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The purpose of this paper is to extend the mixture factor analyzers (MFA) model to handle missing and heavy-tailed data. In this model, the distribution of factors loading and errors arise from the multivariate normal mean-variance mixture of the Birnbaum-Saunders (NMVBS) distribution. By using the structures covariance matrix, we introduce parsimonious MFA based on NMVBS distribution. An Expectation Maximization (EM)-type algorithm is developed for parameter estimation. Simulations study and real data sets represent the efficiency and performance of the proposed model. [ABSTRACT FROM AUTHOR]

This paper introduces an innovative family of statistical models called the multivariate skew scale-shape mixtures of normal distributions. These models serve as a versatile tool in statistical analysis by efficiently characterizing the skewed and leptokurtic nature commonly observed in multivariate datasets. Their applicability shines in real-world scenarios where data often deviate from standard statistical assumptions due to the presence of outliers. We present an EM-type algorithm designed for maximizing likelihood estimation and evaluate the model's effectiveness through real-world data applications. Through rigorous testing against various datasets, we assess the performance and practicality of the proposed algorithm in real statistical scenarios. The results demonstrate the remarkable performance of this new family of distributions. [ABSTRACT FROM AUTHOR]

AbstractThis paper introduces a novel family of distributions known as generalized scale mixtures of generalized normal distributions (GSMGN). These distributions incorporate two additional shape parameters that serve to regulate the shape and tails of the distribution. A finite mixture model based on this family is presented to address clustering heterogeneous data in the presence of leptokurtic and heavy-tailed outcomes. The estimation of the parameters of this model are obtained by developing an ECM-PLA ensemble algorithm which combine the profile likelihood approach (PLA) and the classical Expectation Conditional Maximization (ECM) algorithm, and the observed information matrix is obtained. The applicability of this new family and the numerical performance of the proposed methodology is discussed through simulated and real data examples. [ABSTRACT FROM AUTHOR]

In this paper, we study linear regression models in which the error term has shape mixtures of skew-normal distribution. This type of distribution belongs to the skew-normal (SN) distribution class that can be used for heavy tails and asymmetry data. For the first time, for the classical (non-Bayesian) estimation of the parameters of the SN family, we apply the Markov chains Monte Carlo ECM (MCMC-ECM) algorithm where the samples are generated by Gibbs sampling, denoted by Gibbs-ECM, and also, we extend two other types of the EM algorithm for the above model. Finally, the proposed method is evaluated through a simulation and compared with the Numerical Math-ECM algorithm and Monte Carlo ECM (MC-ECM) using a real data set. [ABSTRACT FROM AUTHOR]

In this paper, we introduced a family of distributions with a very flexible shape named generalized scale mixtures of generalized asymmetric normal distributions (GSMAGN). We investigated the main properties of the new family including moments, skewness, kurtosis coefficients and order statistics. A variant of the expectation maximization (EM)-type algorithm was established by combining the proflie likihood approach (PLA) with the classical expectation conditional maximization (ECM) algorithm for parameter estimation of this model. This approach with analytical expressions in the E-step and tractable M-step can greatly improve the computational speed and efficiency of the algorithm. The performance of the proposed algorithm was assessed by some simulation studies. The feasibility of the proposed methodology was illustrated through two real datasets.

This paper proposes some diagnostics procedures for the skew-t linear regression model with censored response. The skew-t distribution is an attractive family of asymmetrical heavy-tailed densities that includes the normal, skew-normal and student’s-t distributions as special cases. Inspired by the power and wide applicability of the EM-type algorithm, local and global influence analysis, based on the conditional expectation of the complete-data log-likelihood function are developed, following Zhu and Lee’s approach. For the local influence analysis, four specific perturbation schemes are discussed. Two real data sets, from education and economics, which are right and left censoring, respectively, are analyzed in order to illustrate the usefulness of the proposed methodology [ABSTRACT FROM AUTHOR]

The class of normal mean‐variance mixture (NMVM) distributions is a rich family of asymmetric and heavy‐tailed distributions and has been widely considered in parametric modeling of the data for robust statistical inference. This paper proposes an extension of measurement error models by assuming the NMVM distributions for the unobserved covariates and error terms in the model, referred to as the NMVM‐MEM. An expectation conditional maximization either (ECME) algorithm is developed to compute the maximum likelihood (ML) estimates of model parameters. Furthermore, an information‐based approach is performed to derive the asymptotic covariance matrix of ML estimators. The analysis of a blood pressure dataset illustrates the superiority of NMVM‐MEM to accommodate asymmetry and outliers over the normal counterpart. Two simulation studies are undertaken to validate our proposed techniques. [ABSTRACT FROM AUTHOR]

The mixture-of-experts (MoE) paradigm attempts to learn complex models by combining several “experts” via probabilistic mixture models. Each expert in the MoE model handles a small area of the data space in which a gating function controls the data-to-expert assignment. The MoE framework has been used extensively in designing non-linear models in machine learning and statistics to model the heterogeneity in data for the purpose of regression, classification and clustering. The existing MoE of multi-target regression (MoE-MTR) models for continuous data is based on multivariate normal distributions. However, in many practical situations, for a set of data, a group or groups of observations may exhibit asymmetric and heavy-tailed behaviour, and inference based on symmetric distributions in such situations can unduly affect the fit of the regression model. We introduce here a novel robust multivariate nonnormal MoE model by the use of mean mixture of normal distributions. The proposed model can handle the issues of MoE-MTR models regarding possibly skewed, heavytailed and noisy data. Maximum likelihood estimates of model parameters are developed based on an expectation-maximization (EM)-type algorithm. Parsimony is also obtained by imposing suitable constraints on the expert dispersion matrices. The usefulness of the proposed methodology is illustrated using simulated and real data sets. [ABSTRACT FROM AUTHOR]

In the censored data exploration, the classical linear regression model which assumes normally distributed random errors is perhaps one of the commonly used frameworks. However, practical studies have often criticized the classical linear regression model because of its sensitivity to departure from the normality and partial nonlinearity. This paper proposes to solve these potential issues simultaneously in the context of the partial linear regression model by assuming that the random errors follow a scale-mixture of normal (SMN) family of distributions. The postulated method allows us to model data with great flexibility, accommodating heavy tails and outliers. By implementing the B-spline approximation and using the convenient hierarchical representation of the SMN distributions, a computationally analytical EM-type algorithm is developed for obtaining maximum likelihood (ML) parameter estimates. Various simulation studies are conducted to investigate the finite sample properties, as well as the robustness of the model in dealing with the heavy tails distributed datasets. Real-world data examples are finally analyzed for illustrating the usefulness of the proposed methodology. [ABSTRACT FROM AUTHOR]

Functional linear models are important tools for studying the relationship between functional response and covariates. However, if subjects come from an inhomogeneous population that demonstrates different linear relationship between the response and covariates among different subpopulations/clusters, a single functional linear model is no longer adequate for the data. A new class of mixtures of functional linear models for the analysis of heterogeneous functional data is introduced. Identifiability is established for the proposed class of mixture models under mild conditions. The proposed estimation procedures combine the ideas of local kernel regression, functional principal component analysis and EM algorithm. A generalized likelihood ratio test based on a conditional bootstrap is given as to whether the regression coefficient functions are constant. A Monte Carlo simulation study is conducted to examine the finite sample performance of the new methodology. Finally, the analysis of CO2-GDP data reveals the dynamic patterns of relationship between CO2 and GDP among different countries.

The skew-t distribution is an attractive family of asymmetrical heavy-tailed densities that includes the normal, skew-normal and Student’s-t distributions as special cases. In this work, we propose an EM-type algorithm for computing the maximum likelihood estimates for skew-t linear regression models with censored response. In contrast with previous proposals, this algorithm uses analytical expressions at the E-step, as opposed to Monte Carlo simulations. These expressions rely on formulas for the mean and variance of a truncated skew-t distribution, and can be computed using the R library MomTrunc. The standard errors, the prediction of unobserved values of the response and the log-likelihood function are obtained as a by-product. The proposed methodology is illustrated through the analyses of simulated and a real data application on Letter-Name Fluency test in Peruvian students. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce a new family of distributions, namely compound zero-truncated Poisson exponential distribution of which exponential distribution is a special case. The proposed family of distributions represents the zero truncated-Poisson sum of independent and identically distributed exponential random variables. The proposed distribution has two parameters and its probability density function can be skewed and unimodal. It can be used quite effectively in analyzing skewed data. We suggest to use expectation–maximization (EM)-type algorithm to estimate the unknown parameters, and it is observed that it is easy to implement in practice. We further consider the bivariate version of the proposed model which has three parameters and provides different properties. We have performed extensive simulation studies to see the performances of the proposed EM algorithm, and a real data set has been analyzed to see the effectiveness of the proposed models. [ABSTRACT FROM AUTHOR]

The Birnbaum–Saunders (BS) distribution, introduced in 1969, is a popular univariate fatigue life distribution which has been widely used to model right-skewed lifetime and reliability data. In this paper, a new class of generalized multivariate BS distributions is proposed based on mean-variance mixture models to accommodate strongly skewed and heavy tailed multivariate lifetime data. Some special cases of this class as well as their properties are then discussed. We present a hierarchical representation which facilitates an efficient EM-type algorithm for the computation of maximum likelihood estimates. Empirical results from a simulation study and real data analyses show that this class of distributions outperforms many existing extensions of the BS distribution in modeling lifetime data. [ABSTRACT FROM AUTHOR]

This article incorporates a skew-normal distribution into an inverse Gaussian (IG) process to represent the unit-to-unit variability of the degradation rate, while a symmetrical distribution or approximately symmetrical distribution is commonly adopted in the IG process. Then we derive the corresponding lifetime distribution and mean-time-to-failure including two special cases under the condition that the skew-normal distribution reduces to an half-normal distribution or a normal distribution. Moreover, an improved EM-type algorithm is presented to overcome the difficulties in estimating parameters. Finally, two simulation studies and a case application are used to illustrate the advantages of the proposed model. [ABSTRACT FROM AUTHOR]

In spite of widespread use as well as theoretical properties of the multivariate scale mixture of normal distributions, practical studies show a lack of stability and robustness against asymmetric features such as asymmetry and heavy tails. In this paper, we develop a new multivariate model by assuming the Birnbaum-Saunders distribution for the mixing variable in the scale mixture of the restricted skew-normal distribution. An analytically simple and efficient EM-type algorithm is adopted for iteratively computing maximum likelihood estimate of model parameters. To account standard errors, the observed information matrix is derived analytically by offering an information-based approach. Results obtained from real and simulated datasets are reported to illustrate the practical utility of the proposed methodology. [ABSTRACT FROM AUTHOR]

Vector Auto-regressive (VAR) models are commonly used for modelling multivariate time series and the typical distributional form is to assume a multivariate normal. However, the assumption of Gaussian white noise in multivariate time series is often not reasonable in applications where there are extreme and/or skewed observations. In this setting, inference based on using a Gaussian distributional form will provide misleading results. In this paper, we extended the multivariate setting of autoregressive process, by considering the multivariate scale mixture of skew-normal (SMSN) distributions for VAR innovations. The multivariate SMSN family is able to be represented in a hierarchical form which relatively easily facilitates simulation and an EM-type algorithm to estimate the model parameters. The performance of the proposed model is illustrated by using simulated and real datasets. [ABSTRACT FROM AUTHOR]

The family of expectation--maximization (EM) algorithms provides a general approach to fitting flexible models for large and complex data. The expectation (E) step of EM-type algorithms is time-consuming in massive data applications because it requires multiple passes through the full data. We address this problem by proposing an asynchronous and distributed generalization of the EM called the distributed EM (DEM). Using DEM, existing EM-type algorithms are easily extended to massive data settings by exploiting the divide-and-conquer technique and widely available computing power, such as grid computing. The DEM algorithm reserves two groups of computing processes called workers and managers for performing the E step and the maximization step (M step), respectively. The samples are randomly partitioned into a large number of disjoint subsets and are stored on the worker processes. The E step of DEM algorithm is performed in parallel on all the workers, and every worker communicates its results to the managers at the end of local E step. The managers perform the M step after they have received results from a γ-fraction of the workers, where γ is a fixed constant in (0, 1]. The sequence of parameter estimates generated by the DEM algorithm retains the attractive properties of EM: convergence of the sequence of parameter estimates to a local mode and linear global rate of convergence. Across diverse simulations focused on linear mixed-effects models, the DEM algorithm is significantly faster than competing EM-type algorithms while having a similar accuracy. The DEM algorithm maintains its superior empirical performance on a movie ratings database consisting of 10 million ratings. Supplementary material for this article is available online. [ABSTRACT FROM AUTHOR]

Mixture modeling has emerged as a statistical tool to perform unsupervised model-based clustering for heterogeneous data. A framework of using contaminated mean-mixture of normal distributions as the components of the mixture model is designed to accommodate asymmetric data with outliers. Fourteen parsimonious variants of the postulated model are introduced by employing an eigenvalue decomposition of the component scale matrices. Simultaneously clustering and outliers detection is an outstanding advantage of the proposed model in analyzing non-normally distributed data. A computationally feasible and flexible EM-type algorithm is outlined for obtaining maximum likelihood parameter estimates. Moreover, the score vector and empirical information matrix for calculating asymptotic standard errors of the parameter estimates are derived by offering an information-based approach. The applicability of the proposed method is demonstrated through the analysis of simulated and real datasets with varying proportions of outliers. [ABSTRACT FROM AUTHOR]

The standard location and scale unrestricted (or unified) skew-normal (SUN) family studied by Arellano-Valle and Genton [On fundamental skew distributions. J Multivar Anal. 2005;96:93-116] and Arellano-Valle and Azzalini [On the unification of families of skew-normal distributions. Scand J Stat. 2006;33:561-574], allows the modelling of data which is symmetrically or asymmetrically distributed. The family has a number of advantages suitable for the analysis of stochastic processes such as Auto-Regressive Moving-Average (ARMA) models, including being closed under linear combinations, being able to satisfy the consistency condition of Kolmogorov’s theorem and providing the guarantee of the existence of such a SUN stochastic process. The family is able to be represented in a hierarchical form which can be used for the ease of simulation. In addition, it facilitates an EM-type algorithm to estimate the model parameters. The performances and suitability of the proposed model are demonstrated on simulations and using two real data sets in applications. [ABSTRACT FROM AUTHOR]

This paper aims at evaluating different aspects of Monte Carlo expectation – maximization algorithm to estimate heavy-tailed mixed logistic regression (MLR) models. As a novelty it also proposes a multiple chain Gibbs sampler to generate of the latent variables distributions thus obtaining independent samples. In heavy-tailed MLR models, the analytical forms of the full conditional distributions for the random effects are unknown. Four different Metropolis–Hastings algorithms are assumed to generate from them. We also discuss stopping rules in order to obtain more efficient algorithms in heavy-tailed MLR models. The algorithms are compared through the analysis of simulated andAscaris Suumdata. [ABSTRACT FROM PUBLISHER]

We introduce the class of elliptical mixed logistic model with focus on the normal/independent subclass. Parameter interpretation in mixed logistic model is not straightforward since the odds ratio is random. For the proposed models, we obtain the odds ratio distribution and its summaries used to interpret the fixed effects and to measure the heterogeneity among the clusters thus extending previous results. Fisher information is also obtained. A Monte Carlo expectation-maximization algorithm is considered to obtain the maximum likelihood estimates. A simulation study is performed comparing normal and heavy-tailed models. It also address the effect of the misspecification of the random effect distribution and other model aspects in the parameter interpretation. A data analysis is performed showing the utility of heavy-tailed mixed logistic model. Among the main conclusions, we note that the misspecification of the random effect distribution influences the fixed effects interpretation and the quantification of the among clusters heterogeneity. [ABSTRACT FROM AUTHOR]

In this paper the normal-skew-normal distribution, proposed by Gomez et al. (Statistics 47:411-421, ), is extended to the multivariate case. It is also a special case of $$SUN_{n,2}$$ -distribution, recently studied by Arellano-Valle and Genton (Chil J Stat 2:17-34, ). We show that the proposed distribution can be expressed as a shape mixture of the multivariate extended skew-normal distribution. Applying this property leads to deriving stochastic representations for the proposed distribution. Also we give some basic properties for this new family. Computational techniques using EM-type algorithms are employed for iteratively computing maximum likelihood estimates. Finally, an application of the new distribution is illustrated using some real data sets. [ABSTRACT FROM AUTHOR]

We carry out a value-at-risk (VaR) analysis of an extremely popular strategy in the currency markets, namely, 'carry trades,' whereby a position purchased in high interest rate currencies is funded by selling low interest rate currencies. Since the natural outcome of the truncated normal distribution of interest-rate spreads combined with the normal distribution of exchange rate returns is a skew-normal distribution, we consider a skew-normal innovation with zero mean for our analysis of carry trade returns using generalized autoregressive conditional heteroskedasticity (GARCH) models. The stress testing results reveal that skew-normal or densities are suitable for the measurement of VaR for carry trade returns involving, for example, taking up a long position in Australian Dollars or Argentine Peso which are funded by selling Japanese Yen. [ABSTRACT FROM AUTHOR]

In a large, prospective longitudinal study designed to monitor cardiac abnormalities in children born to women who are infected with the human immunodeficiency virus, instead of a single outcome variable, there are multiple binary outcomes (e.g. abnormal heart rate, abnormal blood pressure and abnormal heart wall thickness) considered as joint measures of heart function over time. In the presence of missing responses at some time points, longitudinal marginal models for these multiple outcomes can be estimated by using generalized estimating equations (GEEs), and consistent estimates can be obtained under the assumption of a missingness completely at random mechanism. When the missing data mechanism is missingness at random, i.e. the probability of missing a particular outcome at a time point depends on observed values of that outcome and the remaining outcomes at other time points, we propose joint estimation of the marginal models by using a single modified GEE based on an EM-type algorithm. The method proposed is motivated by the longitudinal study of cardiac abnormalities in children who were born to women infected with the human immunodeficiency virus, and analyses of these data are presented to illustrate the application of the method. Further, in an asymptotic study of bias, we show that, under a missingness at random mechanism in which missingness depends on all observed outcome variables, our joint estimation via the modified GEE produces almost unbiased estimates, provided that the correlation model has been correctly specified, whereas estimates from standard GEEs can lead to substantial bias. [ABSTRACT FROM AUTHOR]

Mendelian randomization (MR) is a statistical method exploiting genetic variants as instrumental variables to estimate the causal effect of modifiable risk factors on an outcome of interest. Despite wide uses of various popular two-sample MR methods based on genome-wide association study summary level data, however, those methods could suffer from potential power loss or/and biased inference when the chosen genetic variants are in linkage disequilibrium (LD), and also have relatively large direct effects on the outcome whose distribution might be heavy-tailed which is commonly referred to as the idiosyncratic pleiotropy phenomenon. To resolve those two issues, we propose a novel Robust Bayesian Mendelian Randomization (RBMR) model that uses the more robust multivariate generalized t $t$ -distribution to model such direct effects in a probabilistic model framework which can also incorporate the LD structure explicitly. The generalized t $t$ -distribution can be represented as a Gaussian scaled mixture so that our model parameters can be estimated by the expectation maximization (EM)-type algorithms. We compute the standard errors by calibrating the evidence lower bound using the likelihood ratio test. Through extensive simulation studies, we show that our RBMR has robust performance compared with other competing methods. We further apply our RBMR method to two benchmark data sets and find that RBMR has smaller bias and standard errors. Using our proposed RBMR method, we find that coronary artery disease is associated with increased risk of critically ill coronavirus disease 2019. We also develop a user-friendly R package RBMR (https://github.com/AnqiWang2021/RBMR) for public use., (© 2022 Wiley Periodicals LLC.)

Despite the widespread use of Gaussian mixture model for clustering datasets, practical applications show that the skewed and leptokurtic mixture models can be considered as promising alternatives. This paper proposes a finite mixture of Birnbaum–Saunders (FM-BS) distributions for analyzing and clustering right-skewed, leptokurtic, and multimodal lifetime datasets. The maximum likelihood (ML) estimates of the proposed model are obtained by developing a computationally analytical expectation–maximization (EM) type algorithm, as well as a fuzzy classification maximum likelihood (FCML) type algorithm, that combines the advantages of fuzzy clustering and robust statistical estimators. Simulation studies demonstrate the accuracy and computational efficiency of the FCML algorithm to estimate parameters of the FM-BS distributions and to cluster samples drawn from the FM-BS distributions. Finally, some real datasets have been analyzed to illustrate how well the proposed FM-BS model estimates the membership values. • A finite mixture model for clustering right-skewed, and multimodal data is proposed. • The EM and FCML type algorithms are implemented for computing ML estimates. • Asymptotic standard errors of parameter estimate are obtained through. [ABSTRACT FROM AUTHOR]