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.     

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.     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

A new generalized Balakrishnan type skewed–normal distribution: properties and associated inference

It is also shown that our proposed skew-normal model subsumes many other well-known skew-normal model that exists in the literature. Recent work on a new two-parameter generalized skew-normal model has received a lot of attention. This paper presents a new generalized Balakrishnan type skew–normal distribution by introducing two shape parameters. We also provide some useful results for this new generalization. It is also shown that our proposed skew–normal model subsumes the original Balakrishnan skew–normal model (2002) as well as other well–known skew–normal models as special cases. The resulting flexible model can be expected to fit a wider variety of data structures than either of the models involving a single skewing mechanism. For illustrative purposes, a famed data set on IQ scores has been used to exhibit the efficacy of the proposed model.     

On the distribution of quotient of random variables conditioned to the positive quadrant

Abstract

Motivated by Caginalp and Caginalp [Physica A—Statistical Mechanics and Its Applications, 499, 2018, 457–471], we derive the exact distribution of X/Y conditioned on X?>?0, Y?>?0 for more than ten classes of distributions, including the bivariate t, bivariate Cauchy, bivariate Lomax, Arnold and Strauss’ bivariate exponential, Balakrishna and Shiji’s bivariate exponential, Mohsin et al.’s bivariate exponential, Morgenstern type bivariate exponential, bivariate gamma exponential and bivariate alpha skew normal distributions. The results can be useful in finance and other areas.     

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.     

Calibration model with skew scale mixtures of normal distributions

This work presents a new linear calibration model with replication by assuming that the error of the model follows a skew scale mixture of the normal distributions family, which is a class of asymmetric thick-tailed distributions that includes the skew normal distribution and symmetric distributions. In the literature, most calibration models assume that the errors are normally distributed. However, the normal distribution is not suitable when there are atypical observations and asymmetry. The estimation of the calibration model parameters are done numerically by the EM algorithm. A simulation study is carried out to verify the properties of the maximum likelihood estimators. This new approach is applied to a real dataset from a chemical analysis.     

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.     

关于斜Laplace分布与Levy偏稳定分布的性质

目前有关重尾或偏态数据的统计分析和理论模型相对较少,基于传统的Laplace分布,提出一种处理偏态和重尾数据的新模型——斜Laplace分布,以研究其参数估计方法。利用数理统计知识推导出该分布与一些常见分布(如正态分布、指数分布)间的统计关系,并给出一种可通过设置不同参数值得到不同分布的Levy偏稳定分布及其稳定性。     

Characteristic function of the SGT distribution

An explicit closed form is derived for the characteristic function for the skew generalized t distribution studied by Arslan and Genç [The skew generalized t (SGT) distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation, Statistics 43(5) (2009), pp. 481–498]. The expression involves the Wright generalized hypergeometric Ψ–function.     

Modified skew-slash distribution

Abstract

In this work, we introduce a new skewed slash distribution. This modification of the skew-slash distribution is obtained by the quotient of two independent random variables. That quotient consists on a skew-normal distribution divided by a power of an exponential distribution with scale parameter equal to two. In this way, the new skew distribution has a heavier tail than that of the skew-slash distribution. We give the probability density function expressed by an integral, but we obtain some important properties useful for making inferences, such as moment estimators and maximum likelihood estimators. By way of illustration and by using real data, we provide maximum likelihood estimates for the parameters of the modified skew-slash and the skew-slash distributions. Finally, we introduce a multivariate version of this new distribution.     

Skew Gaussian Process for Nonlinear Regression

In this article, we extend the Gaussian process for regression model by assuming a skew Gaussian process prior on the input function and a skew Gaussian white noise on the error term. Under these assumptions, the predictive density of the output function at a new fixed input is obtained in a closed form. Also, we study the Gaussian process predictor when the errors depart from the Gaussianity to the skew Gaussian white noise. The bias is derived in a closed form and is studied for some special cases. We conduct a simulation study to compare the empirical distribution function of the Gaussian process predictor under Gaussian white noise and skew Gaussian white noise.     

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 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 generalized Gudermannian distribution: inference and volatility modelling

In this paper, we introduce a new distribution, called generalized Gudermannian (GG) distribution, and its skew extension for GARCH models in modelling daily Value-at-Risk (VaR). Basic structural properties of the proposed distribution are obtained including probability density and cumulative distribution functions, moments, and stochastic representation. The maximum likelihood method is used to estimate unknown parameters of the proposed model and finite sample performance of maximum likelihood estimates are evaluated by means of Monte-Carlo simulation study. The real data application on Nikkei 225 index is given to demonstrate the performance of GARCH model specified under skew extension of GG innovation distribution against normal, Student's-t, skew normal and generalized error and skew generalized error distributions in terms of the accuracy of VaR forecasts. The empirical results show that the GARCH model with GG innovation distribution produces the most accurate VaR forecasts for all confidence levels.     

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.     

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 Podgórski (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. Birkhäuser, 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.     

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 new absolute continuous bivariate generalized exponential distribution

The generalized exponential is the most commonly used distribution for analyzing lifetime data. This distribution has several desirable properties and it can be used quite effectively to analyse several skewed life time data. The main aim of this paper is to introduce absolutely continuous bivariate generalized exponential distribution using the method of Block and Basu (1974). In fact, the Block and Basu exponential distribution will be extended to the generalized exponential distribution. We call the new proposed model as the Block and Basu bivariate generalized exponential distribution, then, discuss its different properties. In this case the joint probability distribution function and the joint cumulative distribution function can be expressed in compact forms. The model has four unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. To compute the maximum likelihood estimators directly, one needs to solve a four dimensional optimization problem. The EM algorithm has been proposed to compute the maximum likelihood estimations of the unknown parameters. One data analysis is provided for illustrative purposes. Finally, we propose some generalizations of the proposed model and compare their models with each other.     

Geeta distribution and its properties

A new discrete distribution defined over all the positive integers and with the name of Geeta distribution is described. It is L-shaped like the logarithmic series distribution, Yule distribution and the discrete Pareto distribution but is far more versatile than them as it has two parameters. It belongs to the classes of location parameter distributions, modified power series distributions, Lagrange series distributions and exponential distributions. Its mean fi, variance a2 and two recurrence formulae for higher central moments are obtained. Convolution theorem and variations in the model with changes in the parameters have been considered. ML estimators, MVU estimators and estimators based of mean and variance and on mean and first frequency have been derived.