Multivariate Skew-t Distributions Research Articles

Flexible Bayesian semiparametric mixed-effects model for skewed longitudinal data

BackgroundIn clinical trials and epidemiological research, mixed-effects models are commonly used to examine population-level and subject-specific trajectories of biomarkers over time. Despite their increasing popularity and application, the specification of these models necessitates a great deal of care when analysing longitudinal data with non-linear patterns and asymmetry. Parametric (linear) mixed-effect models may not capture these complexities flexibly and adequately. Additionally, assuming a Gaussian distribution for random effects and/or model errors may be overly restrictive, as it lacks robustness against deviations from symmetry.MethodsThis paper presents a semiparametric mixed-effects model with flexible distributions for complex longitudinal data in the Bayesian paradigm. The non-linear time effect on the longitudinal response was modelled using a spline approach. The multivariate skew-t distribution, which is a more flexible distribution, is utilized to relax the normality assumptions associated with both random-effects and model errors.ResultsTo assess the effectiveness of the proposed methods in various model settings, simulation studies were conducted. We then applied these models on chronic kidney disease (CKD) data and assessed the relationship between covariates and estimated glomerular filtration rate (eGFR). First, we compared the proposed semiparametric partially linear mixed-effect (SPPLM) model with the fully parametric one (FPLM), and the results indicated that the SPPLM model outperformed the FPLM model. We then further compared four different SPPLM models, each assuming different distributions for the random effects and model errors. The model with a skew-t distribution exhibited a superior fit to the CKD data compared to the Gaussian model. The findings from the application revealed that hypertension, diabetes, and follow-up time had a substantial association with kidney function, specifically leading to a decrease in GFR estimates.ConclusionsThe application and simulation studies have demonstrated that our work has made a significant contribution towards a more robust and adaptable methodology for modeling intricate longitudinal data. We achieved this by proposing a semiparametric Bayesian modeling approach with a spline smoothing function and a skew-t distribution.

Likelihood-based inference for the multivariate skew-[formula omitted] regression with censored or missing responses

Skew-t regression models have been widely used to model and analyze asymmetric heavy-tailed data. Moreover, observations in this kind of data can be missing or subject to some upper and/or lower detection limits because of the restriction of the experimental apparatus. We propose a novel robust regression model for multiple censored or missing data based on the multivariate skew-t distribution for such data structures. This approach allows us to model data with great flexibility, simultaneously accommodating heavy tails and skewness. We develop an analytically simple yet efficient EM-type algorithm to conduct maximum likelihood estimation of the parameters. The algorithm has closed-form expressions at the E-step that rely on formulas for the mean and variance of truncated multivariate Student’s-t, skew-t, and extended skew-t distributions. Furthermore, a general information-based method for approximating the asymptotic covariance matrix of the estimators is also presented. Results obtained from the analysis of both simulated and real datasets are reported to demonstrate the effectiveness of the proposed method.

Measuring systemic risk with high-frequency data: A realized GARCH approach

This paper incorporates high-frequency information to measure systemic risk. Under the Multivariate Realized GARCH framework, we compute the CoVaR measure using a multivariate skew-t distribution. Using 5-minute data of 98 U.S. financial institutions from 2000 to 2022, we show the empirical improvement of the high-frequency measurement. We also investigate the relationship between institutions’ systemic risk contributions and firm-level characteristics. Our empirical findings suggest that firm size and leverage are positively related to institutions’ contributions to systemic risk.

A Semiparametric Bayesian Joint Modelling of Skewed Longitudinal and Competing Risks Failure Time Data: With Application to Chronic Kidney Disease

In clinical and epidemiological studies, when the time-to-event(s) and the longitudinal outcomes are associated, modelling them separately may give biased estimates. A joint modelling approach is required to obtain unbiased results and to evaluate their association. In the joint model, a subject may be exposed to more than one type of failure event (competing risks). Considering the competing event as an independent censoring of the time-to-event process may underestimate the true survival probability and give biased results. Within the joint model, longitudinal outcomes may have nonlinear (irregular) trajectories over time and exhibit skewness with heavy tails. Accordingly, fully parametric mixed-effect models may not be flexible enough to model this type of complex longitudinal data. In addition, assuming a Gaussian distribution for model errors may be too restrictive to adequately represent within-individual variations and may lack robustness against deviation from distributional assumptions. To simultaneously overcome these issues, in this paper, we presented semiparametric joint models for competing risks failure time and skewed-longitudinal data by using a smoothing spline approach and a multivariate skew-t distribution. We also considered different parameterization approaches in the formulation of joint models and used a Bayesian approach to make the statistical inference. We illustrated the proposed methods by analyzing real data on a chronic kidney disease. To evaluate the performance of the methods, we also carried out simulation studies. The results of both the application and simulation studies revealed that the joint modelling approach proposed in this study performed well when the semiparametric, random-effects parameterization, and skew-t distribution specifications were taken into account.

A Southern Photometric Quasar Catalog from the Dark Energy Survey Data Release 2

We present a catalog of 1.4 million photometrically selected quasar candidates in the southern hemisphere over the ∼5000 deg2 Dark Energy Survey (DES) wide survey area. We combine optical photometry from the DES second data release (DR2) with available near-infrared (NIR) and the all-sky unWISE mid-infrared photometry in the selection. We build models of quasars, galaxies, and stars with multivariate skew-t distributions in the multidimensional space of relative fluxes as functions of redshift (or color for stars) and magnitude. Our selection algorithm assigns probabilities for quasars, galaxies, and stars and simultaneously calculates photometric redshifts (photo-z) for quasar and galaxy candidates. Benchmarking on spectroscopically confirmed objects, we successfully classify (with photometry) 94.7% of quasars, 99.3% of galaxies, and 96.3% of stars when all IR bands (NIR YJHK and WISE W1W2) are available. The classification and photo-z regression success rates decrease when fewer bands are available. Our quasar (galaxy) photo-z quality, defined as the fraction of objects with the difference between the photo-z z p and the spectroscopic redshift z s , ∣Δz∣ ≡ ∣z s − z p ∣/(1 + z s ) ≤ 0.1, is 92.2% (98.1%) when all IR bands are available, decreasing to 72.2% (90.0%) using optical DES data only. Our photometric quasar catalog achieves an estimated completeness of 89% and purity of 79% at r < 21.5 (0.68 million quasar candidates), with reduced completeness and purity at 21.5 < r ≲ 24. Among the 1.4 million quasar candidates, 87,857 have existing spectra, and 84,978 (96.7%) of them are spectroscopically confirmed quasars. Finally, we provide quasar, galaxy, and star probabilities for all (0.69 billion) photometric sources in the DES DR2 coadded photometric catalog.

Some Statistical Aspects of the Truncated Multivariate Skew-t Distribution

The multivariate skew-t distribution plays an important role in statistics since it combines skewness with heavy tails, a very common feature in real-world data. A generalization of this distribution is the truncated multivariate skew-t distribution which contains the truncated multivariate t distribution and the truncated multivariate skew-normal distribution as special cases. In this article, we study several distributional properties of the truncated multivariate skew-t distribution involving affine transformations, marginalization, and conditioning. The generation of random samples from this distribution is described.

Efficient recursive computational algorithms for multivariate t and multivariate unified skew-t distributions with applications to inference

Efficient recursive computational algorithms for multivariate t and multivariate unified skew-t distributions with applications to inference

Skewness-Kurtosis Model-Based Projection Pursuit with Application to Summarizing Gene Expression Data

Non-normality is a usual fact when dealing with gene expression data. Thus, flexible models are needed in order to account for the underlying asymmetry and heavy tails of multivariate gene expression measures. This paper addresses the issue by exploring the projection pursuit problem under a flexible framework where the underlying model is assumed to follow a multivariate skew-t distribution. Under this assumption, projection pursuit with skewness and kurtosis indices is addressed as a natural approach for data reduction. The work examines its properties giving some theoretical insights and delving into the computational side in regards to the application to real gene expression data. The results of the theory are illustrated by means of a simulation study; the outputs of the simulation are used in combination with the theoretical insights to shed light on the usefulness of skewness-kurtosis projection pursuit for summarizing multivariate gene expression data. The application to gene expression measures of patients diagnosed with triple-negative breast cancer gives promising findings that may contribute to explain the heterogeneity of this type of tumors.

An Information-Theoretic Approach for Multivariate Skew-t Distributions and Applications

Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew-t (FMST) distributions. This class of flexible distributions allows handling asymmetry and tail weight behavior simultaneously. We find upper and lower bounds of Rényi entropy for these families. Numerical simulations illustrate the results for several scenarios: symmetry/asymmetry and light/heavy-tails. Finally, we present applications of our findings to a swordfish length-weight dataset to illustrate the behavior of entropies of the FMST distribution. Comparisons with the counterparts—the finite mixture of multivariate skew-normal and normal distributions—are also presented.

Sparsity-regularized skewness estimation for the multivariate skew normal and multivariate skew [formula omitted] distributions

The multivariate skew normal (MSN) and multivariate skew t (MST) distributions have received considerable attention in the past two decades because of their appealing mathematical properties and their usefulness for modeling skewed data. We develop sparse regularization methodology for estimating the skewness parameters of these two distributions. This methodology facilitates skewness selection, i.e., the identification of those marginal indices of skewness, if any, that are equal to zero. Obstacles that render skewness selection infeasible for existing parameterizations of the two distributions are described, and a new parameterization that permits the circumvention of those obstacles is introduced. A penalized likelihood method for sparsity-based regularized skewness estimation using the new parameterization is proposed. Model selection consistency and the oracle property of the method are established. A simulation study demonstrates that the method is reasonably effective for skewness selection and, because of inclusion of a ridge penalty, is more effective at preventing the divergence of the shape parameter in small samples than the Q-penalty approach of Azzalini and Arellano-Valle (2013). The simulation study demonstrates further that our method may improve the estimation of all parameters of the MSN and MST distributions, not merely the skewnesses, when some of the skewnesses are zero.

Comparison of Multivariate Outlier Detection Methods for Nearly Elliptical Distributions

In this paper, the performance of outlier detection methods has been evaluated with symmetrically distributed datasets. We choose four estimators, viz. modified Stahel-Donoho (MSD) estimators, blocked adaptive computationally efficient outlier nominators, minimum covariance determinant estimator obtained by a fast algorithm, and nearest-neighbour variance estimator, which are known for their good performance with elliptically distributed data, for practical applications in national survey data processing. We adopt the data model of multivariate skew-t distribution, of which only the direction of the main axis is skewed and contaminated with outliers following another probability distribution for evaluation. We conducted Monte Carlo simulation under the data distribution to compare the performance of outlier detection. We also explore the applicability of the selected methods for several accounting items in small and medium enterprise survey data. Accordingly, it was found that the MSD estimators are the most suitable.

A monotone data augmentation algorithm for longitudinal data analysis via multivariate skew-t, skew-normal or t distributions.

The mixed effects model for repeated measures has been widely used for the analysis of longitudinal clinical data collected at a number of fixed time points. We propose a robust extension of the mixed effects model for repeated measures for skewed and heavy-tailed data on basis of the multivariate skew-t distribution, and it includes the multivariate normal, t, and skew-normal distributions as special cases. An efficient Markov chain Monte Carlo algorithm is developed using the monotone data augmentation and parameter expansion techniques. We employ the algorithm to perform controlled pattern imputations for sensitivity analyses of longitudinal clinical trials with nonignorable dropouts. The proposed methods are illustrated by real data analyses. Sample SAS programs for the analyses are provided in the online supplementary material.

Recent Advances in Spatio‐Temporal Methodology

This special issue consists of a collection of articles that describe innovations in spatio-temporal methodology. Spatio-temporal statistics has been developing at a rapid pace over the past 25 years. The topic is briefly covered in Cressie's (1993) comprehensive book on spatial statistics. Subsequently, Cressie and Wikle (2011) provided the first comprehensive book on spatio-temporal statistics, but it focused primarily on linear, univariate, and Gaussian methods. More recently, there have been numerous significant advancements in non-linear, multivariate, and non-Gaussian spatio-temporal methods, particularly those suited for ‘big’ data problems. Yet, each of these topic areas is relatively underdeveloped, and there are still many research challenges. Historically, spatio-temporal statistics methodology has developed more as an extension of spatial statistics rather than time series. That is, the emphasis has been on specification of models through their second-order structure, typically within a Gaussian process framework. Alternatively, dynamic spatio-temporal models (DSTMs) have been used to model processes for which it is more realistic to think of spatial processes evolving through time, that is, when it is more reasonable to think of the process conditionally (in time) rather than marginally (as with the Gaussian process framework). DSTMs often have model forms as in classical multivariate time series models, but they present unique challenges in that the types of relationships between space and time are often driven by mechanistic processes, and the associated statistical models must attempt to accommodate these relationships. In addition, the dimensionality of the spatial components of these models often prohibits the use of classical multivariate time series methods. Although much of the development of spatio-temporal methodology has been driven by environmental, epidemiological, and ecological applications (e.g. pollution monitoring, weather forecasting, climate, oceanography, disease mapping, invasive species, animal movement, etc.), there is an increasing number of novel methodologies that are motivated by the biological sciences (e.g. brain science), federal statistics (e.g. multivariate surveys, non-Gaussian change of support), sociological statistics (e.g. crime analysis), and econometric (e.g. multivariate panel) applications. Indeed, these processes are often multivariate, non-linear, and/or non-Gaussian. The articles in this special issue provide a representative snapshot of innovative work in these areas. Specifically, the articles selected for this issue highlight non-Gaussian spatio-temporal models, computational efficiency, and/or improving parameterizations of DSTMs. In their article ‘Scalable Inference for Space-Time Gaussian Cox Processes’, Shirota and Banerjee show how one can use an efficient data augmentation approach in conjunction with nearest-neighbor log-Gaussian processes to efficiently model count (crime event) data via spatio-temporal Gaussian Cox processes. In ‘Estimating Spatial Changes Over Time of Artic Sea Ice Using Hidden 2 × 2 Tables’, Zhang and Cressie consider non-Gaussian (Bernoulli) data on the presence/absence of sea ice. The model is cast in an efficient manner by using dimension-reduced spatio-temporal processes in an EM algorithm, cleverly describing the process as time-varying 2 × 2 tables. Tagle, Castruccio, Crippa, and Genton, in their article ‘A Non-Gaussian Spatio-Temporal Model for Daily Wind Speeds Based on a Multivariate Skew-T Distribution’, show how to build an efficient and realistic ‘weather generator’ for daily wind speeds that is able to simulate the type of skewed distributions that are often found in weather variables. In their article ‘On a Semiparametric Data-Driven Nonlinear Model with Penalized Spatio-Temporal Lag Interactions’, Al-Sulami, Jiang, Lu, and Zhu consider a non-linear (semiparametric) regression spatio-temporal model that also includes a dynamic interaction term and consider an adaptive lasso approach for selecting the space–time lag interactions that are most useful (in their example, to model US housing price data). Their methodology does not rely on a Gaussian error assumption. In the spirit of efficient modeling of the spatio-temporal terms beyond a trend and seasonality, Gao and Tsay present an efficient structured factor approach for multivariate time series and spatio-temporal data (e.g. PM2.5 pollutant observations) in their article ‘A Structural-Factor Approach to Modeling High-Dimensional Time Series and Space-Time Data’. Finally, in their article ‘Spatio-Temporal Models for Big Multinomial Data Using the Conditional Multivariate Logit-Beta Distribution’, Bradley, Wikle and Holan develop a novel efficient conjugate Bayesian algorithm to model high-dimensional spatio-temporal multinomial data, which includes a spatio-temporal dynamic process. They apply this model to public-use Quarterly Workforce Indicators from the Longitudinal Employer Household Dynamics program of the US Census Bureau. These articles provide excellent examples of how spatio-temporal statistical models can be applied to address complex non-Gaussian, non-linear, and multivariate data with efficient computational algorithms. These problems are by no means ‘solved’, but it is our hope that the readers of the Journal of Time Series Analysis may see connections to their own work that can be applied to this rich set of problems and this growing and important area of statistics. Finally, we would like to end this Editorial by thanking Prof. A. M. R. Taylor for encouraging us to edit this special issue.

Mixtures of generalized hyperbolic distributions and mixtures of skew-t distributions for model-based clustering with incomplete data

Robust clustering from incomplete data is an important topic because, in many practical situations, real datasets are heavy-tailed, asymmetric, and/or have arbitrary patterns of missing observations. Flexible methods and algorithms for model-based clustering are presented via mixture of the generalized hyperbolic distributions and its limiting case, the mixture of multivariate skew-t distributions. An analytically feasible EM algorithm is formulated for parameter estimation and imputation of missing values for mixture models employing missing at random mechanisms. The proposed methodologies are investigated through a simulation study with varying proportions of synthetic missing values and illustrated using a real dataset. Comparisons are made with those obtained from the traditional mixture of generalized hyperbolic distribution counterparts by filling in the missing data using the mean imputation method.

Maximum likelihood estimation of skew-t copulas with its applications to stock returns

ABSTRACTThe multivariate Student-t copula family is used in statistical finance and other areas when there is tail dependence in the data. It often is a good-fitting copula but can be improved on when there is tail asymmetry. Multivariate skew-t copula families can be considered when there is tail dependence and tail asymmetry, and we show how a fast numerical implementation for maximum likelihood estimation is possible. For the copula implicit in a multivariate skew-t distribution, the fast implementation makes use of (i) monotone interpolation of the univariate marginal quantile function and (ii) a re-parametrization of the correlation matrix. Our numerical approach is tested with simulated data with data-driven parameters. A real data example involves the daily returns of three stock indices: the Nikkei225, S&P500 and DAX. With both unfiltered returns and GARCH/EGARCH filtered returns, we compare the fits of the Azzalini–Capitanio skew-t, generalized hyperbolic skew-t, Student-t, skew-Normal and Normal copulas.

Multivariate measures of skewness for the scale mixtures of skew-normal distributions

Several measures of multivariate skewness for scale mixtures of skew-normal distributions are derived. As a special case, those of multivariate skew-t distribution are considered in detail. Furthermore, the similarities, differences, and behavior of these measures are explored for cases of some specific members of the multivariate skew-normal and skew-t distributions using a simulation study. Since some measures are vectors, it is better to take all measures in the same scale when comparing them. In order to attain such a set of comparable indices, the sample version is considered for each of the skewness measures that are taken as test statistics for the hypothesis of t distribution against skew-t distribution. An application is reported for the data set consisting of 71 total glycerol and magnesium contents in Grignolino wine.

Mixtures of restricted skew-t factor analyzers with common factor loadings

Mixtures of common t factor analyzers (MCtFA) have been shown its effectiveness in robustifying mixtures of common factor analyzers (MCFA) when handling model-based clustering of the high-dimensional data with heavy tails. However, the MCtFA model may still suffer from a lack of robustness against observations whose distributions are highly asymmetric. This paper presents a further robust extension of the MCFA and MCtFA models, called the mixture of common restricted skew-t factor analyzers (MCrstFA), by assuming a restricted multivariate skew-t distribution for the common factors. The MCrstFA model can be used to accommodate severely non-normal (skewed and leptokurtic) random phenomena while preserving its parsimony in factor-analytic representation and performing graphical visualization in low-dimensional plots. A computationally feasible expectation conditional maximization either algorithm is developed to carry out maximum likelihood estimation. The numbers of factors and mixture components are simultaneously determined based on common likelihood penalized criteria. The usefulness of our proposed model is illustrated with simulated and real datasets, and experimental results signify its superiority over some existing competitors.

Quasar Photometric Redshifts and Candidate Selection: A New Algorithm Based on Optical and Mid-infrared Photometric Data

Abstract We present a new algorithm to estimate quasar photometric redshifts (photo-zs), by considering the asymmetries in the relative flux distributions of quasars. The relative flux models are built with multivariate Skew-t distributions in the multidimensional space of relative fluxes as a function of redshift and magnitude. For 151,392 quasars in the SDSS, we achieve a photo-z accuracy, defined as the fraction of quasars with the difference between the photo-z z p and the spectroscopic redshift z s , within 0.1, of 74%. Combining the WISE W1 and W2 infrared data with the SDSS data, the photo-z accuracy is enhanced to 87%. Using the Pan-STARRS1 or DECaLS photometry with WISE W1 and W2 data, the photo-z accuracies are 79% and 72%, respectively. The prior probabilities as a function of magnitude for quasars, stars, and galaxies are calculated, respectively, based on (1) the quasar luminosity function, (2) the Milky Way synthetic simulation with the Besançon model, and (3) the Bayesian Galaxy Photometric Redshift estimation. The relative fluxes of stars are obtained with the Padova isochrones, and the relative fluxes of galaxies are modeled through galaxy templates. We test our classification method to select quasars using the DECaLS g, r, z, and WISE W1 and W2 photometry. The quasar selection completeness is higher than 70% for a wide redshift range , and a wide magnitude range mag. Our photo-z regression and classification method has the potential to extend to future surveys. The photo-z code will be publicly available.

Robust skew-t 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.

Mixture modeling of freeway speed and headway data using multivariate skew-t distributions

ABSTRACTThe knowledge of vehicle speed and headway distributions is very useful for developing microscopic traffic simulation models. Traditionally, speed and headway distributions are often not studied jointly and some microscopic traffic simulation models consider vehicle speeds and arrival times as independent inputs to the traffic simulation process. However, the traditional approaches ignore the possible correlation between freeway vehicle speed and headway. Recently, a Farlie–Gumbel–Morgenstern (FGM) approach was used to construct bivariate distributions to describe the characteristics of speed and headway. The FGM approach only allows a weak statistical dependency and lacks the ability to consider the dynamic correlation structure for speed and headway data collected under different traffic conditions. The objective of this study is to explore the applicability of the finite mixtures of multivariate skew-t distributions to capture the heterogeneity in speed and headway data. The proposed bivariate mixture modeling approach is applied to the 24-hour traffic data collected on IH-35 in Austin, Texas. The results of this study show that the bivariate skew-t mixture model can provide an excellent fit to the multimodal speed and headway distribution. Moreover, the mixture modeling approach can naturally accommodate the varying correlation coefficient by assigning different covariance matrixes for each component in the finite mixture model. The findings in this study can also improve car-following models for simulation purposes.