Skew Laplace Distribution Research Articles

Alternative skew Laplace scale mixtures for modeling data exhibiting high-peaked and heavy-tailed traits

The search and construction of appropriate and flexible models for describing and modelling empirical data sets incongruent with normality retains a sustained interest. This paper focuses on proposing flexible skew Laplace scale mixture distributions to model these types of data sets. Each member of the collection of distributions is obtained by dividing the scale parameter of a conditional skew Laplace distribution by a purposefully chosen mixing random variable. Highly-peaked, heavy-tailed skew models with relevance and impact in different fields are obtained and investigated, and elegant sampling schemes to simulate from this collection of developed models are proposed. Finite mixtures consisting of the members of the skew Laplace scale mixture models are illustrated, further extending the flexibility of the distributions by being able to account for multimodality. The maximum likelihood estimates of the parameters for all the members of the developed models are described via a developed EM algorithm. Real-data examples highlight select models’ performance and emphasize their viability compared to other commonly considered candidates, and various goodness-of-fit measures are used to endorse the performance of the proposed models as reasonable and viable candidates for the practitioner. Finally, an outline is discussed for future work in the multivariate realm for these models.

Tests for skewness parameter of skew log Laplace distribution

Laplace probability density function with additional shape parameter that regulates the degree of skewness is a skew Laplace distribution. The various forms of skew Laplace distribution are found in the literature, the distributions defined by Mc Gill (1962), Holla and Bhattacharya (1968), Lingappaiah (1988), Fernandez and Steel (1998). The skew log Laplace distribution is the probability distribution of a random variable whose logarithm follows a skew Laplace distribution. In this paper, the classical optimum tests for skewness parameter of skew log Laplace distribution (SLLD) derived from Lingappaiah (1988) distribution are discussed. Uniformly most powerful test, uniformly most powerful unbiased test and Wald’s sequential probability ratio test for skewness parameter are compared. The exact likelihood ratio test and Neyman structure test for testing skewness parameter when scale parameter is known are derived. Finally, the underreported income of Road Transport Company is analysed on the basis of the tests derived in this paper.

Modified Quantile Regression for Modeling the Low Birth Weight

This study aims to identify the best model of low birth weight by applying and comparing several methods based on the quantile regression method's modification. The birth weight data is violated with linear model assumptions; thus, quantile approaches are used. The quantile regression is adjusted by combining it with the Bayesian approach since the Bayesian method can produce the best model in small size samples. Three kinds of the modified quantile regression methods considered here are the Bayesian quantile regression, the Bayesian Lasso quantile regression, and the Bayesian Adaptive Lasso quantile regression. This article implements the skewed Laplace distribution as the likelihood function in Bayesian analysis. The cross-sectional study collected the primary data of 150 birth weights in West Sumatera, Indonesia. This study indicated that Bayesian Adaptive Lasso quantile regression performed well compared to the other two methods based on a smaller absolute bias and a shorter Bayesian credible interval based on the simulation study. This study also found that the best model of birth weight is significantly affected by maternal education, the number of pregnancy problems, and parity.

On transmuted skewed Laplace distribution and its application

The aim of this study is to add a shape parameter into the existing 3-parameter skewed Laplace distribution using the method of transmutation and apply the proposed distribution on two life simulated data generated using R-statistical software. The parameters of the proposed distribution are obtained using the method of maximum likelihood estimation. The fitness of the proposed distribution was compared with that of existing skewed Laplace distribution and four parameter Weibull distribution using measures of fitness criteria, and the results obtained showed that the proposed model can be used effectively to model real life situations than the Skewed Laplace Distribution. The results also showed that the Four Parameter Weibull Distribution which is a competing model performed better than the Skewed Laplace Distribution and Transmuted Skewed Laplace Distribution.

Multivariate tail covariance risk measure for generalized skew-elliptical distributions

In this paper, the multivariate tail covariance (MTCov) for generalized skew-elliptical distributions is considered. Some special cases for this distribution, such as generalized skew-normal, generalized skew student-t, generalized skew-logistic and generalized skew-Laplace distributions, are also considered. In order to test the theoretical feasibility of our results, the MTCov for skewed and non skewed normal distributions is computed and compared. Finally, we give a special formula of the MTCov for generalized skew-elliptical distributions.

Extended Generalized Skew Laplace Random Field: Spatial Autoregressive and Moving Average Model for Prediction of Missing Data in Skew and Heavy Tailed Data

In this paper, we define a spatial skew and heavy-tailed random field by an extended version of multivariate generalized skew Laplace distribution. The Bayesian spatial regression model is developed to explain the spatial data. A simulation study is then carried out to validate and evaluate the performance of the proposed model. The application of this model is also demonstrated in an analysis of a geological real data set.

Generalized Skew Laplace Random Fields: Bayesian Spatial Prediction for Skew and Heavy Tailed Data

Earlier works on spatial prediction issue often assume that the spatial data are realization of Gaussian random field. However, this assumption is not applicable to the skewed and kurtosis distributed data. The closed skew normal distribution has been used in these circumstances. As another alternative, we apply generalized skew Laplace distributions for defining a skew and heavy tailed random field for Bayesian prediction. Simulation study and a real problem are then applied to evaluate the performance of this model.

TAIL CONDITIONAL EXPECTATIONS FOR GENERALIZED SKEW-ELLIPTICAL DISTRIBUTIONS

This paper deals with the multivariate tail conditional expectation (MTCE) for generalized skew-elliptical distributions. We present tail conditional expectation for univariate generalized skew-elliptical distributions and MTCE for generalized skew-elliptical distributions. There are many special cases for generalized skew-elliptical distributions, such as generalized skew-normal, generalized skew Student-t, generalized skew-logistic and generalized skew-Laplace distributions.

Rényi Entropy for Mixture Model of Multivariate Skew Laplace distributions

Rényi entropy is an important concept developed by Rényi in information theory. In this paper, we study in detail this measure of information in cases multivariate skew Laplace distributions then we extend this study to the class of mixture model of multivariate skew Laplace distributions. The upper and lower bounds of Rényi entropy of mixture model are determined. In addition, an asymptotic expression for Rényi entropy is given by the approximation. Finally, we give a real data example to illustrate the behavior of entropy of the mixture model under consideration.

Tail variance for Generalized Skew-Elliptical distributions

Notable changes in financial markets have required the development of a standard structure for risk measurement, and obtaining an appropriate risk measurement from historical data is the challenge that is addressed in this article. In recent years, insurance and investment experts are interested in focusing on the use of the tail conditional expectation (TCE) because it has usable and desirable features in different situations. It is well-known that the tail conditional expectation as a risk measurement provides information about the mean of the tail of the loss distribution, while the tail variance (TV) measures the deviation of the loss from this mean along the tail of the distribution. In this paper, we present a theorem that extends the tail variance formula from the elliptical distributions to a rich class of Generalized Skew-Elliptical (GSE) distributions. We develop this theory for the four main classes of skew-elliptical distributions, including the Skew-Normal, Skew-Student-t, Skew-Logistic and Skew-Laplace distributions and obtain the proposed TV measure for them.

Bayesian single-index quantile regression for ordinal data

A Bayesian single-index quantile estimation approach for ordinal data is proposed. A simple and efficient MCMC algorithm was developed for posterior computation using a normal-exponential mixture representation of the skewed Laplace distribution. The proposed method is then demonstrated via simulated studies and two real data sets. Results show that the proposed method performs well under the simulated studies and real data analysis.

Gene selection for microarray gene expression classification using Bayesian Lasso quantile regression

Gene selection has been proven to be an effective way to improve the results of many classification methods. However, existing gene selection techniques in binary classification regression are sensitive to outliers of the data, heteroskedasticity or other anomalies of the latent response. In this paper, we propose a new Bayesian hierarchical model to overcome these problems in a relatively straightforward way. In particular, we propose a new Bayesian Lasso method that employs a skewed Laplace distribution for the errors and a scaled mixture of uniform distribution for the regression parameters, together with Bayesian MCMC estimation. Comprehensive comparisons between our proposed gene selection method and other competitor methods are performed experimentally, depending on four benchmark gene expression datasets. The experimental results prove that the proposed method is very effective for selecting the most relevant genes with high classification accuracy.

Bayesian composite Tobit quantile regression

ABSTRACTComposite quantile regression models have been shown to be effective techniques in improving the prediction accuracy [H. Zou and M. Yuan, Composite quantile regression and the oracle model selection theory, Ann. Statist. 36 (2008), pp. 1108–1126; J. Bradic, J. Fan, and W. Wang, Penalized composite quasi-likelihood for ultrahighdimensional variable selection, J. R. Stat. Soc. Ser. B 73 (2011), pp. 325–349; Z. Zhao and Z. Xiao, Efficient regressions via optimally combining quantile information, Econometric Theory 30(06) (2014), pp. 1272–1314]. This paper studies composite Tobit quantile regression (TQReg) from a Bayesian perspective. A simple and efficient MCMC-based computation method is derived for posterior inference using a mixture of an exponential and a scaled normal distribution of the skewed Laplace distribution. The approach is illustrated via simulation studies and a real data set. Results show that combine information across different quantiles can provide a useful method in efficient statistical estimation. This is the first work to discuss composite TQReg from a Bayesian perspective.

Bayesian Analysis of Discrete Skewed Laplace Distribution

The discrete skewed Laplace distribution is a flexible distribution with integer domain and simple closed form that can be applied to model count data. Parameters are estimated under empirical Bayes (EB) analysis and comparison are made between the Bayesian parameter estimation and classical parameter estimation, i.e. the maximum likelihood (ML) approach. The results show that the Bayesian parameter estimations are preferable.

Bayesian Analysis of Composite Quantile Regression

This paper introduces a Bayesian approach for composite quantile regression employing the skewed Laplace distribution for the error distribution. We use a two-level hierarchical Bayesian model for coefficient estimation and future selection which assumes a prior distribution that favors sparseness. An efficient Gibbs sampling algorithm is developed to update the unknown quantities from the posteriors. The proposed approach is illustrated via simulation studies and two real datasets. Results indicate that the proposed approach performs quite good in comparison to the other approaches.

Tail dependence for skew Laplace distribution and skew Cauchy distribution

ABSTRACTCoefficient 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.

Variance-mean mixture of the multivariate skew normal distribution

In this paper, we introduce a new class of multivariate distributions as an extension of the normal variance–mean mixture distributions class. The new class results from a variance-mean mixture of the skew normal and the generalized inverse Gaussian distributions. The new class is very flexible in terms of heavy tails and skewness and many of the widely used distributions, such as generalized hyperbolic, skew t, and skew Laplace distributions are included as special or limiting cases of the new class. An explicit expression for the density function of the new class is given and some of its distributional properties, such as moment generating function, linear transformations, quadratic forms, marginal and conditional distributions are examined. We give a simulation algorithm to generate random variates from the new class and propose an EM algorithm for maximum likelihood estimation of its parameters. We provide some examples to demonstrate the modeling strength of the proposed class.

Confidence limits for genome DNA copy number variations in HR-CGH array measurements

Estimation of the genome copy number variations (CNVs) measured using the high-resolution array-comparative genomic hybridization (HR-CGH) microarray is commonly provided in the presence of large Gaussian noise having white properties with different segmental variances. Medical experts must thus be highly concerned about the confidence limits for CNVs in order to make correct decisions about genomic changes. We carry out a probabilistic analysis of CNVs in HR-CGH microarray measurements and show that jitter in the breakpoints can be approximated with the discrete skew Laplace distribution. Using this distribution, we find the confidence upper and lower boundaries to guarantee an existence of genomic changes in the confidence interval of 99.73%. We suggest combining these boundaries with the estimates to give medical experts more information about actual CNVs. Experimental verification of the theory is provided by simulation and using real HR-CGH microarray-based measurements.

A global optimisation approach for parameter estimation of a mixture of double Pareto lognormal and lognormal distributions

The double Pareto Lognormal (dPlN) statistical distribution, defined in terms of both an exponentiated skewed Laplace distribution and a lognormal distribution, has proven suitable for fitting heavy tailed data. In this work we investigate inference for the mixture of a dPlN component and (k−1) lognormal components for k fixed, a model for extreme and skewed data which additionally captures multimodality.The optimisation criterion based on the likelihood maximisation is considered, which yields a global optimisation problem with an objective function difficult to evaluate and optimise. Variable Neighbourhood Search (VNS) is proven to be a powerful tool to overcome such difficulties. Our approach is illustrated with both simulated and real data, in which our VNS and a standard multistart are compared. The computational experience shows that the VNS is more stable numerically and provides slightly better objective values.

Tail Conditional Expectations for Generalized Skew — Elliptical Distributions

Tail conditional expectation (TCE) has properties which are considered desirable and applicable in a variety of situations. (In particular, it satis es requirements of a coherent risk measure in the spirit developed by Artzner et al. (1999)). Consequently, there has been growing interest among risk measures, actuaries, and investment experts in this risk measure. This paper derives formulae for computing the TCE for Generalized Skew Elliptical distributions, a family of non-symmetric distributions that includes, among many, the more familiar skew normal, skew student-t and skew laplace distributions.