) the selection of an optimal con guration of the grid points where the empirical and theoretical characteristic functions are compared, and (ii) the estimation of the spectral measure through which the distributions are de ned in Rd. The problems are resolved successfully and certain new approximations to the spectral measure are proposed and implemented. E fficient proposal/importance distributions are constructed, and tested thoroughly, to ensure good performance in connection with ABC. The new techniques are applied to exchange rate and stock return data and are supplemented by Monte Carlo simulations. In addition, critical values are provided for a closeness statistic between the empirical and theoretical characteristic functions, resolving successfully another major problem in ABC-related inference.

Bayesian Inference in Multivariate Stable Distributions Using Copulae

In this paper we take up Bayesian inference in multivariate stable distributions through innovative multivariate stable copulae. The problem that the characteristic function is defined through a difficult object, the spectral measure is completely bypassed by our approach. The new methods are applied to major exchange rates with encouraging results. The copula-based technique is based on non-parametric margins (both data-estimated as well as Dirichlet process priors) and we compare with a multivariate stable copula whose margins can be normal, Student-t or univariate stable.

Bayesian Analysis of Multivariate Stable Distributions Using One-Dimensional Projections

In this paper we take up Bayesian inference in general multivariate stable distributions. We exploit the representation of Matsui and Takemura (2009) for univariate projections, and the representation of the distributions in terms of their spectral measure. We present efficient MCMC schemes to perform the computations when the spectral measure is approximated discretely or, as we propose, by a normal distribution. Appropriate latent variables are introduced to implement MCMC. In relation to the discrete approximation, we propose efficient computational schemes based on the characteristic function.

Multivariate stable distributions and generating densities

The probability density function of multivariate stable distributions only applies to special accessible cases. Consequently, because of the absence of an explicit solution for their probability distribution function, applications have been limited. In this paper, we present an analytic method for generating densities to resolve this problem. Some examples and special cases are discussed.

Characterizing Decentralized Wireless Networks with Temporal Correlation in the Low Outage Regime

Communication in decentralized wireless networks is limited by interference. Because transmissions typically last for more than a single contention time slot, interference often exhibits a strong statistical dependence over time that results in temporally correlated communication performance. The temporal dependence in interference increases as user mobility decreases and/or the total transmission time increases. We propose a network model that spans the extremes of temporal independence to long-term temporal dependence. Using the proposed model, closed-form single hop communication performance metrics are derived that are asymptotically exact in the low outage regime. The primary contributions are (i) deriving the joint temporal statistics of network interference and showing that it follows a multivariate symmetric alpha stable distribution; (ii) utilizing the joint interference statistics to derive closed-form expressions for local delay, throughput outage probability, and average network throughput; and (iii) using the joint interference statistics to redefine and analyze the network transmission capacity that captures the throughput-delay-reliability tradeoffs in single hop transmissions. Simulation results verify the closed-form expressions derived in this paper and we demonstrate up to 2× gain in network throughput and reliability by optimizing certain parameters of medium access control layer protocol in view of the temporal correlations.

Estimation for multivariate stable distributions with generalized empirical likelihood

This paper considers the generalized empirical likelihood (GEL) method for estimating the parameters of the multivariate stable distribution. The GEL method is considered to be an extension of the generalized method of moments (GMM). The multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails. We treat the spectral measure, which summarizes scale and asymmetry, by discretization. In order to estimate all the model parameters simultaneously, we apply the estimating function constructed by equating empirical and theoretical characteristic functions. The efficacy of the proposed GEL method is demonstrated in Monte Carlo studies. An illustrative example involving daily returns of market indexes is also included.

Modeling fat tails in stock returns: a multivariate stable-GARCH approach

In this paper a new multivariate volatility model is proposed. It combines the appealing properties of the stable Paretian distribution to model the heavy tails with the GARCH model to capture the volatility clustering. Returns on assets are assumed to follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. In this way the characteristic function of the fitted returns has a tractable expression and the density function can be recovered by numerical methods. A multivariate GARCH structure is then adopted to model the covariance matrix of the Gaussian vectors underlying the sub-Gaussian system. The model is applied to a bivariate series of daily U.S. stock returns. Value-at-risk for long and short positions is computed and compared with the one obtained using the multivariate normal and the multivariate Student’s t distribution. Finally, exploiting the recent developments in the vast dimensional time-varying covariances modeling, possible feasible extensions of our model to higher dimensions are suggested and an illustrative example using the Dow Jones index components is presented.

Modeling Fat Tails in Stock Returns: A Multivariate Stable-GARCH Approach

In this paper a new multivariate volatility model is proposed. It combines the appealing properties of the stable Paretian distribution to model the heavy tails with the GARCH model to capture the volatility clustering. We assume that multivariate asset-returns of financial stocks follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. In this way the characteristic function of the fitted returns has a tractable expression and the density function can be recovered by numerical methods. A multivariate GARCH structure is then adopted to model the covariance matrix of the Gaussian vectors underlying the sub-Gaussian system. The model is applied to a bivariate series of daily U.S. stock returns. Value-at-Risk for long and short positions is computed and compared with the one obtained using the multivariate normal and the multivariate Student's t distribution. Finally, exploiting the recent developments in the vast dimensional time-varying covariances modeling, possible feasible extensions to higher dimensions are suggested and an illustrative example using the Dow Jones index components is presented.