Tail dependence for skew Laplace distribution and skew Cauchy distribution

ABSTRACT

Coefficient of tail dependence measures the strength of dependence in the tail of a bivariate distribution and it has been found useful in the risk management. In this paper, we derive the upper tail dependence coefficient for a random vector following the skew Laplace distribution and the skew Cauchy distribution, respectively. The result shows that skew Laplace distribution is asymptotically independent in upper tail, however, skew Cauchy distribution has asymptotic upper tail dependence.     

The flexible skew Laplace distribution

ABSTRACT

Skew-symmetric distributions have been discussed by several research-ers. In this article we construct a skew-symmetric Laplace distribution, which is the generalization of distribution given by Ali et al. (2009 Ali, M., Pal, M., Woo, J. (2009). Skewed reflected distributions generated by the Laplace kernel. Aust. J. Statist. 38:45–58. [Google Scholar]) and Nekoukhou and Alamatsaz (2012 Nekoukhou, V., Alamatsaz, M.H. (2012). A family of skew-symmetric-Laplace distributions. Statist. Papers. 53(3):685–696.[Crossref], [Web of Science ®] , [Google Scholar]). This new distribution contains more parameters, and this induces flexibility properties, such as unimodality or bimodality. We study on some properties of this distribution. In the last section we also provide an application with a real data. Concerning example has recently been discussed by Nekoukhou et al. (2013 Nekoukhou, V., Alamatsaz, M.H., Aghajani, A.H. (2013). A flexible skew-generalized normal distribution. Commun. Statist. Theory Methods. 42(13):2324–2334.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to apply to their model. We compare the behavior of our distribution to their distribution on this example.     

An alternative skew exponential power distribution formulation

In this note we propose a newly formulated skew exponential power distribution that behaves substantially better than previously defined versions. This new model performs very well in terms of the large sample behavior of the maximum likelihood estimation procedure when compared to the classically defined four parameter model defined by Azzalini. More recently, approaches to defining a skew exponential power distribution have used five or more parameters. Our approach improves upon previous attempts to extend the symmetric power exponential family to include skew alternatives by maintaining a minimum set of four parameters corresponding directly to location, scale, skewness and kurtosis. We illustrate the utility of our proposed model using translational and clinical data sets.     

Statistical applications of the multivariate skew normal distribution

Azzalini and Dalla Valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.     

A new family of multivariate skew slash distribution

In this article, the new family of multivariate skew slash distribution is defined. According to the definition, a stochastic representation of the multivariate skew slash distribution is derived. The first four moments and measures of skewness and kurtosis of a random vector with the multivariate skew slash distribution are obtained. The distribution of quadratic forms for the multivariate skew slash distribution and the non central skew slash χ² distribution are studied. Maximum likelihood inference and real data illustration are discussed. In the end, the potential extension of multivariate skew slash distribution is discussed.     

Bayesian analysis of multivariate stochastic volatility with skew return distribution

Multivariate stochastic volatility models with skew distributions are proposed. Exploiting Cholesky stochastic volatility modeling, univariate stochastic volatility processes with leverage effect and generalized hyperbolic skew t-distributions are embedded to multivariate analysis with time-varying correlations. Bayesian modeling allows this approach to provide parsimonious skew structure and to easily scale up for high-dimensional problem. Analyses of daily stock returns are illustrated. Empirical results show that the time-varying correlations and the sparse skew structure contribute to improved prediction performance and Value-at-Risk forecasts.     

Parameter estimation for mixtures of skew Laplace normal distributions and application in mixture regression modeling

In this article, we propose mixtures of skew Laplace normal (SLN) distributions to model both skewness and heavy-tailedness in the neous data set as an alternative to mixtures of skew Student-t-normal (STN) distributions. We give the expectation–maximization (EM) algorithm to obtain the maximum likelihood (ML) estimators for the parameters of interest. We also analyze the mixture regression model based on the SLN distribution and provide the ML estimators of the parameters using the EM algorithm. The performance of the proposed mixture model is illustrated by a simulation study and two real data examples.     

The alpha–beta skew normal distribution: properties and applications

This paper introduces a new class of skew distributions by extending the alpha skew normal distribution proposed by Elal-Olivero [Elal-Olivero, D. Alpha-skew-normal distribution. Proyecciones. 2010;29:224–240]. Statistical properties of the new family are studied in details. In particular, explicit expressions for the moments and the shape parameters including the skewness and the kurtosis coefficients and the moment generating function are derived. The problem of estimating parameters on the basis of a random sample coming from the new class of distribution is considered. To examine the performance of the obtained estimators, a Monte Carlo simulation study is conducted. Flexibility and usefulness of the proposed family of distributions are illustrated by analysing three real data sets.     

A robust multivariate Birnbaum–Saunders distribution: EM estimation

We propose here a robust multivariate extension of the bivariate Birnbaum–Saunders (BS) distribution derived by Kundu et al. [Bivariate Birnbaum–Saunders distribution and associated inference. J Multivariate Anal. 2010;101:113–125], based on scale mixtures of normal (SMN) distributions that are used for modelling symmetric data. This resulting multivariate BS-type distribution is an absolutely continuous distribution whose marginal and conditional distributions are of BS-type distribution of Balakrishnan et al. [Estimation in the Birnbaum–Saunders distribution based on scalemixture of normals and the EM algorithm. Stat Oper Res Trans. 2009;33:171–192]. Due to the complexity of the likelihood function, parameter estimation by direct maximization is very difficult to achieve. For this reason, we exploit the nice hierarchical representation of the proposed distribution to propose a fast and accurate EM algorithm for computing the maximum likelihood (ML) estimates of the model parameters. We then evaluate the finite-sample performance of the developed EM algorithm and the asymptotic properties of the ML estimates through empirical experiments. Finally, we illustrate the obtained results with a real data and display the robustness feature of the estimation procedure developed here.     

Value-at-risk estimation with new skew extension of generalized normal distribution

In this paper, we introduce a new distribution, called the alpha-skew generalized normal (ASGN), for GARCH models in modeling daily Value-at-Risk (VaR). Basic structural properties of the proposed distribution are derived including probability and cumulative density functions, moments and stochastic representation. The real data application based on ISE-100 index is given to show the performance of GARCH model specified under ASGN innovation distribution with respect to normal, Student’s-t, skew normal and generalized normal models in terms of the VaR accuracy. The empirical results show that GARCH model with ASGN innovation distribution generates the most accurate VaR forecasts for all confidence levels.     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.     

An asymmetric multivariate weibull distribution

Abstract

A class of multivariate laws as an extension of univariate Weibull distribution is presented. A well known representation of the asymmetric univariate Laplace distribution is used as the starting point. This new family of distributions exhibits some similarities to the multivariate normal distribution. Properties of this class of distributions are explored including moments, correlations, densities and simulation algorithms. The distribution is applied to model bivariate exchange rate data. The fit of the proposed model seems remarkably good. Parameters are estimated and a bootstrap study performed to assess the accuracy of the estimators.     

An alternative approach to multivariate EWMA control chart

This Paper proposes a multivariate EWMA scheme that is alternative to the traditional EWMA-M. The distribution of the chart statistic is derived from Box quadratic form and the sensitivity of the chart is examined. The average run lengths of the M-EWMA scheme are numerically computed with the integral equation method. The exponential weight of 0.2 is found to be the optimal choice for the sensitive chart to detect assignable causes in the mean vector of processes.     

A continuous general multivariate distribution and its properties

Starting from two known continuous univariate distributions, a bivariate distribution is constructed depending on a parameter which measures the degree of stochastic dependence between the two random variables. From the foregoing construction we then pass to a multivariate-type distribution, constructed using only univariate distributions and an association matrix. Some properties of the multivariate and bivariate case are studied.     

A generalized two-piece skew normal distribution and some of its properties

In this paper, we develop a generalized version of the two-piece skew normal distribution of Kim [On a class of two-piece skew-normal distributions, Statistics 39(6) (2005), pp. 537–553] and derive explicit expressions for its distribution function and characteristic function and discuss some of its important properties. Further estimation of the parameters of the generalized distribution is carried out.     

Heteroscedastic and heavy-tailed regression with mixtures of skew Laplace normal distributions

Joint modelling skewness and heterogeneity is challenging in data analysis, particularly in regression analysis which allows a random probability distribution to change flexibly with covariates. This paper, based on a skew Laplace normal (SLN) mixture of location, scale, and skewness, introduces a new regression model which provides a flexible modelling of location, scale and skewness parameters simultaneously. The maximum likelihood (ML) estimators of all parameters of the proposed model via the expectation-maximization (EM) algorithm as well as their asymptotic properties are derived. Numerical analyses via a simulation study and a real data example are used to illustrate the performance of the proposed model.     

A skew logistic distribution as an alternative to the model of van Staden and King

The logistic distribution is a simple distribution possessing many useful properties and has been used extensively for analyzing growth. Recently, van Staden and King proposed a quantile-based skew logistic distribution. In this paper, we introduce an alternative skew logistic distribution. We then establish recurrence relations for the computation of the single and product moments of order statistics from the standard skew logistic distribution by using the moments of order statistics from the standard half logistic distribution. These enable an efficient computation of means, variances and covariances of order statistics from the skew logistic distibution for all sample sizes. The results become useful in determining the best linear unbiased estimators of the location and scale paramters of the skew logistic distribution. Finally, we provide an example to illustrate the usefulness of the developed model and then compare its fit with that provided by the model of van Staden and King.     

Shrinkage estimation of location parameters in a multivariate skew-normal distribution

Abstract

This paper studies decision theoretic properties of Stein type shrinkage estimators in simultaneous estimation of location parameters in a multivariate skew-normal distribution with known skewness parameters under a quadratic loss. The benchmark estimator is the best location equivariant estimator which is minimax. A class of shrinkage estimators improving on the best location equivariant estimator is constructed when the dimension of the location parameters is larger than or equal to four. An empirical Bayes estimator is also derived, and motivated from the Bayesian procedure, we suggest a simple skew-adjusted shrinkage estimator and show its dominance property. The performances of these estimators are investigated by simulation.