Browsing by Author "Genton, Marc G."

Now showing 1 - 15 of 15
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    A multivariate modified skew-normal distribution
    (2024) Mondal, Sagnik; Arellano-Valle, Reinaldo B.; Genton, Marc G.
    We introduce a multivariate version of the modified skew-normal distribution, which contains the multivariate normal distribution as a special case. Unlike the Azzalini multivariate skew-normal distribution, this new distribution has a nonsingular Fisher information matrix when the skewness parameters are all zero, and its profile log-likelihood of the skewness parameters is always a non-monotonic function. We study some basic properties of the proposed family of distributions and present an expectation-maximization (EM) algorithm for parameter estimation that we validate through simulation studies. Finally, we apply the proposed model to the univariate frontier data and to a trivariate wind speed data, and compare its performance with the Azzalini skew-normal model.
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    A unified view on skewed distributions arising from selections
    (WILEY, 2006) Arellano Valle, Reinaldo B.; Branco, Marcia D.; Genton, Marc G.
    Parametric families of multivariate nonnormal distributions have received considerable attention in the past few decades. The authors propose a new definition of a selection distribution that encompasses many existing families of multivariate skewed distributions. Their work is motivated by examples that involve various forms of selection mechanisms and lead to skewed distributions. They give the main properties of selection distributions and show how various families of multivariate skewed distributions, such as the skew-normal and skew-elliptical distributions, arise as special cases. The authors further introduce several methods of constructing selection distributions based on linear and nonlinear selection mechanisms.
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    An invariance property of quadratic forms in random vectors with a selection distribution, with application to sample variogram and covariogram estimators
    (2010) Arellano-Valle, Reinaldo B.; Genton, Marc G.
    We study conditions under which an invariance property holds for the class of selection distributions. First, we consider selection distributions arising from two uncorrelated random vectors. In that setting, the invariance holds for the so-called C-class and for elliptical distributions. Second, we describe the invariance property for selection distributions arising from two correlated random vectors. The particular case of the distribution of quadratic forms and its invariance, under various selection distributions, is investigated in more details. We describe the application of our invariance results to sample variogram and covariogram estimators used in spatial statistics and provide a small simulation study for illustration. We end with a discussion about other applications, for example such as linear models and indices of temporal/spatial dependence.
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    Bayesian Inference for Shape Mixtures of Skewed Distributions, with Application to Regression Analysis
    (INT SOC BAYESIAN ANALYSIS, 2008) Arellano Valle, Reinaldo B.; Castro, Luis M.; Genton, Marc G.; Gomez, Hector W.
    We introduce a class of shape mixtures of skewed distributions and study some of its main properties. We discuss a Bayesian interpretation and some invariance results of the proposed class. We develop a Bayesian analysis of the skew-normal, skew-generalized-normal, skew-normal-t and skew-t-normal linear regression models under some special prior specifications for the model parameters. In particular, we show that the full posterior of the skew-normal regression model parameters is proper under an arbitrary proper prior for the shape parameter and noninformative prior for the other parameters. We implement a convenient hierarchical representation in order to obtain the corresponding posterior analysis. We illustrate our approach with an application to a real dataset on characteristics of Australian male athletes.
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    Multivariate unified skew-t distributions and their properties
    (2024) Wang, Kesen; Karling, Maicon J.; Arellano-Valle, Reinaldo B.; Genton, Marc G.
    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.
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    On the exact distribution of linear combinations of order statistics from dependent random variables
    (ELSEVIER INC, 2007) Arellano Valle, Reinaldo B.; Genton, Marc G.
    We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions. (c) 2007 Elsevier Inc. All rights reserved.
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    On the exact distribution of the maximum of absolutely continuous dependent random variables
    (ELSEVIER SCIENCE BV, 2008) Arellano Valle, Reinaldo B.; Genton, Marc G.
    We derive the exact probability density function of the maximum of arbitrary absolutely continuous dependent random variables and of absolutely continuous exchangeable random variables. We show this density is related to the family of fundamental skew distributions. In particular, we examine the case where the random variables have an elliptically contoured distribution. We study some particular examples based on the multivariate normal and multivariate Student t distributions, and discuss numerical computation issues. We illustrate our results on a genetic selection problem and on an autoregressive time series model of order one. (C) 2007 Elsevier B.V. All rights reserved.
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    On the non-identifiability of unified skew-normal distributions
    (2023) Wang, Kesen; Arellano-Valle, Reinaldo B.; Azzalini, Adelchi; Genton, Marc G.
    We investigate the non-identifiability of the multivariate unified skew-normal distribution under permutation of its latent variables. We show that the non-identifiability issue also holds with other parameterizations and extends to the family of unified skew-elliptical distributions and more generally to selection distributions. We provide several suggestions to make the unified skew-normal model identifiable and describe various sub-models that are identifiable.
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    Perturbation of Numerical Confidential Data via Skew-t Distributions
    (INFORMS, 2010) Lee, Seokho; Genton, Marc G.; Arellano Valle, Reinaldo B.
    We propose a new data perturbation method for numerical database security problems based on skew-t distributions. Unlike the normal distribution, the more general class of skew-t distributions is a flexible parametric multivariate family that can model skewness and heavy tails in the data. Because databases having a normal distribution are seldom encountered in practice, the newly proposed approach, coined the skew-t data perturbation (STDP) method, is of great interest for database managers. We also discuss how to preserve the sample mean vector and sample covariance matrix exactly for any data perturbation method. We investigate the performance of the STDP method by means of a Monte Carlo simulation study and compare it with other existing perturbation methods. Of particular importance is the ability of STDP to reproduce characteristics of the joint tails of the distribution in order for database users to answer higher-level questions. We apply the STDP method to a medical database related to breast cancer.
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    Scale and shape mixtures of multivariate skew-normal distributions
    (2018) Arellano Valle, Reinaldo Boris; Ferreira, Clécio S.; Genton, Marc G.
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    Shannon entropy and mutual information for multivariate skew-elliptical distributions
    (2013) Arellano Valle, Reinaldo Boris; Contreras Reyes, Javier E.; Genton, Marc G.
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    Shape mixtures of multivariate skew-normal distributions
    (ELSEVIER INC, 2009) Arellano Valle, Reinaldo B.; Genton, Marc G.; Loschi, Rosangela H.
    Classes of shape mixtures of independent and dependent multivariate skew-normal distributions are considered and some of their main properties are studied. If interpreted from a Bayesian point of view, the results obtained in this paper bring tractability to the problem of inference for the shape parameter, that is, the posterior distribution can be written in analytic form. Robust inference for location and scale parameters is also obtained under particular conditions. (C) 2008 Elsevier Inc. All rights reserved.
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    Skewed factor models using selection mechanisms
    (2016) Kim, Hyoung-Moon.; Maadooliat, Mehdi.; Arellano Valle, Reinaldo Boris; Genton, Marc G.
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    Sub-dimensional Mardia measures of multivariate skewness and kurtosis
    (2022) Chowdhury, Joydeep; Dutta, Subhajit; Arellano-Valle, Reinaldo B.; Genton, Marc G.
    The Mardia measures of multivariate skewness and kurtosis summarize the respective characteristics of a multivariate distribution with two numbers. However, these mea-sures do not reflect the sub-dimensional features of the distribution. Consequently, test-ing 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 Mar-dia 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. (c) 2022 Elsevier Inc. All rights reserved.
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    Tractable Bayes of skew-elliptical link models for correlated binary data
    (2023) Zhang, Zhongwei; Arellano-Valle, Reinaldo B.; Genton, Marc G.; Huser, Raphael
    Correlated binary response data with covariates are ubiquitous in longitudinal or spatial studies. Among the existing statistical models, the most well-known one for this type of data is the multivariate probit model, which uses a Gaussian link to model dependence at the latent level. However, a symmetric link may not be appropriate if the data are highly imbalanced. Here, we propose a multivariate skew-elliptical link model for correlated binary responses, which includes the multivariate probit model as a special case. Furthermore, we perform Bayesian inference for this new model and prove that the regression coefficients have a closed-form unified skew-elliptical posterior with an elliptical prior. The new methodology is illustrated by an application to COVID-19 data from three different counties of the state of California, USA. By jointly modeling extreme spikes in weekly new cases, our results show that the spatial dependence cannot be neglected. Furthermore, the results also show that the skewed latent structure of our proposed model improves the flexibility of the multivariate probit model and provides a better fit to our highly imbalanced dataset.