Multivariate Exponential Power Research Articles

An alternative multivariate exponential power distribution

In the existing multivariate Exponential Power Distribution (EPD), the shape parameter has the tendency to affect the variability in the data leading to distorted features of the distribution. This study resolves this problem by obtaining a generalized expression in place of the constant coefficient (0.5) of the quadratic form of the multivariate EPD as a function of the shape parameter, such that the variance-covariance matrix of the data is exactly equal to the scale parameter. The density function thus obtained is a more generalized multivariate EPD that provides flexibility in depicting the true underlying characteristics of the distribution for all sample sizes.

Multivariate exponential power Lévy processes and random fields

Two multivariate exponential power Lévy processes are introduced in this paper via compound Poisson representations, which are so termed because their characteristic exponents are the linear combination of certain positive or negative power functions. They can be represented as the time changed multivariate Brownian motions using the time changes that are two exponential power subordinators, respectively. Two multivariate exponential power random fields are also proposed as elliptically contoured random fields, whose finite-dimensional characteristic functions are made up of the exponential power functions and enjoy the infinitely divisible property.

New Insights on the Multivariate Skew Exponential Power Distribution

ABSTRACT The multivariate exponential power is a useful distribution for modeling departures from normality in data by means of a tail weight scalar parameter that regulates the non-normality of the model. The incorporation of a shape asymmetry vector into the model serves to account for potential asymmetries and gives rise to the multivariate skew exponential power distribution. This work is aimed at revisiting the skew exponential power distribution taking as a starting point its formulation as a scale mixture of skew-normal distributions. The paper provides some highlights and theoretical insights on the role played by its parameters to assess two complementary aspects of the multivariate non-normality such as directional asymmetry and tail weight behavior regardless of the asymmetry. The intuition behind both issues relies on well-known mathematical ideas about skewness maximization and convex transform stochastic orderings.

Estimation and order selection for multivariate exponential power mixture models

Finite mixture model is a promising statistical model in investigating the heterogeneity of population. For multivariate non-Gaussian density estimation and approximation, in this paper, we consider to use multivariate exponential power mixture models. We propose the penalized-likelihood method with a generalized EM algorithm to estimate locations, scale matrices, shape parameters, and mixing probabilities. Order selection is achieved simultaneously. Properties of the estimated order have been derived. Although we mainly focus on the unconstrained scale matrix type in multivariate exponential power mixture models, three more parsimonious types of scale matrix have also been considered. Performance based on simulation and real data analysis implies the parsimony of the exponential power mixture models, and verifies the consistency of order selection.

Implementation of Multivariate Exponential Power Distribution in Discrimination and Classification of Psychological Data and Other applications

Recent advances have shown that some multivariate psychological data are deviating from usual normal assumption either in the tails or kurtosis. Thereby, allowing the call for modelling of such data using more robust elliptically contoured density which includes the normal distribution as a special case. This allowed more flexibility at the kurtosis and tail regions, which is better in handling non-normality in data analysis and also lower the cost of misclassification. The present study employed a robust model for such cases in the context of discrimination and classification of multivariate psychological disorder data using multivariate exponential distribution as an underlining model. Parameters were estimated using the method of maximum likelihood estimation and the discrimination and classification were based on the log likelihood ratio approach. The resulting models relied solidly on the shape parameter, which regulate the tails and the kurtosis, thereby allowed flexibility. This method enable us to lower the cost of misclassification. Some other areas of applications were also considered in the paper.

Some Properties of Univariate and Multivariate Exponential Power Distributions and Related Topics

In the paper, a survey of the main results concerning univariate and multivariate exponential power (EP) distributions is given, with main attention paid to mixture representations of these laws. The properties of mixing distributions are considered and some asymptotic results based on mixture representations for EP and related distributions are proved. Unlike the conventional analytical approach, here the presentation follows the lines of a kind of arithmetical approach in the space of random variables or vectors. Here the operation of scale mixing in the space of distributions is replaced with the operation of multiplication in the space of random vectors/variables under the assumption that the multipliers are independent. By doing so, the reasoning becomes much simpler, the proofs become shorter and some general features of the distributions under consideration become more vivid. The first part of the paper concerns the univariate case. Some known results are discussed and simple alternative proofs for some of them are presented as well as several new results concerning both EP distributions and some related topics including an extension of Gleser’s theorem on representability of the gamma distribution as a mixture of exponential laws and limit theorems on convergence of the distributions of maximum and minimum random sums to one-sided EP distributions and convergence of the distributions of extreme order statistics in samples with random sizes to the one-sided EP and gamma distributions. The results obtained here open the way to deal with natural multivariate analogs of EP distributions. In the second part of the paper, we discuss the conventionally defined multivariate EP distributions and introduce the notion of projective EP (PEP) distributions. The properties of multivariate EP and PEP distributions are considered as well as limit theorems establishing the conditions for the convergence of multivariate statistics constructed from samples with random sizes (including random sums of random vectors) to multivariate elliptically contoured EP and projective EP laws. The results obtained here give additional theoretical grounds for the applicability of EP and PEP distributions as asymptotic approximations for the statistical regularities observed in data in many fields.

Discrimination and Classification model from Multivariate Exponential Power Distribution

It is common to assume a normal distribution when discriminating and classifying a multivariate data based on some attributes. But when such data is lighter or heavier in both tails than the normal distribution, then the probability of misclassification becomes higher giving unreliable result. This study proposed multivariate exponential power distribution a family of elliptically contoured model as underlining model for discrimination and classification. The distribution has a shape parameter which regulate the tail of the symmetric distribution to mitigate the problem of both lighter and heavier tails data, this generalizes the normal distribution and thus will definitely gives a lower misclassification error in discrimination and classification. The resulting discriminant model was compared with fisher linear discriminant function when applying to real data.

Inferences from Asymmetric Multivariate Exponential Power Distribution

In this paper, we develop the asymptotic distribution of the covariance structure of an asymmetric multivariate exponential power distribution (AMEPD) by extending the existing Hotelling t2 distribution to its generalized form. The obtained results ‘generalised Hotelling t2 test statistic’, accommodates for the existing Mahalanobis distance between two multivariate data sets. The hazard rate and the characteristics function of the desired test statistics are established. The study further establishes the test of hypothesis and simultaneous confidence interval for the population mean vectors of p th attributes from AMEPD by using the Neymar-Pearson lemma and Pivotal quantity approaches. The obtained results, which are dependent on shape parameter flexibility and skewness parameter, generalize the test of hypothesis and simultaneous confidence intervals of various distributions which are members of the exponential power family. Simulated and real life data were used to demonstrate the usefulness of the obtained results.

Employment and energy uncertainty

This paper examines the effect of uncertainty on aggregate employment, private sector employment and employment in sectors that produce goods and services. Our empirical model is a structural vector autoregression with the multivariable GARCH-in-Mean, which we extend to permit the errors to be distributed multivariate exponential power. This distribution includes a shape parameter to characterize the degree of excess kurtosis. We find that increased uncertainty, measured from energy prices, has strong negative effects on employment, with the magnitude of the effect greater for employment in the private sector and goods producing sectors. We find that energy uncertainty contributes to a reduction in private sector employment growth more than 30 basis points on average, and by as much as 120 basis points during periods of extreme volatility. Once we account for the effects of uncertainty, we find that the dynamic response of employment to higher energy prices is unambiguously negative after about three months.

Forecasting Value-at-Risk under Different Distributional Assumptions

Financial asset returns are known to be conditionally heteroskedastic and generally non-normally distributed, fat-tailed and often skewed. These features must be taken into account to produce accurate forecasts of Value-at-Risk (VaR). We provide a comprehensive look at the problem by considering the impact that different distributional assumptions have on the accuracy of both univariate and multivariate GARCH models in out-of-sample VaR prediction. The set of analyzed distributions comprises the normal, Student, Multivariate Exponential Power and their corresponding skewed counterparts. The accuracy of the VaR forecasts is assessed by implementing standard statistical backtesting procedures used to rank the different specifications. The results show the importance of allowing for heavy-tails and skewness in the distributional assumption with the skew-Student outperforming the others across all tests and confidence levels.

Construction of Value-at-Risk Forecasts Under Different Distributional Assumptions within a BEKK Framework

Financial asset returns are known to be conditionally heteroskedastic and generally non-normally distributed, fat-tailed and often skewed. In order to account for both the skewness and the excess kurtosis in returns, we combine the BEKK model from the multivariate GARCH literature with different multivariate densities for the returns. The set of distributions we consider comprises the normal, Student, Multivariate Exponential Power and their skewed counterparts. Applying this framework to a sample of ten assets from the Dow Jones Industrial Average Index, we compare the performance of equally- weighted portfolios derived from the symmetric and skewed distributions in forecasting out-of-sample Value-at-Risk. The accuracy of the VaR forecasts is assessed by implementing standard statistical backtesting procedures. The results unanimously show that the inclusion of fat-tailed densities into the model specification yields more accurate VaR forecasts, while the further addition of skewness does not lead to significant improvements.

Construction of Value-at-Risk Forecasts Under Different Distributional Assumptions within a BEKK Framework

Financial asset returns are known to be conditionally heteroskedastic and generally non-normally distributed, fat-tailed and often skewed. In order to account for both the skewness and the excess kurtosis in returns, we combine the BEKK model from the multivariate GARCH literature with different multivariate densities for the returns. The set of distributions we consider comprises the normal, Student, Multivariate Exponential Power and their skewed counterparts. Applying this framework to a sample of ten assets from the Dow Jones Industrial Average Index, we compare the performance of equally- weighted portfolios derived from the symmetric and skewed distributions in forecasting out-of-sample Value-at-Risk. The accuracy of the VaR forecasts is assessed by implementing standard statistical backtesting procedures. The results unanimously show that the inclusion of fat-tailed densities into the model specification yields more accurate VaR forecasts, while the further addition of skewness does not lead to significant improvements.

Assessing the effect of kurtosis deviations from Gaussianity on conditional distributions

The multivariate exponential power family is considered for n-dimensional random variables, Z, with a known partition Z≡(Y,X) of dimensions p and n-p, respectively, with interest focusing on the conditional distribution Y|X. An infinitesimal variation of any parameter of the joint distribution produces perturbations in both the conditional and marginal distributions. The aim of the study was to determine the local effect of kurtosis deviations using the Kullback–Leibler divergence measure between probability distributions. The additive decomposition of this measure in terms of the conditional and marginal distributions, Y|X and X, is used to define a relative sensitivity measure of the conditional distribution family {Y|X=x}. Finally, simulated results suggest that for large dimensions, the measure is approximately equal to the ratio p/n, and then the effect of non-normality with respect to kurtosis depends only on the relative size of the variables considered in the partition of the random vector.

Relative Sensitivity of Conditional Distributions to Kurtosis Deviations in the Joint Model

The Multivariate Exponential Power family is considered for n-dimensional random variables, Z, with a known partition Z (Y,X) of dimensions p and n-p, respectively. An infinitesimal variation of any parameter produces both conditional and marginal distributions perturbations. The aim of our study is to determine the local effect of kurtosis deviations by means of the Kullback-Leibler divergence measure between probability distributions. The additive decomposition of this measure in terms of the conditional and marginal distributions, YǀX and X, has been used to define the relative sensitivity of the conditional distributions family {YǀX=x}. The obtained results show that, for large dimensions, it is nearly p/n.

Analyzing the effect of introducing a kurtosis parameter in Gaussian Bayesian networks

Gaussian Bayesian networks are graphical models that represent the dependence structure of a multivariate normal random variable with a directed acyclic graph (DAG). In Gaussian Bayesian networks the output is usually the conditional distribution of some unknown variables of interest given a set of evidential nodes whose values are known. The problem of uncertainty about the assumption of normality is very common in applications. Thus a sensitivity analysis of the non-normality effect in our conclusions could be necessary. The aspect of non-normality to be considered is the tail behavior. In this line, the multivariate exponential power distribution is a family depending on a kurtosis parameter that goes from a leptokurtic to a platykurtic distribution with the normal as a mesokurtic distribution. Therefore a more general model can be considered using the multivariate exponential power distribution to describe the joint distribution of a Bayesian network, with a kurtosis parameter reflecting deviations from the normal distribution. The sensitivity of the conclusions to this perturbation is analyzed using the Kullback–Leibler divergence measure that provides an interesting formula to evaluate the effect.

An alternative multivariate skew Laplace distribution: properties and estimation

A special case of the multivariate exponential power distribution is considered as a multivariate extension of the univariate symmetric Laplace distribution. In this paper, we focus on this multivariate symmetric Laplace distribution, and extend it to a multivariate skew distribution. We call this skew extension of the multivariate symmetric Laplace distribution the “multivariate skew Laplace (MSL) distribution” to distinguish between the asymmetric multivariate Laplace distribution proposed by Kozubowski and Podgorski (Comput Stat 15:531–540, 2000a) Kotz et al. (The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Chap. 6. Birkhauser, Boston, 2001) and Kotz et al. (An asymmetric multivariate Laplace Distribution, Working paper, 2003). One of the advantages of (MSL) distribution is that it can handle both heavy tails and skewness and that it has a simple form compared to other multivariate skew distributions. Some fundamental properties of the multivariate skew Laplace distribution are discussed. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of the multivariate skew Laplace distribution is given. Some examples are provided to demonstrate the modeling strength of the skew Laplace distribution.

A characterisation of scale mixtures of the uniform distribution

Certain univariate symmetric distributions are representable as scale mixtures of the uniform distribution (SMU). We give a characterisation theorem for the more general multivariate spherical distributions, and illustrate by application to the multivariate exponential power distribution.

Multivariate Exponential Power Distributions as Mixtures of Normal Distributions with Bayesian Applications

This paper shows that a multivariate exponential power distribution is a scale mixture of normal distributions, with respect to a probability distribution function, when its kurtosis parameter belongs to the interval (0, 1]. The corresponding mixing probability distribution function is presented. This result is used to design, through a Bayesian hierarchical model, an algorithm to generate samples of the posterior distribution; this is applied to a problem of quantitative genetics.

Robustness of Parameter Estimation Procedures in Multilevel Models When Random Effects are MEP Distributed

In this paper we examine maximum likelihood estimation procedures in multilevel models for two level nesting structures. Usually, for fixed effects and variance components estimation, level-one error terms and random effects are assumed to be normally distributed. Nevertheless, in some circumstances this assumption might not be realistic, especially as concerns random effects. Thus we assume for random effects the family of multivariate exponential power distributions (MEP); subsequently, by means of Monte Carlo simulation procedures, we study robustness of maximum likelihood estimators under normal assumption when, actually, random effects are MEP distributed.

On Parameters Estimation Procedures in Multilevel Models

We examine maximum likelihood estimation procedures in multilevel models relatedbto two-level hierarchically structured data. Usually, for fixed effects and variance components estimation, multivariate normal distribution is assumed. Here we consider for random effects multivariate exponential power distribution (MEP), which represents one of the possible generalizations of multivariate normal distribution. We examine robustness of maximum likelihood estimators under normal assumption when, indeed, random effects are MEP distributed. The study is conducted through MC simulation procedures.