Finie mixures of canonical fundamenal skew $$$$-disribuions

This paper introduces a finite mixture of canonical fundamental skew \(t\) (CFUST) distributions for a model-based approach to clustering where the clusters are asymmetric and possibly long-tailed (in: Lee and McLachlan, arXiv:1401.8182 [statME], 2014b). The family of CFUST distributions includes the restricted multivariate skew \(t\) and unrestricted multivariate skew \(t\) distributions as special cases. In recent years, a few versions of the multivariate skew \(t\) (MST) mixture model have been put forward, together with various EM-type algorithms for parameter estimation. These formulations adopted either a restricted or unrestricted characterization for their MST densities. In this paper, we examine a natural generalization of these developments, employing the CFUST distribution as the parametric family for the component distributions, and point out that the restricted and unrestricted characterizations can be unified under this general formulation. We show that an exact implementation of the EM algorithm can be achieved for the CFUST distribution and mixtures of this distribution, and present some new analytical results for a conditional expectation involved in the E-step.     

A class of weighted multivariate distributions related to doubly truncated multivariate t-distribution

This article introduces a class of weighted multivariate t-distributions, which includes the multivariate generalized Student t and multivariate skew t as its special members. This class is defined as the marginal distribution of a doubly truncated multivariate generalized Student t-distribution and studied from several aspects such as weighting of probability density functions, inequality constrained multivariate Student t-distributions, scale mixtures of multivariate normal and probabilistic representations. The relationships among these aspects are given, and various properties of the class are also discussed. Necessary theories and two applications are provided.     

Parameter estimation for mixtures of skew Laplace normal distributions and application in mixture regression modeling

In this article, we propose mixtures of skew Laplace normal (SLN) distributions to model both skewness and heavy-tailedness in the neous data set as an alternative to mixtures of skew Student-t-normal (STN) distributions. We give the expectation–maximization (EM) algorithm to obtain the maximum likelihood (ML) estimators for the parameters of interest. We also analyze the mixture regression model based on the SLN distribution and provide the ML estimators of the parameters using the EM algorithm. The performance of the proposed mixture model is illustrated by a simulation study and two real data examples.     

Generalized linear mixed models for correlated binary data with t-link

A critical issue in modeling binary response data is the choice of the links. We introduce a new link based on the Student’s t-distribution (t-link) for correlated binary data. The t-link relates to the common probit-normal link adding one additional parameter which controls the heaviness of the tails of the link. We propose an interesting EM algorithm for computing the maximum likelihood for generalized linear mixed t-link models for correlated binary data. In contrast with recent developments (Tan et al. in J. Stat. Comput. Simul. 77:929–943, 2007; Meza et al. in Comput. Stat. Data Anal. 53:1350–1360, 2009), this algorithm uses closed-form expressions at the E-step, as opposed to Monte Carlo simulation. Our proposed algorithm relies on available formulas for the mean and variance of a truncated multivariate t-distribution. To illustrate the new method, a real data set on respiratory infection in children and a simulation study are presented.     

Flexible mixture modelling using the multivariate skew-t-normal distribution

This paper presents a robust probabilistic mixture model based on the multivariate skew-t-normal distribution, a skew extension of the multivariate Student’s t distribution with more powerful abilities in modelling data whose distribution seriously deviates from normality. The proposed model includes mixtures of normal, t and skew-normal distributions as special cases and provides a flexible alternative to recently proposed skew t mixtures. We develop two analytically tractable EM-type algorithms for computing maximum likelihood estimates of model parameters in which the skewness parameters and degrees of freedom are asymptotically uncorrelated. Standard errors for the parameter estimates can be obtained via a general information-based method. We also present a procedure of merging mixture components to automatically identify the number of clusters by fitting piecewise linear regression to the rescaled entropy plot. The effectiveness and performance of the proposed methodology are illustrated by two real-life examples.     

On approximations to the inverse student’s-t distribution function

The inverse of the Student's t-distribution is often needed in computer simulation and applied statistics, e.g, in generating random variates from t-distributions and in computing tables needed for statistical procedures which do not assume known variances. The t-distribution algorithm of Dudewicz and Dalal (1972) can be used to approximate the inverset t distribution function. The author notes an algorithm for evaluation of this inverse d.f. which can be implemented in a fast, accurate and short computer program. The error analysis is also reported. An application is considered for the problem of testing the hypothesis that a sequence of random variates follows Student's-t distribution.     

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.     

Mini-batch learning of exponential family finite mixture models

Mini-batch algorithms have become increasingly popular due to the requirement for solving optimization problems, based on large-scale data sets. Using an existing online expectation–maximization (EM) algorithm framework, we demonstrate how mini-batch (MB) algorithms may be constructed, and propose a scheme for the stochastic stabilization of the constructed mini-batch algorithms. Theoretical results regarding the convergence of the mini-batch EM algorithms are presented. We then demonstrate how the mini-batch framework may be applied to conduct maximum likelihood (ML) estimation of mixtures of exponential family distributions, with emphasis on ML estimation for mixtures of normal distributions. Via a simulation study, we demonstrate that the mini-batch algorithm for mixtures of normal distributions can outperform the standard EM algorithm. Further evidence of the performance of the mini-batch framework is provided via an application to the famous MNIST data set.     

Analysis of local influence in geostatistics using Student's t-distribution

This article aims to estimate parameters of spatial variability with Student's t-distribution by the EM algorithm and present the study of local influence by means of two methods known as likelihood displacement and Q-displacement of likelihood, both using Student's t-distribution with fixed degrees of freedom (ν). The results showed that both methods are effective in the identification of influential points.     

Robust mixture modeling using multivariate skew <Emphasis Type="Italic">t</Emphasis> distributions

This paper presents a robust mixture modeling framework using the multivariate skew t distributions, an extension of the multivariate Student’s t family with additional shape parameters to regulate skewness. The proposed model results in a very complicated likelihood. Two variants of Monte Carlo EM algorithms are developed to carry out maximum likelihood estimation of mixture parameters. In addition, we offer a general information-based method for obtaining the asymptotic covariance matrix of maximum likelihood estimates. Some practical issues including the selection of starting values as well as the stopping criterion are also discussed. The proposed methodology is applied to a subset of the Australian Institute of Sport data for illustration.     

Testing for sub-models of the skew <Emphasis Type="Italic">t</Emphasis>-distribution

The skew t-distribution includes both the skew normal and the normal distributions as special cases. Inference for the skew t-model becomes problematic in these cases because the expected information matrix is singular and the parameter corresponding to the degrees of freedom takes a value at the boundary of its parameter space. In particular, the distributions of the likelihood ratio statistics for testing the null hypotheses of skew normality and normality are not asymptotically \(\chi ^2\). The asymptotic distributions of the likelihood ratio statistics are considered by applying the results of Self and Liang (J Am Stat Assoc 82:605–610, 1987) for boundary-parameter inference in terms of reparameterizations designed to remove the singularity of the information matrix. The Self–Liang asymptotic distributions are mixtures, and it is shown that their accuracy can be improved substantially by correcting the mixing probabilities. Furthermore, although the asymptotic distributions are non-standard, versions of Bartlett correction are developed that afford additional accuracy. Bootstrap procedures for estimating the mixing probabilities and the Bartlett adjustment factors are shown to produce excellent approximations, even for small sample sizes.     

A generalized skew two-piece skew-elliptical distribution

We present a new generalized family of skew two-piece skew-elliptical (GSTPSE) models and derive some its statistical properties. It is shown that the new family of distribution may be written as a mixture of generalized skew elliptical distributions. Also, a new representation theorem for a special case of GSTPSE-distribution is given. Next, we will focus on t kernel density and prove that it is a scale mixture of the generalized skew two-piece skew normal distributions. An explicit expression for the central moments as well as a recurrence relations for its cumulative distribution function and density are obtained. Since, this special case is a uni-/bimodal distribution, a sufficient condition for each cases is given. A real data set on heights of Australian females athletes is analysed. Finally, some concluding remarks and open problems are discussed.     

Robust skew-<Emphasis Type="Italic">t</Emphasis> factor analysis models for handling missing data

This paper presents a novel framework for maximum likelihood (ML) estimation in skew-t factor analysis (STFA) models in the presence of missing values or nonresponses. As a robust extension of the ordinary factor analysis model, the STFA model assumes a restricted version of the multivariate skew-t distribution for the latent factors and the unobservable errors to accommodate non-normal features such as asymmetry and heavy tails or outliers. An EM-type algorithm is developed to carry out ML estimation and imputation of missing values under a missing at random mechanism. The practical utility of the proposed methodology is illustrated through real and synthetic data examples.     

Robust mixture modeling using the skew <Emphasis Type="Italic">t</Emphasis> distribution

A finite mixture model using the Student's t distribution has been recognized as a robust extension of normal mixtures. Recently, a mixture of skew normal distributions has been found to be effective in the treatment of heterogeneous data involving asymmetric behaviors across subclasses. In this article, we propose a robust mixture framework based on the skew t distribution to efficiently deal with heavy-tailedness, extra skewness and multimodality in a wide range of settings. Statistical mixture modeling based on normal, Student's t and skew normal distributions can be viewed as special cases of the skew t mixture model. We present analytically simple EM-type algorithms for iteratively computing maximum likelihood estimates. The proposed methodology is illustrated by analyzing a real data example.     

Box–Cox realized asymmetric stochastic volatility models with generalized Student's t-error distributions

This study proposes a class of non-linear realized stochastic volatility (SV) model by applying the Box–Cox (BC) transformation, instead of the logarithmic transformation, to the realized estimator. The non-Gaussian distributions such as Student's t, non-central Student's t, and generalized hyperbolic skew Student's t-distributions are applied to accommodate heavy-tailedness and skewness in returns. The proposed models are fitted to daily returns and realized kernel of six stocks: SP500, FTSE100, Nikkei225, Nasdaq100, DAX, and DJIA using an Markov chain Monte Carlo Bayesian method, in which the Hamiltonian Monte Carlo (HMC) algorithm updates BC parameter and the Riemann manifold HMC algorithm updates latent variables and other parameters that are unable to be sampled directly. Empirical studies provide evidence against both the logarithmic transformation and raw versions of realized SV model.     

Characterizations of Student's t-distribution via regressions of order statistics

Utilizing regression properties of order statistics, we characterize a family of distributions introduced by Akhundov et al. [New characterizations by properties of midrange and related statistics, Commun. Stat. Theory Methods 33(12) (2004), pp. 3133–3143], which includes the t-distribution with two degrees of freedom as one of its members. Then we extend this characterization result to t-distribution with more than two degrees of freedom.     

Domains of convergence for the EM algorithm: a cautionary tale in a location estimation problem

The EM algorithm is a popular method for maximizing a likelihood in the presence of incomplete data. When the likelihood has multiple local maxima, the parameter space can be partitioned into domains of convergence, one for each local maximum. In this paper we investigate these domains for the location family generated by the t-distribution. We show that, perhaps somewhat surprisingly, these domains need not be connected sets. As an extreme case we give an example of a domain which consists of an infinite union of disjoint open intervals. Thus the convergence behaviour of the EM algorithm can be quite sensitive to the starting point.     

A Note on Robustness of Normal Variance Estimators Under t-Distributions

ABSTRACT

In many real life problems one assumes a normal model because the sample histogram looks unimodal, symmetric, and/or the standard tests like the Shapiro-Wilk test favor such a model. However, in reality, the assumption of normality may be misplaced since the normality tests often fail to detect departure from normality (especially for small sample sizes) when the data actually comes from slightly heavier tail symmetric unimodal distributions. For this reason it is important to see how the existing normal variance estimators perform when the actual distribution is a t-distribution with k degrees of freedom (d.f.) (t _k -distribution). This note deals with the performance of standard normal variance estimators under the t _k -distributions. It is shown that the relative ordering of the estimators is preserved for both the quadratic loss as well as the entropy loss irrespective of the d.f. and the sample size (provided the risks exist).     

Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts

A new method to calculate the multivariate t-distribution is introduced. We provide a series of substitutions, which transform the starting q-variate integral into one over the (q—1)-dimensional hypercube. In this situation standard numerical integration methods can be applied. Three algorithms are discussed in detail. As an application we derive an expression to calculate the power of multiple contrast tests assuming normally distributed data.     

A robust multivariate Birnbaum–Saunders distribution: EM estimation

We propose here a robust multivariate extension of the bivariate Birnbaum–Saunders (BS) distribution derived by Kundu et al. [Bivariate Birnbaum–Saunders distribution and associated inference. J Multivariate Anal. 2010;101:113–125], based on scale mixtures of normal (SMN) distributions that are used for modelling symmetric data. This resulting multivariate BS-type distribution is an absolutely continuous distribution whose marginal and conditional distributions are of BS-type distribution of Balakrishnan et al. [Estimation in the Birnbaum–Saunders distribution based on scalemixture of normals and the EM algorithm. Stat Oper Res Trans. 2009;33:171–192]. Due to the complexity of the likelihood function, parameter estimation by direct maximization is very difficult to achieve. For this reason, we exploit the nice hierarchical representation of the proposed distribution to propose a fast and accurate EM algorithm for computing the maximum likelihood (ML) estimates of the model parameters. We then evaluate the finite-sample performance of the developed EM algorithm and the asymptotic properties of the ML estimates through empirical experiments. Finally, we illustrate the obtained results with a real data and display the robustness feature of the estimation procedure developed here.