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Modelling skewness and heavy-tailedness in heterogeneous data sets is a compelling problem, particularly in regression analysis. The main goal of this study is to propose a mixture regression model based on the shape mixtures of skew Laplace normal (SMSLN) distribution for modelling skewness and heavy-tailedness simultaneously. The SMSLN distribution has been introduced by Doğru and Arslan [Finite mixtures of skew Laplace normal distributions with random skewness. 11th International Statistics Congress (ISC2019); Bodrum/Turkey; Finite mixtures of skew Laplace normal distributions with random skewness. Comput Stat. 2021;36(1):423–447] as a flexible extension of the skew Laplace normal (SLN) distribution and includes an extra shape parameter that controls skewness and kurtosis. The maximum likelihood estimators for the parameters of interest with the help of the expectation-maximization (EM) algorithm are obtained. The performance of the proposed mixture model is demonstrated via a simulation study and a real data example. [ABSTRACT FROM AUTHOR]

In this paper, the shape mixtures of the skew Laplace normal (SMSLN) distribution is introduced as a flexible extension of the skew Laplace normal distribution which is also a heavy-tailed distribution. The SMSLN distribution includes an extra shape parameter, which controls skewness and kurtosis. Some distributional properties of this distribution are derived. Besides, we propose finite mixtures of SMSLN distributions to model both skewness and heavy-tailedness in heterogeneous data sets. The maximum likelihood estimators for parameters of interests are obtained via the expectation–maximization algorithm. We also give a simulation study and examine a real data example for the numerical illustration of proposed estimators. [ABSTRACT FROM AUTHOR]

Variable selection in elliptical Linear Mixed Models (LMMs) with a shrinkage penalty function (SPF) is the main scope of this study. SPFs are applied for parameter estimation and variable selection simultaneously. The smoothly clipped absolute deviation penalty (SCAD) is one of the SPFs and it is adapted into the elliptical LMM in this study. The proposed idea is highly applicable to a variety of models which are set up with different distributions such as normal, student-t, Pearson VII, power exponential and so on. Simulation studies and real data example with one of the elliptical distributions show that if the variable selection is also a concern, it is worthwhile to carry on the variable selection and the parameter estimation simultaneously in the elliptical LMM. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Linear mixed modeling (LMM) is a comprehensive technique used for clustered, panel and longitudinal data. The main assumption of classical LMM is having normally distributed random effects and error terms. However, there are several situations for that we need to use heavier tails distributions than the (multivariate) normal to handle outliers and/or heavy tailness in data. In this study, we focus on LMM using the multivariate Laplace distribution which is known as the heavy tailed alternative to the normal distribution. The parameter estimators of interest are generated with EM algorithm for the proposed model. A simulation study is provided to illustrate the performance of the Laplace distribution over the normal distribution for LMM. Also, a real data example is used to explore the behavior of the proposed estimators over the counterparts. [ABSTRACT FROM AUTHOR]

In the analysis of repeated or clustered measurements, it is crucial to determine the dynamics that affect the mean, variance, and correlations of the data, which will be possible using appropriate models. One of these models is the joint mean–covariance model, which is a multivariate heteroscedastic regression model with autoregressive covariance structures. In these models, parameter estimation is usually carried on under normality assumption, but the resulting estimators will be very sensitive to the outliers or non-normality of data. In this study, we propose a robust alternative method and an EM algorithm for estimating the parameters of joint mean–covariance models. Robustification is achieved using a multivariate heavy-tailed distribution with the same number of parameters as the multivariate normal distribution. To simplify the estimation procedure, a modified Cholesky decomposition is adopted to factorize the dependence structure in terms of unconstrained autoregressive and scale innovation parameters. Also, the technique for the prediction of future responses is given. An intensive simulation study and a real data example are provided to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

One-stepM(OSM)-estimator needs some initial/preliminary estimates at the beginning of the calculation process. In this study, we propose to use new initial estimates for the calculation of the OSM-estimator. We consider simple location and simple linear regression models when the distribution of the error terms is Jones and Faddy's skewedt. Monte-Carlo simulation study shows that the OSM estimator(s) based on the proposed initial estimates is/are more efficient than the OSM estimator(s) based on the traditional initial estimates especially for the skewed cases. We also analyze some real data sets taken from the literature at the end of the paper. [ABSTRACT FROM PUBLISHER]

This article proposes a new family of multivariate distributions as the variance-mean mixture of the multivariate Kotz-type distribution with the extended generalized inverse Gaussian as the mixing distribution. Definition and some of the main properties are given. [ABSTRACT FROM AUTHOR]

Finite mixtures of multivariate t distributions (Peel and McLachlan (2000)) were introduced as an alternative to the finite mixtures of multivariate normal distributions to model data sets with heavy tails. In this study, we define the finite mixtures of matrix variate t distributions as an extension of finite mixtures of multivariate t distributions. Mixtures of matrix variate t distributions can provide an alternative robust model to the mixtures of matrix variate normal distributions (Viroli (2011)) for modeling matrix variate data sets with heavy tails. We give an Expectation Maximization (EM) algorithm to find the maximum likelihood (ML) estimators for the parameters of interest. We also provide a small simulation study to illustrate the performance of the proposed EM algorithm for finding estimates. [ABSTRACT FROM AUTHOR]