Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

Inference and diagnostics in skew scale mixtures of normal regression models

Skew scale mixtures of normal distributions are often used for statistical procedures involving asymmetric data and heavy-tailed. The main virtue of the members of this family of distributions is that they are easy to simulate from and they also supply genuine expectation-maximization (EM) algorithms for maximum likelihood estimation. In this paper, we extend the EM algorithm for linear regression models and we develop diagnostics analyses via local influence and generalized leverage, following Zhu and Lee's approach. This is because Cook's well-known approach cannot be used to obtain measures of local influence. The EM-type algorithm has been discussed with an emphasis on the skew Student-t-normal, skew slash, skew-contaminated normal and skew power-exponential distributions. Finally, results obtained for a real data set are reported, illustrating the usefulness of the proposed method.     

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.     

Nonlinear semiparametric autoregressive model with finite mixtures of scale mixtures of skew normal innovations

We propose data generating structures which can be represented as the nonlinear autoregressive models with single and finite mixtures of scale mixtures of skew normal innovations. This class of models covers symmetric/asymmetric and light/heavy-tailed distributions, so provide a useful generalization of the symmetrical nonlinear autoregressive models. As semiparametric and nonparametric curve estimation are the approaches for exploring the structure of a nonlinear time series data set, in this article the semiparametric estimator for estimating the nonlinear function of the model is investigated based on the conditional least square method and nonparametric kernel approach. Also, an Expectation–Maximization-type algorithm to perform the maximum likelihood (ML) inference of unknown parameters of the model is proposed. Furthermore, some strong and weak consistency of the semiparametric estimator in this class of models are presented. Finally, to illustrate the usefulness of the proposed model, some simulation studies and an application to real data set are considered.     

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.     

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.     

Statistical diagnostics for nonlinear regression models based on scale mixtures of skew-normal distributions

The purpose of this paper is to develop diagnostics analysis for nonlinear regression models (NLMs) under scale mixtures of skew-normal (SMSN) distributions introduced by Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124]. This novel class of models provides a useful generalization of the symmetrical NLM [Vanegas LH, Cysneiros FJA. Assessment of diagnostic procedures in symmetrical nonlinear regression models. Comput. Statist. Data Anal. 2010;54:1002–1016] since the random terms distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as the skew-t, skew-slash, skew-contaminated normal distributions, among others. Motivated by the results given in Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124], we presented a score test for testing the homogeneity of the scale parameter and its properties are investigated through Monte Carlo simulations studies. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. The newly developed procedures are illustrated considering a real data set.     

Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model.     

Mean-shift outliers model in skew scale-mixtures of normal distributions

ABSTRACT

Asymmetric models have been discussed quite extensively in recent years, in situations where the normality assumption is suspected due to lack of symmetry in the data. Techniques for assessing the quality of fit and diagnostic analysis are important for model validation. This paper presents a study of the mean-shift method for the detection of outliers in regression models under skew scale-mixtures of normal distributions. Analytical solutions for the estimators of the parameters are obtained through the use of Expectation–Maximization algorithm. The observed information matrix for the calculation of standard errors is obtained for each distribution. Simulation studies and an application to the analysis of a data have been carried out, showing the efficiency of the proposed method in detecting outliers.     

Heteroscedastic and heavy-tailed regression with mixtures of skew Laplace normal distributions

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.     

Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.     

Two-Piece location-scale distributions based on scale mixtures of normal family

In this work, we study the maximum likelihood (ML) estimation problem for the parameters of the two-piece (TP) distribution based on the scale mixtures of normal (SMN) distributions. This is a family of skewed distributions that also includes the scales mixtures of normal class, and is flexible enough for modeling symmetric and asymmetric data. The ML estimates of the proposed model parameters are obtained via an expectation-maximization (EM)-type algorithm.     

Robust Bayesian analysis of loss reserving data using scale mixtures distributions

It is vital for insurance companies to have appropriate levels of loss reserving to pay outstanding claims and related settlement costs. With many uncertainties and time lags inherently involved in the claims settlement process, loss reserving therefore must be based on estimates. Existing models and methods cannot cope with irregular and extreme claims and hence do not offer an accurate prediction of loss reserving. This paper extends the conventional normal error distribution in loss reserving modeling to a range of heavy-tailed distributions which are expressed by certain scale mixtures forms. This extension enables robust analysis and, in addition, allows an efficient implementation of Bayesian analysis via Markov chain Monte Carlo simulations. Various models for the mean of the sampling distributions, including the log-Analysis of Variance (ANOVA), log-Analysis of Covariance (ANCOVA) and state space models, are considered and the straightforward implementation of scale mixtures distributions is demonstrated using OpenBUGS.     

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 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.     

Stochastic volatility in mean models with scale mixtures of normal distributions and correlated errors: A Bayesian approach

A stochastic volatility in mean model with correlated errors using the symmetrical class of scale mixtures of normal distributions is introduced in this article. The scale mixture of normal distributions is an attractive class of symmetric distributions that includes the normal, Student-t, slash and contaminated normal distributions as special cases, providing a robust alternative to estimation in stochastic volatility in mean models in the absence of normality. Using a Bayesian paradigm, an efficient method based on Markov chain Monte Carlo (MCMC) is developed for parameter estimation. The methods developed are applied to analyze daily stock return data from the São Paulo Stock, Mercantile & Futures Exchange index (IBOVESPA). The Bayesian predictive information criteria (BPIC) and the logarithm of the marginal likelihood are used as model selection criteria. The results reveal that the stochastic volatility in mean model with correlated errors and slash distribution provides a significant improvement in model fit for the IBOVESPA data over the usual normal model.     

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.