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.     

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.     

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.     

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.     

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.     

Tail dependence of skew t-copulas

We examine tail behavior of skew t-copula in the bivariate case. The tail dependence coefficient is calculated for different skewing parameter values and compared with the corresponding coefficient for the t-copula. It is shown that depending on skewing parameter values, the tail dependence coefficient can differ considerably from the tail dependence of the t-copula. The speed of convergence of the estimator of tail dependence coefficient to its theoretical value is examined in a simulation experiment. Method of moments and maximum likelihood method are compared by simulation either. In the considered cases, maximum likelihood method converged faster to the theoretical value.     

The power series skew normal class of distributions

A class of power series skew normal distributions is introduced by generalizing the geometric skew normal distribution of Kundu. Various mathematical properties are derived and estimation addressed by the method of maximum likelihood. The data application of Kundu [Sankhyā B, 76, 2014, 167–189] is revisited and the proposed class is shown to provide a better fit.     

The inverse problem of multivariate and matrix-variate skew normal distributions

In this paper, we prove that the joint distribution of random vectors Z ₁ and Z ₂ and the distribution of Z ₂ are skew normal provided that Z ₁ is skew normally distributed and Z ₂ conditioning on Z ₁ is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.     

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.     

Moments and quadratic forms of matrix variate skew normal distributions

ABSTRACT

In 2007, Domínguez-Molina et al. obtained the moment generating function (mgf) of the matrix variate closed skew normal distribution. In this paper, we use their mgf to obtain the first two moments and some additional properties of quadratic forms for the matrix variate skew normal distributions. The quadratic forms are particularly interesting because they are essentially correlation tests that introduce a new type of orthogonality condition.     

Balakrishnan skew logistic distribution

Abstract

Based on Balakrishnan's (2002 Balakrishnan, N. (2002). Discussion on “Skew multivariate models related to hidden truncation and/or selective reporting” by B.C. Arnold and R.J. Beaver. Test 11: 37–39.[Web of Science ®] , [Google Scholar]) novel idea, a new skew logistic distribution is proposed. It contains Azzalini's skew logistic and standard logistic distributions as particular cases. Several mathematical properties of the proposed distribution (including characteristic function, cumulative distribution function, stochastic orderings, and simulation schemes) are derived. A real-data application is used to illustrate practical usefulness.     

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.     

Inference of R = P[X < Y] for skew normal distribution

This article studies the estimation of R = P[X < Y] when X and Y are two independent skew normal distribution with different parameters. When the scale parameter is unknown, the maximum likelihood estimator of R is proposed. The maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimation, and confidence interval of R are obtained when the common scale parameter is known. In the general case, the maximum likelihood estimator of R is also discussed. To compare the different proposed methods, Monte Carlo simulations are performed. At last, the analysis of a real dataset has been presented for illustrative purposes too.     

The density of the skew normal sample mean and its applications

This paper focuses on the distribution of the skew normal sample mean. For a random sample drawn from a skew normal population, we derive the density function and the moment generating function of the sample mean. The density function derived can be used for statistical inference on the disease occurrence time of twins in epidemiology, in which the skew normal model plays a key role.     

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.     

A multivariate two-factor skew model

It is well known that many data, such as the financial or demographic data, exhibit asymmetric distributions. In recent years, researchers have concentrated their efforts to model this asymmetry. Skew normal model is one of such models that are skew and yet possess many properties of the normal model. In this paper, a new multivariate skew model is proposed, along with its statistical properties. It includes the multivariate normal distribution and multivariate skew normal distribution as special cases. The quadratic form of this random vector follows a χ² distribution. The roles of the parameters in the model are investigated using contour plots of bivariate densities.     

A method to estimate power parameter in exponential power distribution via polynomial regression

The exponential power distribution (EPD), also known as generalized error distribution, is a flexible symmetrical unimodal family that belongs to the exponential one. The EPD becomes the density function of a range of symmetric distributions with different values of its power parameter β. A closed-form estimator for β does not exist, so the power parameter is usually estimated numerically. Unfortunately, the optimization algorithms do not always converge, especially when the true value of β is close to its parametric space frontier. In this paper, we present an alternative method to estimate β. Our proposal is based on the normal standardized Q–Q plot, and it exploits the relationship between β and the kurtosis. Furthermore, it is a direct method which does not require computational efforts nor the use of optimization algorithms.     

A multimodal skewed extension of normal distribution: its properties and applications

A multimodal skewed extension of normal distribution is proposed by applying the general method as in [Huang WJ, Chen YH. Generalized skew-Cauchy distribution. Stat Probab Lett. 2007;77:1137–1147] for the construction of skew-symmetric distributions by using a trigonometric periodic skew function. Some of its distributional properties are investigated. Properties of maximum likelihood estimation of the parameters are studied numerically by simulation. The suitability of the proposed distribution in empirical data modelling is investigated by carrying out comparative fitting of two real-life data sets.     

The beta generalized linear exponential distribution

In this paper, a new five-parameter lifetime distribution called beta generalized linear exponential distribution (BGLED) is introduced. It includes at least 17 popular sub-models as special cases such as the beta linear exponential, the beta generalized exponential, and the exponentiated generalized linear distributions. Mathematical and statistical properties of the proposed distribution are discussed in details. In particular, explicit expression for the density function, moments, asymptotics distributions for sample extreme statistics, and other statistical measures are obtained. The estimation of the parameters by the method of maximum-likelihood is discussed and the finite sample properties of the maximum-likelihood estimators (MLEs) are investigated numerically. A real data set is used to demonstrate its superior performance fit over several existing popular lifetime models.