Skewness and mixtures of normal distributions

The effect of skewness on hypothesis tests for the existence of a mixture of univariate and bivariate normal distributions is examined through a Monte Carlo study. A likelihood ratio test based on results of the simultaneous estimation of skewness parameters, derived from power transformations, with mixture parameters is proposed. This procedure detects the difference between inherent distributional skewness and the apparent skewness which is a manifestation of the mixture of several distributions. The properties of this test are explored through a simulation study.     

A new class of multivariate skew distributions with applications to bayesian regression models

Abstract: The authors develop a new class of distributions by introducing skewness in multivariate elliptically symmetric distributions. The class, which is obtained by using transformation and conditioning, contains many standard families including the multivariate skew‐normal and t distributions. The authors obtain analytical forms of the densities and study distributional properties. They give practical applications in Bayesian regression models and results on the existence of the posterior distributions and moments under improper priors for the regression coefficients. They illustrate their methods using practical examples.     

On discrete weighted distributions and their use in model choice for observed data

This paper provides a brief structural perspective of discrete weighted distributions in theory and practice.. It develops a unified view of previous work involving univariate and bivariate models with some new results pertaining to mixtures, form-invariance and Bayesian inference     

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.     

Multivariate discrete exponential family of distributions and their properties

The paper generalizes the univariate discrete exponential family of distributions to the multivariate situation, and this generalization includes the multivariate power series distributions, the multivariate Lagrangian distributions, and the modified multivariate power-series distributions. This provides a unified approach for the study of these three classes of distributions. We obtain recurrence relations for moments and cumulants, and the maximum likelihood estimation for the discrete exponential family. These results are applied to some multivariate discrete distributions like the Lagrangian Poisson, Lagrangian (negative) multinomial, logarithmic series distributions and multivariate Lagrangian negative binomial distribution.     

Properties and applications of the sarmanov family of bivariate distributions

We discuss properties of the bivariate family of distributions introduced by Sarmanov (1966). It is shown that correlation coefficients of this family of distributions have wider range than those of the Farlie-Gumbel-Morgenstern distributins. Possible applications of this family of bivariate distributions as prior distributins in Bayesian inference are discussed. The density of the bivariate Sarmanov distributions with beta marginals can be expressed as a linear combination of products of independent beta densities. This pseudoconjugate property greatly reduces the complexity of posterior computations when this bivariate beta distribution is used as a prior. Multivariate extensions are derived.     

Optimal direction Gibbs sampler for truncated multivariate normal distributions

Generalized Gibbs samplers simulate from any direction, not necessarily limited to the coordinate directions of the parameters of the objective function. We study how to optimally choose such directions in a random scan Gibbs sampler setting. We consider that optimal directions will be those that minimize the Kullback–Leibler divergence of two Markov chain Monte Carlo steps. Two distributions over direction are proposed for the multivariate Normal objective function. The resulting algorithms are used to simulate from a truncated multivariate Normal distribution, and the performance of our algorithms is compared with the performance of two algorithms based on the Gibbs sampler.     

EM algorithms for beta kernel distributions

The EM algorithm is employed to compute maximum-likelihood estimates for beta kernel distributions. Estimation is considered under two censoring schemes: the progressive Type-I censoring and progressive Type-II right censoring schemes. As an application, the EM algorithm is executed to obtain maximum-likelihood estimates for the beta Weibull distribution under the two censoring schemes. A simulation study and two real data sets are used to show the efficiency of the EM algorithm.     

Multi-modal tempered stable distributions and prosses with applications to finance

Abstract

The assumption of underlying return distribution plays an important role in asset pricing models. While the return distribution used in the traditional theories of asset pricing is the unimodal distribution, numerous studies which have investigated the empirical behavior of asset returns in financial markets use multi-modal distribution. We introduce a new parsimonious multi-modal distribution, referred to as the multi-modal tempered stable (MMTS) distribution. In this article we also generate the exponential Lévy market models and derive the value-at-risk (VaR) induced from them. To demonstrate the advantages, we will present the results of the parameter estimation and the VaRs for financial data.     

Bayesian multivariate GARCH models with dynamic correlations and asymmetric error distributions

The main goal in this paper is to develop and apply stochastic simulation techniques for GARCH models with multivariate skewed distributions using the Bayesian approach. Both parameter estimation and model comparison are not trivial tasks and several approximate and computationally intensive methods (Markov chain Monte Carlo) will be used to this end. We consider a flexible class of multivariate distributions which can model both skewness and heavy tails. Also, we do not fix tail behaviour when dealing with fat tail distributions but leave it subject to inference.     

Quantitative comparisons between finitary posterior distributions and Bayesian posterior distributions

The main object of Bayesian statistical inference is the determination of posterior distributions. Sometimes these laws are given for quantities devoid of empirical value. This serious drawback vanishes when one confines oneself to considering a finite horizon framework. However, assuming infinite exchangeability gives rise to fairly tractable a posteriori quantities, which is very attractive in applications. Hence, with a view to a reconciliation between these two aspects of the Bayesian way of reasoning, in this paper we provide quantitative comparisons between posterior distributions of finitary parameters and posterior distributions of allied parameters appearing in usual statistical models.     

隐非齐次马尔可夫多元线性回归模型及其贝叶斯估计

隐马尔可夫模型对于异质纵向数据的处理有良好的效果,因此被广泛应用于工程技术、生物医学、经济管理等领域。文章引入了一种特殊的非齐次隐马尔可夫状态转移方式,并将其与经典的多元线性回归相结合,提出了隐非齐次马尔可夫多元线性回归模型,介绍了对该模型进行贝叶斯推断的方法原理和技术细节。最后,通过两个模拟实验说明了推断方法的结果是可靠的。     

Multivariate extreme-value distributions with applications to environmental data

Some parametric families of multivariate extreme-value distributions have been proposed in recent years; several additional parametric families are derived here. The parametric models are fitted, using numerical maximum likelihood, to some environmental multivariate extreme data sets consisting of extreme concentrations of a pollutant at several monitoring stations in a region. Some multivariate nonnormal data analysis techniques are proposed to aid in the likelihood analysis. The new models, together with previous models, appear to be adequate for inferences in that they cover a wide range of possible dependence patterns.     

Multivariate distributions defined in terms of contours

The paper deals with the problem of using contours as the basis for defining probability distributions. First, the most general probability densities with given contours are obtained and the particular cases of circular and elliptical contours are dealt with. It is shown that the so-called elliptically contoured distributions do not include all possible cases. Next, the case of contours defined by polar coordinates is analyzed including its simulation and parameter estimation. Finally, the case of cumulative distribution functions with given contours is discussed. Several examples are used for illustrative purposes.     

Generalised data augmentation and posterior inferences

This paper explores the use of data augmentation in settings beyond the standard Bayesian one. In particular, we show that, after proposing an appropriate generalised data-augmentation principle, it is possible to extend the range of sampling situations in which fiducial methods can be applied by constructing Markov chains whose stationary distributions represent valid posterior inferences on model parameters. Some properties of these chains are presented and a number of open questions are discussed. We also use the approach to draw out connections between classical and Bayesian approaches in some standard settings.     

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 symmetry of finite mixtures of normal distributions

In this note we derive a necessary and sufficient condition for a distribution obtained by taking a finite mixture of multivariate normal distributions to be symmetric about zero. The result derived also holds for mixtures of symmetric stable distributions, including the Cauchy distribution.     

Multivariate Poisson regression with covariance structure

In recent years the applications of multivariate Poisson models have increased, mainly because of the gradual increase in computer performance. The multivariate Poisson model used in practice is based on a common covariance term for all the pairs of variables. This is rather restrictive and does not allow for modelling the covariance structure of the data in a flexible way. In this paper we propose inference for a multivariate Poisson model with larger structure, i.e. different covariance for each pair of variables. Maximum likelihood estimation, as well as Bayesian estimation methods are proposed. Both are based on a data augmentation scheme that reflects the multivariate reduction derivation of the joint probability function. In order to enlarge the applicability of the model we allow for covariates in the specification of both the mean and the covariance parameters. Extension to models with complete structure with many multi-way covariance terms is discussed. The method is demonstrated by analyzing a real life data set.     

Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution

This article is concerned with modifications of both maximum likelihood and moment estimators for parameters of the three-parameter gamma distribution. Modifications employed here are essentially the same as those previously considered by the authors (1980, 1981) in connection with the lognormal distribution. Sampling behavior of the estimates is indicated by a Monte Carlo simulation. For certain combinations of parameter values, these new estimators appear better than both maximum likelihood and moment estimators with respect to bias, variance and/or ease of calculation.     

Numerical Bayesian inference with arbitrary prior

The purpose of the paper, is to explain how recent advances in Markov Chain Monte Carlo integration can facilitate the routine Bayesian analysis of the linear model when the prior distribution is completely user dependent. The method is based on a Metropolis-Hastings algorithm with a Student-t source distribution that can generate posterior moments as well as marginal posterior densities for model parameters. The method is illustrated with numerical examples where the combination of prior and likelihood information leads to multimodal posteriors due to prior-likelihood conflicts, and to cases where prior information can be summarized by symmetric stable Paretian distributions.