Multivariate Skewness Research Articles

Multivariate unified skew-[formula omitted] distributions and their properties

The unified skew-t (SUT) is a flexible parametric multivariate distribution that accounts for skewness and heavy tails in the data. A few of its properties can be found scattered in the literature or in a parameterization that does not follow the original one for unified skew-normal (SUN) distributions, yet a systematic study is lacking. In this work, explicit properties of the multivariate SUT distribution are presented, such as its stochastic representations, moments, SUN-scale mixture representation, linear transformation, additivity, marginal distribution, canonical form, quadratic form, conditional distribution, change of latent dimensions, Mardia measures of multivariate skewness and kurtosis, and non-identifiability issue. These results are given in a parameterization that reduces to the original SUN distribution as a sub-model, hence facilitating the use of the SUT for applications. Several models based on the SUT distribution are provided for illustration.

The skewness of mean–variance normal mixtures

Mean–variance mixtures of normal distributions are very flexible: they model many nonnormal features, such as skewness, kurtosis and multimodality. Special cases include generalized asymmetric Laplace distributions, mixtures of two normal distributions with proportional covariance matrices, scale mixtures of normal distributions and normal distributions. This paper investigates the skewness of multivariate mean–variance normal mixtures. The special case of mixtures of two normal distributions with proportional covariance matrices is treated in greater detail. The paper derives the analytical forms of prominent measures of multivariate skewness and applies them to model-based clustering, normalizing linear transformations, projection pursuit and normality testing. The practical relevance of the theoretical results is assessed with both real and simulated data.

Face, Fingerprint, and Signature based Multimodal Biometric System using Score Level and Decision Level Fusion Approaches

Principal Component Analysis (PCA) is the best face recognition method. This research suggests PCA for fingerprint and signature recognition. Simple image processing transforms like DCT, 2D-DCT, DWT, SWT, 2D-SWT, SVD (Singular Vector Decomposition), Entropy, and Rank can be used for feature extraction. These transforms and measures are utilized with PCA as a feature extraction module to construct uni-modal and multimodal biometric systems using face, fingerprint, and signature modalities. Most PCA biometrics systems compare stored template to claimed identification using Euclidean distance. This paper proposes matching modules using similarity and dissimilarity measures viz. Absolute Pearson's Correlation Coefficient (APCC), Absolute Uncentered Pearson's Correlation Coefficient (AUPCC), Bray Curtis Distance (BC), Canberra distance (CB), Chebyshev Distance (CBS), Chessboard Distance (CSB), City block or Manhattan distance (CTB), Cross Correlation (CC), Dot product (DP), Euclidean distance (EUC), Extended Jaccard Distance (EJ), Hamming Distance (HM), Harmonically Summed Euclidean distance (HSEUC), Kendall Correlation Coefficient (KCC), Mahalanobis Distance (MH), Minimum Coordinate Difference (MCD), Minkowiski distance (MNK), Multivariate Kurtosis Coefficient (MVK), Multivariate Skew (MVS), Normalized City Block or Manhattan distance (NCTB), Normalized Cross-correlation (NCC), Normalized Euclidean distance (NEUC), Pearson’s Cosine Distance (PCOS), Pearson's Correlation Coefficient (PCC), Pearson's Absolute Value Dissimilarity (PAVD), Pearson's Linear Dissimilarity (PLDISS), Spearman Correlation Coefficient (SCC), Standardized Euclidean Distance (SEUC), Uncentered Pearson's Correlation Coefficient (UPCC), Wave-Hedges Distance (WVH). This study again discusses score level fusion of face, fingerprint, and signature using sum and max rules, z-score normalization, and decision level fusion using AND rule.

Dynamics diagnosis of the COVID-19 deaths using the Pearson diagram

The pandemic COVID-19 brings with it the need for studies and tools to help those in charge make decisions. Working with classical time series methods such as ARIMA and SARIMA has shown promising results in the first studies of COVID-19. We advance in this branch by proposing a risk factor map induced by the well-known Pearson diagram based on multivariate kurtosis and skewness measures to analyze the dynamics of deaths from COVID-19. In particular, we combine bootstrap for time series with SARIMA modeling in a new paradigm to construct a map on which one can analyze the dynamics of a set of time series. The proposed map allows a risk analysis of multiple countries in the four different periods of the pandemic COVID-19 in 55 countries. Our empirical evidence suggests a direct relationship between the multivariate skewness and kurtosis. We observe that the multivariate kurtosis increase leads to the rise of the multivariate skewness. Our findings reveal that the countries with high risk from the behavior of the number of deaths tend to have pronounced skewness and kurtosis values.

Fast inference methods for high-dimensional factor copulas

Abstract Gaussian factor models allow the statistician to capture multivariate dependence between variables. However, they are computationally cumbersome in high dimensions and are not able to capture multivariate skewness in the data. We propose a copula model that allows for arbitrary margins, and multivariate skewness in the data by including a non-Gaussian factor whose dependence structure is the result of a one-factor copula model. Estimation is carried out using a two-step procedure: margins are modelled separately and transformed to the normal scale, after which the dependence structure is estimated. We develop an estimation procedure that allows for fast estimation of the model parameters in a high-dimensional setting. We first prove the theoretical results of the model with up to three Gaussian factors. Then, simulation results confirm that the model works as the sample size and dimensionality grow larger. Finally, we apply the model to a selection of stocks of the S&P500, which demonstrates that our model is able to capture cross-sectional skewness in the stock market data.

Sub-dimensional Mardia measures of multivariate skewness and kurtosis

The Mardia measures of multivariate skewness and kurtosis summarize the respective characteristics of a multivariate distribution with two numbers. However, these measures do not reflect the sub-dimensional features of the distribution. Consequently, testing procedures based on these measures may fail to detect skewness or kurtosis present in a sub-dimension of the multivariate distribution. We introduce sub-dimensional Mardia measures of multivariate skewness and kurtosis, and investigate the information they convey about all sub-dimensional distributions of some symmetric and skewed families of multivariate distributions. The maxima of the sub-dimensional Mardia measures of multivariate skewness and kurtosis are considered, as these reflect the maximum skewness and kurtosis present in the distribution, and also allow us to identify the sub-dimension bearing the highest skewness and kurtosis. Asymptotic distributions of the vectors of sub-dimensional Mardia measures of multivariate skewness and kurtosis are derived, based on which testing procedures for the presence of skewness and of deviation from Gaussian kurtosis are developed. The performances of these tests are compared with some existing tests in the literature on simulated and real datasets.

Using (ECM) Algorithm to Estimate the Missing Values and Make Comparison between (MLE) and (GA) Algorithm for Estimating Parameters of Multivariate Skew Normal Distribution (MSN)

The estimation of statistical parameters for multivariate data leads to waste in the information if the missing values are neglected, which will subsequently lead to inaccurate estimates. Therefore, the incomplete data must be estimated using one of the statistical estimation methods to obtain accurate results and thus obtaining good estimates for the parameters.The aim of this paper is to estimate the missing values for the multivariate skew normal distribution function using the Expectation Conditional Maximization (ECM) algorithm. After estimating the missing values, the parameters are estimated using Maximum Likelihood Estimation (MLE) with the Newton-Raphson algorithm, as well as using the Genetic Algorithm (GA). Using simulation, the Mean Squared Error (MSE) was calculated to find out which method is the best for estimation by comparing the two methods using different sample sizes (400, 600, and 800). The (GA) that is based on the (ECM) algorithm to estimate the missing values proved to be better and more efficient than the (MLE) method in terms of the results.

On a Vector-Valued Measure of Multivariate Skewness

The canonical skewness vector is an analytically simple function of the third-order, standardized moments of a random vector. Statistical applications of this skewness measure include semiparametric modeling, independent component analysis, model-based clustering, and multivariate normality testing. This paper investigates some properties of the canonical skewness vector with respect to representations, transformations, and norm. In particular, the paper shows its connections with tensor contraction, scalar measures of multivariate kurtosis and Mardia’s skewness, the best-known scalar measure of multivariate skewness. A simulation study empirically compares the powers of tests for multivariate normality based on the squared norm of the canonical skewness vector and on Mardia’s skewness. An example with financial data illustrates the statistical applications of the canonical skewness vector.

A New Robust Class of Skew Elliptical Distributions

A new robust class of multivariate skew distributions is introduced. Practical aspects such as parameter estimation method of the proposed class are discussed, we show that the proposed class can be fitted under a reasonable time frame. Our study shows that the class of distributions is capable to model multivariate skewness structure and does not suffer from the curse of dimensionality as heavily as other distributions of similar complexity do, such as the class of canonical skew distributions. We also derive a nested form of the proposed class which appears to be the most flexible class of multivariate skew distributions in literature that has a closed-form density function. Numerical examples on two data sets, i) a data set containing daily river flow data recorded in the UK; and ii) a data set containing biomedical variables of athletes collected by the Australian Institute of Sports, are demonstrated. These examples further support the practicality of the proposed class on moderate dimensional data sets.

Some properties of the unified skew-normal distribution

For the family of multivariate probability distributions variously denoted as unified skew-normal, closed skew-normal and other names, a number of properties are already known, but many others are not, even some basic ones. The present contribution aims at filling some of the missing gaps. Specifically, the moments up to the fourth order are obtained, and from here the expressions of the Mardia’s measures of multivariate skewness and kurtosis. Other results concern the property of log-concavity of the distribution, closure with respect to conditioning on intervals, and a possible alternative parameterization.

Asymptotically valid Bayesian inference in the presence of distributional misspecification in VAR models

Inaccurately imposing Gaussian distributional assumptions in standard multivariate time series models does not affect inference on the autoregressive coefficients but distorts both classical and Bayesian inference on the volatility matrix whenever the true error distribution has excess kurtosis relative to the multivariate normal density. Inference on the intercept is also affected whenever the innovations are generated from a non-symmetric distribution. As a result of distributional misspecification, Bayesian methods lead to asymptotically invalid posterior inference for the intercept and the volatility matrix and, consequently, invalid posterior credible sets for quantities such as impulse responses, variance decompositions and density forecasts. We propose a robust and computationally fast Bayesian procedure which delivers asymptotically correct posterior credible sets without the need for distributional assumptions. The proposed corrected Bayesian posteriors for the volatility matrix and the intercept vector are based on the asymptotic covariance of the QML estimators and admit a closed form. Implementation of the procedure requires consistent estimation of the multivariate skewness and kurtosis of the innovations, and we propose novel shrinkage estimators designed to shrink these large dimensional objects towards the skewness and kurtosis of a Gaussian vector. We extend our robust Bayesian analysis to accommodate non-Gaussian disturbances in the presence of parameter instability, by combining the estimators of the current paper with semi-parametric kernel-type methods. We demonstrate that our estimators deliver correct posterior coverage rates in an extensive Monte Carlo exercise under a variety of distributional specifications. Finally, we present empirical evidence that imposing Gaussianity or homoskedasticity assumptions on financial and uncertainty shocks is not justified and may lead to misleading empirical conclusions.

On the canonical form of scale mixtures of skew-normal distributions

The canonical form of scale mixtures of multivariate skew-normal distribution is defined, emphasizing its role in summarizing some key properties of this class of distributions. It is also shown that the canonical form corresponds to an affine invariant co-ordinate system as defined in Tyler et al. (2009), and a method for obtaining the linear transform that converts a scale mixture of multivariate skew-normal distribution into a canonical form is presented. Related results, where the particular case of the multivariate skew t distribution is considered in greater detail, are the general expression of the Mardia indices of multivariate skewness and kurtosis and the reduction of dimensionality in calculating the mode.

Omnibus tests for multivariate normality based on Mardia’s skewness and kurtosis using normalizing transformation

Mardia (Biometrika, 57, 519-530, 1970) defined measures of multivariate skewness and kurtosis. Based on these measures, omnibus test statistics of multivariate normality are proposed using normalizing transformations. The transformations we consider are normal approximation and aWilson-Hilferty transformation. The normalizing transformation proposed by Enomoto et al. (Communications in Statistics-Simulation and Computation, 49, 684-698, 2019) for the Mardia’s kurtosis is also considered. A comparison of power is conducted by a simulation study. As a result, sum of squares of the normal approximation to the Mardia’s skewness and the Enomoto’s normalizing transformation to the Mardia’s kurtosis seems to have relatively good power over the alternatives that are considered.

Family of mean-mixtures of multivariate normal distributions: Properties, inference and assessment of multivariate skewness

In this paper, a new mixture family of multivariate normal distributions, formed by mixing multivariate normal distribution and a skewed distribution, is constructed. Some properties of this family, such as characteristic function, moment generating function, and the first four moments are derived. The distributions of affine transformations and canonical forms of the model are also derived. An EM-type algorithm is developed for the maximum likelihood estimation of model parameters. Some special cases of the family, using standard gamma and standard exponential mixture distributions, denoted by MMNG and MMNE, respectively, are considered. For the proposed family of distributions, different multivariate measures of skewness are computed. In order to examine the performance of the developed estimation method, some simulation studies are carried out to show that the maximum likelihood estimates do provide a good performance. For different choices of parameters of MMNE distribution, several multivariate measures of skewness are computed and compared. Because some measures of skewness are scalar and some are vectors, in order to evaluate them properly, a simulation study is carried out to determine the power of tests, based on sample versions of skewness measures as test statistics for testing the fit of the MMNE distribution. Finally, two real data sets are used to illustrate the usefulness of the proposed model and the associated inferential methods.

Kalman Filter Based on Multiple Scaled Multivariate Skew Normal Variance Mean Mixture Distributions With Application to Target Tracking

In this brief, we first propose a multiple scaled multivariate skew normal variance-mean mixture (MSMSNVMM) distribution to model heavy-tailed and/or skew measurement noises (HTSMN) whose each dimension has different tail and skewness behaviors. The MSMSNVMM distribution has more flexible tail behaviors and richer skewness features than Gaussian scale mixture (GScM) distribution, generalized Gaussian scale mixture (GGScM) distribution and scale mixtures of skew normal (SMSN) distribution. Furthermore, we derive a robust Kalman filter based on variational Bayesian (VB) method. The superiority of the new filter is demonstrated in a maneuvering target tracking example.

On Multivariate Skewness and Kurtosis

A unified treatment of all currently available cumulant-based indexes of multivariate skewness and kurtosis is provided here, expressing them in terms of the third and fourth-order cumulant vectors respectively. Such a treatment helps reveal many subtle features and inter-connections among the existing indexes as well as some deficiencies, which are hitherto unknown. Computational formulae for obtaining these measures are provided for spherical and elliptically-symmetric, as well as skew-symmetric families of multivariate distributions, yielding several new results and a systematic exposition of many known results.

Rényi Entropy for Mixture Model of Multivariate Skew Laplace distributions

Rényi entropy is an important concept developed by Rényi in information theory. In this paper, we study in detail this measure of information in cases multivariate skew Laplace distributions then we extend this study to the class of mixture model of multivariate skew Laplace distributions. The upper and lower bounds of Rényi entropy of mixture model are determined. In addition, an asymptotic expression for Rényi entropy is given by the approximation. Finally, we give a real data example to illustrate the behavior of entropy of the mixture model under consideration.

Behavioral data-driven analysis with Bayesian method for risk management of financial services

Time-varying behavioral features and non-linear dependence are widely observed in big data and challenge the operating systems and processes of risk management in financial services. In order to improve the operational accuracy of risk measures and incorporate customer behavior analytics, we propose a Bayesian approach to efficiently estimate the multivariate risk measures in a dynamic framework. The proposed method can carry the prior information into the Bayesian analysis and fully describe the risk measures’ behavior after utilizing the Cornish–Fisher (CF) approximation with Markov Chain Monte Carlo (MCMC) sampling. Therefore, the operating systems and processes of risk management can be well performed either based on the first four conditional moments of the underlying model employed to consider some specific behavioral features (e.g., the time-varying conditional multivariate skewness) or the characteristics extracted from the big data. We conduct a simulation study to distinguish the applications of CF approximation and MCMC sampling after comparing them with the classic likelihood based method. We then provide a robust procedure for empirical investigation by using the real data of U.S. DJIA stocks. Both simulation and empirical results confirm that the Bayesian method can significantly improve the operations of risk management.

Some remarks on Koziol’s kurtosis

James Koziol, in his 1987 and 1989 papers, proposed to use the sums of either the squared fourth-order cumulants or moments as test statistics for multivariate normality. His proposals are by far less popular than Mardia’s measure of multivariate kurtosis, that is the fourth moment of the Mahalanobis distance of a random vector from its mean. We investigate some properties of Koziol’s measures of multivariate kurtosis which motivate their use in statistical practice. Firstly, we show some of their connections with Mahalanobis angles. Secondly, we use inequalities to highlight their connections with other measures of multivariate skewness and kurtosis. Thirdly, we obtain their analytical formulae for some well-known multivariate statistical models. Simple examples illustrate the interpretation of Koziol’s measures of multivariate kurtosis and detect a wrong statement about them which appeared in the statistical literature. We suggest that Mardia’s and Koziol’s measures of kurtosis should be used together to detect interesting data structures.

Modelling of Correlated Ordinal Responses, by Using Multivariate Skew Probit with Different Types of Variance Covariance Structures

In this paper, a multivariate fundamental skew probit (MFSP) model is used to model correlated ordinal responses which are constructed from the multivariate fundamental skew normal (MFSN) distribution originate to the greater flexibility of MFSN. To achieve an appropriate VC structure for reaching reliable statistical inferences, many types of variance covariance (VC) structures are considered to model MFSN. Simulation methods are used to find the properties of the parameters estimate. The Schizophrenia Collaborative Study data invokes the proposed MFSN model. The results confirm that the first-order autoregressive (AR(1)) structure substantially enhances the estimation of the parameters. Furthermore, over time the drugs effect the schizophrenia treatment, noticeably.