Multivariate T-distribution Research Articles

Bivariate asymmetric GARCH models with heavy tails and dynamic conditional correlations

A bivariate generalized autoregressive conditional heteroskedastic model with dynamic conditional correlation and leverage effect (DCC-GJR-GARCH) for modelling financial time series data is considered. For robustness it is helpful to assume a multivariate Student-t distribution for the innovation terms. This paper proposes a new modified multivariate t-distribution which is a robustifying distribution and offers independent marginal Student-t distributions with different degrees of freedom, thereby highlighting the relationship among different assets. A Bayesian approach with adaptive Markov chain Monte Carlo methods is used for statistical inference. A simulation experiment illustrates good performance in estimation over reasonable sample sizes. In the empirical studies, the pairwise relationship between the Australian stock market and foreign exchange market, and between the US stock market and crude oil market are investigated, including out-of-sample volatility forecasts.

Model-Based Learning Using a Mixture of Mixtures of Gaussian and Uniform Distributions

We introduce a mixture model whereby each mixture component is itself a mixture of a multivariate Gaussian distribution and a multivariate uniform distribution. Although this model could be used for model-based clustering (model-based unsupervised learning) or model-based classification (model-based semi-supervised learning), we focus on the more general model-based classification framework. In this setting, we fit our mixture models to data where some of the observations have known group memberships and the goal is to predict the memberships of observations with unknown labels. We also present a density estimation example. A generalized expectation-maximization algorithm is used to estimate the parameters and thereby give classifications in this mixture of mixtures model. To simplify the model and the associated parameter estimation, we suggest holding some parameters fixed-this leads to the introduction of more parsimonious models. A simulation study is performed to illustrate how the model allows for bursts of probability and locally higher tails. Two further simulation studies illustrate how the model performs on data simulated from multivariate Gaussian distributions and on data from multivariate t-distributions. This novel approach is also applied to real data and the performance of our approach under the various restrictions is discussed.

A Package to Study the Performance of Step-Up and Step-Down Test Procedures

The package can be used to analyze the performance of step-up and step-down procedures. It can be used to compare powers, calculate the “false discovery rate”, to study the effects of reduced step procedures, and to calculate P [U ≤ k], where U is the number of rejected true hypotheses. It can be used to determine the maximum number of steps that can be made and still guarantee (with a given probability) that the number of false rejections will not exceed some specified number. The test statistics are assumed to have a multivariate-t distribution. Examples are included.

Dynamic Conditional Correlations for Asymmetric Processes

The paper develops two Dynamic Conditional Correlation (DCC) models, namely the Wishart DCC (WDCC) model and the Matrix-Exponential Conditional Correlation (MECC) model. The paper applies the WDCC approach to the exponential GARCH (EGARCH) and GJR models to propose asymmetric DCC models. We use the standardized multivariate t-distribution to accommodate heavy-tailed errors. The paper presents an empirical example using the trivariate data of the Nikkei 225, Hang Seng and Straits Times Indices for estimating and forecasting the WDCC-EGARCH and WDCC-GJR models, and compares the performance with the asymmetric BEKK model. The empirical results show that AIC and BIC favour the WDCC-EGARCH model to the WDCC-GJR and asymmetric BEKK models. Moreover, the empirical results indicate that the WDCC-EGARCH-t model produces reasonable VaR threshold forecasts, which are very close to the nominal 1% to 3% values.

Clustering gene expression time course data using mixtures of multivariate t-distributions

Clustering gene expression time course data is an important problem in bioinformatics because understanding which genes behave similarly can lead to the discovery of important biological information. Statistically, the problem of clustering time course data is a special case of the more general problem of clustering longitudinal data. In this paper, a very general and flexible model-based technique is used to cluster longitudinal data. Mixtures of multivariate t-distributions are utilized, with a linear model for the mean and a modified Cholesky-decomposed covariance structure. Constraints are placed upon the covariance structure, leading to a novel family of mixture models, including parsimonious models. In addition to model-based clustering, these models are also used for model-based classification, i.e., semi-supervised clustering. Parameters, including the component degrees of freedom, are estimated using an expectation-maximization algorithm and two different approaches to model selection are considered. The models are applied to simulated data to illustrate their efficacy; this includes a comparison with their Gaussian analogues—the use of these Gaussian analogues with a linear model for the mean is novel in itself. Our family of multivariate t mixture models is then applied to two real gene expression time course data sets and the results are discussed. We conclude with a summary, suggestions for future work, and a discussion about constraining the degrees of freedom parameter.

Comparing Correlation Matrix Estimators Via Kullback-Leibler Divergence

We use a self-averaging measure called Kullback-Leibler divergence to evaluate the performance of four different correlation estimators: Fourier, Pearson, Maximum Likelihood and Hayashi-Yoshida estimator. The study uses simulated transaction prices for a large number of stocks and different data generating mechanisms, including synchronous and non-synchronous transactions, homogeneous and heterogeneous inter-transaction time. Different distributions of stock returns, i.e. multivariate Normal and multivariate Student's t-distribution, are also considered. We show that Fourier and Pearson estimators are equivalent proxies of the `true' correlation matrix within all the settings under analysis, and that both methods are outperformed by the Maximum Likelihood estimator when prices are synchronously sampled and price fluctuations follow a multivariate Student's t-distribution. Finally, we suggest to solve the singularity problem affecting the Hayashi-Yoshida estimator by shrinking the correlation matrix towards either Pearson or Fourier matrices, and provide evidence that the resulting combination leads to an improved estimator with respect to its single components.

Robust estimation of the correlation matrix of longitudinal data

We propose a double-robust procedure for modeling the correlation matrix of a longitudinal dataset. It is based on an alternative Cholesky decomposition of the form Σ=DLL ⊤ D where D is a diagonal matrix proportional to the square roots of the diagonal entries of Σ and L is a unit lower-triangular matrix determining solely the correlation matrix. The first robustness is with respect to model misspecification for the innovation variances in D, and the second is robustness to outliers in the data. The latter is handled using heavy-tailed multivariate t-distributions with unknown degrees of freedom. We develop a Fisher scoring algorithm for computing the maximum likelihood estimator of the parameters when the nonredundant and unconstrained entries of (L,D) are modeled parsimoniously using covariates. We compare our results with those based on the modified Cholesky decomposition of the form LD 2 L ⊤ using simulations and a real dataset.

Functional Mapping of Dynamic Traits with Robust t-Distribution

Functional mapping has been a powerful tool in mapping quantitative trait loci (QTL) underlying dynamic traits of agricultural or biomedical interest. In functional mapping, multivariate normality is often assumed for the underlying data distribution, partially due to the ease of parameter estimation. The normality assumption however could be easily violated in real applications due to various reasons such as heavy tails or extreme observations. Departure from normality has negative effect on testing power and inference for QTL identification. In this work, we relax the normality assumption and propose a robust multivariate -distribution mapping framework for QTL identification in functional mapping. Simulation studies show increased mapping power and precision with the distribution than that of a normal distribution. The utility of the method is demonstrated through a real data analysis.

Model-based clustering, classification, and discriminant analysis via mixtures of multivariate t-distributions

The last decade has seen an explosion of work on the use of mixture models for clustering. The use of the Gaussian mixture model has been common practice, with constraints sometimes imposed upon the component covariance matrices to give families of mixture models. Similar approaches have also been applied, albeit with less fecundity, to classification and discriminant analysis. In this paper, we begin with an introduction to model-based clustering and a succinct account of the state-of-the-art. We then put forth a novel family of mixture models wherein each component is modeled using a multivariate t-distribution with an eigen-decomposed covariance structure. This family, which is largely a t-analogue of the well-known MCLUST family, is known as the tEIGEN family. The efficacy of this family for clustering, classification, and discriminant analysis is illustrated with both real and simulated data. The performance of this family is compared to its Gaussian counterpart on three real data sets.

Robust graphical modeling of gene networks using classical and alternative t-distributions

Graphical Gaussian models have proven to be useful tools for exploring network structures based on multivariate data. Applications to studies of gene expression have generated substantial interest in these models, and resulting recent progress includes the development of fitting methodology involving penalization of the likelihood function. In this paper we advocate the use of multivariate t-distributions for more robust inference of graphs. In particular, we demonstrate that penalized likelihood inference combined with an application of the EM algorithm provides a computationally efficient approach to model selection in the t-distribution case. We consider two versions of multivariate t-distributions, one of which requires the use of approximation techniques. For this distribution, we describe a Markov chain Monte Carlo EM algorithm based on a Gibbs sampler as well as a simple variational approximation that makes the resulting method feasible in large problems.

Missing covariates in logistic regression, estimation and distribution selection

We derive explicit formulae for estimation in logistic regression models where some of the covariates are missing. Our approach allows for modelling the distribution of the missing covariates either as a multivariate normal or as a multivariate t-distribution. A main advantage of this method is that it is fast and does not require the use of iterative procedures. A model selection method is derived which allows to choose among these distributions. In addition, we consider versions of Akaike’s information criterion that are based on the expectation–maximization algorithm and multiple imputation methods that have a wide applicability to model selection in likelihood models in general.

A Dunnett-Type Procedure for Multiple Endpoints

This paper describes a method for comparisons of several treatments with a control, simultaneously for multiple endpoints. These endpoints are assumed to be normally distributed with different scales and variances. An approximate multivariate t-distribution is used to obtain quantiles for test decisions, multiplicity-adjusted p-values, and simultaneous confidence intervals. Simulation results show that this approach controls the family-wise error type I over both the comparisons and the endpoints in an admissible range. The approach will be applied to a randomized clinical trial comparing two new sets of extracorporeal circulations with a standard for three primary endpoints. A related R package is available.

Robust Curve Clustering Based on a Multivariate $t$-Distribution Model

This brief presents a curve clustering technique based on a new multivariate model. Instead of the usual Gaussian random effect model, our method uses the multivariate t-distribution model which has better robustness to outliers and noise. In our method, we use the B-spline curve to model curve data and apply the mixed-effects model to capture the randomness and covariance of all curves within the same cluster. After fitting the B-spline-based mixed-effects model to the proposed multivariate t -distribution, we derive an expectation-maximization algorithm for estimating the parameters of the model, and apply the proposed approach to the simulated data and the real dataset. The experimental results show that our model yields better clustering results when compared to the conventional Gaussian random effect model.

Relative Potency Estimation in Parallel-Line Assays – Method Comparison and Some Extensions

Relative potency estimations in both multiple parallel-line and slope-ratio assays involve construction of simultaneous confidence intervals for ratios of linear combinations of general linear model parameters. The key problem here is that of determining multiplicity adjusted percentage points of a multivariate t-distribution, the correlation matrix R of which depends on the unknown relative potency parameters. Several methods have been proposed in the literature on how to deal with R . In this article, we introduce a method based on an estimate of R (also called the plug-in approach) and compare it with various methods including conservative procedures based on probability inequalities. Attention is restricted to parallel-line assays though the theory is applicable for any ratios of coefficients in the general linear model. Extension of the plug-in method to linear mixed effect models is also discussed. The methods will be compared with respect to their simultaneous coverage probabilities via Monte Carlo simulations. We also evaluate the methods in terms of confidence interval width through application to data on multiple parallel-line assay.

Maximum Likelihood Estimation of Tobit Factor Analysis for Multivariate t-Distribution

We propose multivariate tobit factor analysis models by assuming multivariate t-distribution error in tobit factor analysis. To circumvent direct calculation of the intractable likelihood function, we apply the MCEM algorithm to obtain the MLEs with its E-step made feasible by Monte Carlo estimates and its M-step greatly simplified under Conditional Maximization. Standard errors are evaluated by using Louis's method. The algorithm was illustrated through a simulation study.

On EM Estimation for Mixture of Multivariate t-Distributions

This paper formulates a novel expectation maximization (EM) algorithm for the mixture of multivariate t-distributions. By introducing a new kind of "missing" data, we show that the empirically improved iterative algorithm, in literature, for the mixture of multivariate t-distributions is in fact a type of EM algorithm; thus a theoretical analysis is established, which guarantees the empirical algorithm converges to the maximization likelihood estimates of the mixture parameters. Simulated experiment and real experiments on classification and image segmentation confirm the effectiveness of the improved EM algorithm.

Simultaneous confidence intervals using ordinal effect measures for ordered categorical outcomes

When outcomes are ordered categorical, a model using an ordinal effect size measure is a good alternative of the cumulative logit model to compare several independent group differences. We present a method of constructing simultaneous confidence intervals for the ordinal effect size measures, using the studentized range distribution with the score test statistic. A simulation study shows that the proposed method performs well in terms of coverage probability, and it seems better than the method using a Bonferroni correction for Wald-type statistics and methods that account for the dependencies among pairwise ordinal effect size measures using the multivariate normal distribution (or the multivariate t-distribution for small samples).

Introducing robustness to maximum-likelihood refinement of electron-microscopy data.

An expectation-maximization algorithm for maximum-likelihood refinement of electron-microscopy images is presented that is based on fitting mixtures of multivariate t-distributions. The novel algorithm has intrinsic characteristics for providing robustness against atypical observations in the data, which is illustrated using an experimental test set with artificially generated outliers. Tests on experimental data revealed only minor differences in two-dimensional classifications, while three-dimensional classification with the new algorithm gave stronger elongation factor G density in the corresponding class of a structurally heterogeneous ribosome data set than the conventional algorithm for Gaussian mixtures.

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.

A Bayesian Reliability Approach to Multiple Response Optimization with Seemingly Unrelated Regression Models

This paper presents a Bayesian predictive approach to multiresponse optimization experiments. It generalizes the work of Peterson [33] in two ways that make it more flexible for use in applications. First, a multivariate posterior predictive distribution of seemingly unrelated regression models is used to determine optimum factor levels by assessing the reliability of a desired multivariate response. It is shown that it is possible for optimal mean response surfaces to appear satisfactory yet be associated with unsatisfactory overall process reliabilities. Second, the use of a multivariate normal distribution for the vector of regression error terms is generalized to that of the (heavier tailed) multivariate t-distribution. This provides a Bayesian sensitivity analysis with regard to moderate outliers. The effect of adding design points is also considered through a preposterior analysis. The advantages of this approach are illustrated with two real examples.