A robust multivariate measurement error model with skew-normal/independent distributions and Bayesian MCMC implementation

Skew-normal/independent distributions are a class of asymmetric thick-tailed distributions that include the skew-normal distribution as a special case. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis in multivariate measurement errors models. We propose the use of skew-normal/independent distributions to model the unobserved value of the covariates (latent variable) and symmetric normal/independent distributions for the random errors term, providing an appealing robust alternative to the usual symmetric process in multivariate measurement errors models. Among the distributions that belong to this class of distributions, we examine univariate and multivariate versions of the skew-normal, skew-t, skew-slash and skew-contaminated normal distributions. The results and methods are applied to a real data set.     

Multivariate geometric skew-normal distribution

Azzalini [A class of distributions which include the normal. Scand J Stat. 1985;12:171–178] introduced a skew-normal distribution of which normal distribution is a special case. Recently, Kundu [Geometric skew normal distribution. Sankhya Ser B. 2014;76:167–189] introduced a geometric skew-normal distribution and showed that it has certain advantages over Azzalini's skew-normal distribution. In this paper we discuss about the multivariate geometric skew-normal (MGSN) distribution. It can be used as an alternative to Azzalini's skew-normal distribution. We discuss different properties of the proposed distribution. It is observed that the joint probability density function of the MGSN distribution can take a variety of shapes. Several characterization results have been established. Generation from an MGSN distribution is quite simple, hence the simulation experiments can be performed quite easily. The maximum likelihood estimators of the unknown parameters can be obtained quite conveniently using the expectation–maximization (EM) algorithm. We perform some simulation experiments and it is observed that the performances of the proposed EM algorithm are quite satisfactory. Furthermore, the analyses of two data sets have been performed, and it is observed that the proposed methods and the model work very well.     

A generalized skew two-piece skew-normal distribution

In this paper, we discuss a general class of skew two-piece skew-normal distributions, denoted by GSTPSN(λ₁, λ₂, ρ). We derive its moment generating function and discuss some simple and interesting properties of this distribution. We then discuss the modes of these distributions and present a useful representation theorem as well. Next, we focus on a different generalization of the two-piece skew-normal distribution which is a symmetric family of distributions and discuss some of its properties. Finally, three well-known examples are used to illustrate the practical usefulness of this family of distributions.     

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.     

Multivariate tail conditional expectation for scale mixtures of skew-normal distribution

In practice, a financial or actuarial data set may be a skewed or heavy-tailed and this motivates us to study a class of distribution functions in risk management theory that provide more information about these characteristics resulting in a more accurate risk analysis. In this paper, we consider a multivariate tail conditional expectation (MTCE) for multivariate scale mixtures of skew-normal (SMSN) distributions. This class of distributions contains skewed distributions and some members of this class can be used to analyse heavy-tailed data sets. We also provide a closed form for TCE in a univariate skew-normal distribution framework. Numerical examples are also provided for illustration.     

Order statistics from trivariate normal and -distributions in terms of generalized skew-normal and skew- distributions

We consider here a generalization of the skew-normal distribution, GSN(λ₁,λ₂,ρ), defined through a standard bivariate normal distribution with correlation ρ, which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics.Next, we introduce a generalized skew-t_ν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367–389] univariate skew-t_ν form. We then use the relationship between the generalized skew-normal and skew-t_ν distributions to discuss some properties of generalized skew-t_ν as well as distributions of order statistics from bivariate and trivariate t_ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-t_ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.     

Bayesian analysis of skew-normal independent linear mixed models with heterogeneity in the random-effects population

We present a new class of models to fit longitudinal data, obtained with a suitable modification of the classical linear mixed-effects model. For each sample unit, the joint distribution of the random effect and the random error is a finite mixture of scale mixtures of multivariate skew-normal distributions. This extension allows us to model the data in a more flexible way, taking into account skewness, multimodality and discrepant observations at the same time. The scale mixtures of skew-normal form an attractive class of asymmetric heavy-tailed distributions that includes the skew-normal, skew-Student-t, skew-slash and the skew-contaminated normal distributions as special cases, being a flexible alternative to the use of the corresponding symmetric distributions in this type of models. A simple efficient MCMC Gibbs-type algorithm for posterior Bayesian inference is employed. In order to illustrate the usefulness of the proposed methodology, two artificial and two real data sets are analyzed.     

Likelihood-based inference for censored linear regression models with scale mixtures of skew-normal distributions

In many studies, the data collected are subject to some upper and lower detection limits. Hence, the responses are either left or right censored. A complication arises when these continuous measures present heavy tails and asymmetrical behavior; simultaneously. For such data structures, we propose a robust-censored linear model based on the scale mixtures of skew-normal (SMSN) distributions. The SMSN is an attractive class of asymmetrical heavy-tailed densities that includes the skew-normal, skew-t, skew-slash, skew-contaminated normal and the entire family of scale mixtures of normal (SMN) distributions as special cases. We propose a fast estimation procedure to obtain the maximum likelihood (ML) estimates of the parameters, using a stochastic approximation of the EM (SAEM) algorithm. This approach allows us to estimate the parameters of interest easily and quickly, obtaining as a byproducts the standard errors, predictions of unobservable values of the response and the log-likelihood function. The proposed methods are illustrated through real data applications and several simulation studies.     

Flexible Class of Skew-Symmetric Distributions

Abstract.  We propose a flexible class of skew-symmetric distributions for which the probability density function has the form of a product of a symmetric density and a skewing function. By constructing an enumerable dense subset of skewing functions on a compact set, we are able to consider a family of distributions, which can capture skewness, heavy tails and multimodality systematically. We present three illustrative examples for the fibreglass data, the simulated data from a mixture of two normal distributions and the Swiss bills data.     

Inference in flexible families of distributions with normal kernel

This paper addresses the inference problem for a flexible class of distributions with normal kernel known as skew-bimodal-normal family of distributions. We obtain posterior and predictive distributions assuming different prior specifications. We provide conditions for the existence of the maximum-likelihood estimators (MLE). An EM-type algorithm is built to compute them. As a by product, we obtain important results related to classical and Bayesian inferences for two special subclasses called bimodal-normal and skew-normal (SN) distribution families. We perform a Monte Carlo simulation study to analyse behaviour of the MLE and some Bayesian ones. Considering the frontier data previously studied in the literature, we use the skew-bimodal-normal (SBN) distribution for density estimation. For that data set, we conclude that the SBN model provides as good a fit as the one obtained using the location-scale SN model. Since the former is a more parsimonious model, such a result is shown to be more attractive.     

On diagnostics in multivariate measurement error models under asymmetric heavy-tailed distributions

In this paper, we discuss the extension of some diagnostic procedures to multivariate measurement error models with scale mixtures of skew-normal distributions (Lachos et?al., Statistics 44:541?C556, 2010c). This class provides a useful generalization of normal (and skew-normal) measurement error models since the random term distributions cover symmetric, asymmetric and heavy-tailed distributions, such as skew-t, skew-slash and skew-contaminated normal, among others. Inspired by the EM algorithm proposed by Lachos et?al. (Statistics 44:541?C556, 2010c), we develop a local influence analysis for measurement error models, following Zhu and Lee??s (J R Stat Soc B 63:111?C126, 2001) approach. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex and Cook??s well-known approach can be very difficult to apply to achieve local influence measures. Some useful perturbation schemes are also discussed. In addition, a score test for assessing the homogeneity of the skewness parameter vector is presented. Finally, the methodology is exemplified through a real data set, illustrating the usefulness of the proposed methodology.     

Nonlinear mixed-effects models with scale mixture of skew-normal distributions

Aiming to avoid the sensitivity in the parameters estimation due to atypical observations or skewness, we develop asymmetric nonlinear regression models with mixed-effects, which provide alternatives to the use of normal distribution and other symmetric distributions. Nonlinear models with mixed-effects are explored in several areas of knowledge, especially when data are correlated, such as longitudinal data, repeated measures and multilevel data, in particular, for their flexibility in dealing with measures of areas such as economics and pharmacokinetics. The random components of the present model are assumed to follow distributions that belong to scale mixtures of skew-normal (SMSN) distribution family, that encompasses distributions with light and heavy tails, such as skew-normal, skew-Student-t, skew-contaminated normal and skew-slash, as well as symmetrical versions of these distributions. For the parameters estimation we obtain a numerical solution via the EM algorithm and its extensions, and the Newton-Raphson algorithm. An application with pharmacokinetic data shows the superiority of the proposed models, for which the skew-contaminated normal distribution has shown to be the most adequate distribution. A brief simulation study points to good properties of the parameter vector estimators obtained by the maximum likelihood method.     

An appropriate empirical version of skew-normal density

In this paper, we consider a constructive representation of skewed distributions, which proposed by Ferreira and Steel (J Am Stat Assoc 101:823–829, 2006), and its basic properties is presented. We study the five versions of skew- normal distributions in this general setting. An appropriate empirical model for a skewed distribution is introduced. In data analysis, we compare this empirical model with the other four versions of skew-normal distributions, via some reasonable criteria. It is shown that the proposed empirical model has a better fit for density estimation.     

Change Point Detection in The Skew-Normal Model Parameters

Bayesian inference under the skew-normal family of distributions is discussed using an arbitrary proper prior for the skewness parameter. In particular, we review some results when a skew-normal prior distribution is considered. Considering this particular prior, we provide a stochastic representation of the posterior of the skewness parameter. Moreover, we obtain analytical expressions for the posterior mean and variance of the skewness parameter. The ultimate goal is to consider these results to one change point identification in the parameters of the location-scale skew-normal model. Some Latin American emerging market datasets are used to illustrate the methodology developed in this work.     

Finite mixtures of matrix normal distributions for classifying three-way data

Matrix-variate distributions represent a natural way for modeling random matrices. Realizations from random matrices are generated by the simultaneous observation of variables in different situations or locations, and are commonly arranged in three-way data structures. Among the matrix-variate distributions, the matrix normal density plays the same pivotal role as the multivariate normal distribution in the family of multivariate distributions. In this work we define and explore finite mixtures of matrix normals. An EM algorithm for the model estimation is developed and some useful properties are demonstrated. We finally show that the proposed mixture model can be a powerful tool for classifying three-way data both in supervised and unsupervised problems. A simulation study and some real examples are presented.     

Local influence analysis for regression models with scale mixtures of skew-normal distributions

The robust estimation and the local influence analysis for linear regression models with scale mixtures of multivariate skew-normal distributions have been developed in this article. The main virtue of considering the linear regression model under the class of scale mixtures of skew-normal distributions is that they have a nice hierarchical representation which allows an easy implementation of inference. Inspired by the expectation maximization algorithm, we have developed a local influence analysis based on the conditional expectation of the complete-data log-likelihood function, which is a measurement invariant under reparametrizations. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex and with Cook's well-known approach it can be very difficult to obtain measures of the local influence. Some useful perturbation schemes are discussed. In order to examine the robust aspect of this flexible class against outlying and influential observations, some simulation studies have also been presented. Finally, a real data set has been analyzed, illustrating the usefulness of the proposed methodology.     

Bimodality based on the generalized skew-normal distribution

This paper focuses on the development of a new extension of the generalized skew-normal distribution introduced in Gómez et al. [Generalized skew-normal models: properties and inference. Statistics. 2006;40(6):495–505]. To produce the generalization a new parameter is introduced, the signal of which has the flexibility of yielding unimodal as well as bimodal distributions. We study its properties, derive a stochastic representation and state some expressions that facilitate moments derivation. Maximum likelihood is implemented via the EM algorithm which is based on the stochastic representation derived. We show that the Fisher information matrix is singular and discuss ways of getting round this problem. An illustration using real data reveals that the model can capture well special data features such as bimodality and asymmetry.     

A study of geometric and statistical curvatures for the skew-normal family of distributions

With special reference to the family of skew-normal distributions, we consider geometric curvature of a probability density function as a means to define and identify rare or catastrophic events—a phenomenon common in studying the financial instruments. Further, we study the statistical curvature properties of this family of distributions and discuss the sample size issue, to assess, to what extent the linear and likelihood-based inference of exponential family of distribution can be applicable for the skew-normal family.     

A robust class of homoscedastic nonlinear regression models

In this paper, we examine a nonlinear regression (NLR) model with homoscedastic errors which follows a flexible class of two-piece distributions based on the scale mixtures of normal (TP-SMN) family. The objective of using this family is to develop a robust NLR model. The TP-SMN is a rich class of distributions that covers symmetric/asymmetric and lightly/heavy-tailed distributions and is an alternative family to the well-known scale mixtures of skew-normal (SMSN) family studied by Branco and Dey [35]. A key feature of this study is using a new suitable hierarchical representation of the family to obtain maximum-likelihood estimates of model parameters via an EM-type algorithm. The performances of the proposed robust model are demonstrated using simulated and some natural real datasets and also compared to other well-known NLR models.     

Odd-Burr generalized family of distributions with some applications

We study a new family of continuous distributions with two extra shape parameters called the Burr generalized family of distributions. We investigate the shapes of the density and hazard rate function. We derive explicit expressions for some of its mathematical quantities. The estimation of the model parameters is performed by maximum likelihood. We prove the flexibility of the new family by means of applications to two real data sets. Furthermore, we propose a new extended regression model based on the logarithm of the Burr generalized distribution. This model can be very useful to the analysis of real data and provide more realistic fits than other special regression models.