Statistical applications of the multivariate skew-normal distribution
TL;DR: Azzalini and Dalla Valle as mentioned in this paper have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter.
Abstract: Azzalini & Dalla Valle (1996) have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.
Citations
More filters
TL;DR: This work considers approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non‐Gaussian response variables and can directly compute very accurate approximations to the posterior marginals.
Abstract: Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.
4,164 citations
TL;DR: In this paper, a general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities.
Abstract: A fairly general procedure is studied to perturbate a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is general enough to encompass a number of recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew $t$ density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.
1,174 citations
29 Sep 2014
TL;DR: In this article, the authors present a concise review of developments on various continuous multivariate distributions and present some basic definitions and notations, and present several important continuous multi-dimensional distributions and their significant properties and characteristics.
Abstract: In this article, we present a concise review of developments on various continuous multivariate distributions. We first present some basic definitions and notations. Then, we present several important continuous multivariate distributions and list their significant properties and characteristics.
Keywords:
generating function;
moments;
conditional distribution;
truncated distribution;
regression;
bivariate normal;
multivariate normal;
multivariate exponential;
multivariate gamma;
dirichlet;
inverted dirichlet;
liouville;
multivariate logistic;
multivariate pareto;
multivariate extreme value;
multivariate t;
wishart translated systems;
multivariate exponential families
1,106 citations
TL;DR: These estimates can help countries and the international community gauge the need for appropriate diagnoses and genetic counselling to reduce the number of neonates affected by HbS and could be used for other inherited disorders.
838 citations
TL;DR: In this paper, the authors provide an introductory overview of a portion of distribution theory which is currently under intense development and illustrate connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more.
Abstract: . This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so-called skew-normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew-elliptical families, and some forms of semi-parametric formulations, extending the relevance of the field much beyond the original theme of ‘skewness’. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more.
657 citations
References
More filters
Book•
01 Jan 1982
TL;DR: In this paper, the authors present a set of standard tests on Covariance Matrices and Mean Vectors, and test independence between k Sets of Variables and Canonical Correlation Analysis.
Abstract: Tables. Commonly Used Notation. 1. The Multivariate Normal and Related Distributions. 2. Jacobians, Exterior Products, Kronecker Products, and Related Topics. 3. Samples from a Multivariate Normal Distribution, and the Wishart and Multivariate BETA Distributions. 4. Some Results Concerning Decision-Theoretic Estimation of the Parameters of a Multivariate Normal Distribution. 5. Correlation Coefficients. 6. Invariant Tests and Some Applications. 7. Zonal Polynomials and Some Functions of Matrix Argument. 8. Some Standard Tests on Covariance Matrices and Mean Vectors. 9. Principal Components and Related Topics. 10. The Multivariate Linear Model. 11. Testing Independence Between k Sets of Variables and Canonical Correlation Analysis. Appendix: Some Matrix Theory. Bibliography. Index.
4,343 citations
TL;DR: In this article, the authors developed measures of multivariate skewness and kurtosis by extending certain studies on robustness of the t statistic, and the asymptotic distributions of the measures for samples from a multivariate normal population are derived and a test for multivariate normality is proposed.
Abstract: SUMMARY Measures of multivariate skewness and kurtosis are developed by extending certain studies on robustness of the t statistic. These measures are shown to possess desirable properties. The asymptotic distributions of the measures for samples from a multivariate normal population are derived and a test of multivariate normality is proposed. The effect of nonnormality on the size of the one-sample Hotelling's T2 test is studied empirically with the help of these measures, and it is found that Hotelling's T2 test is more sensitive to the measure of skewness than to the measure of kurtosis. measures have proved useful (i) in selecting a member of a family such as from the Karl Pearson family, (ii) in developing a test of normality, and (iii) in investigating the robustness of the standard normal theory procedures. The role of the tests of normality in modern statistics has recently been summarized by Shapiro & Wilk (1965). With these applications in mind for the multivariate situations, we propose measures of multivariate skewness and kurtosis. These measures of skewness and kurtosis are developed naturally by extending certain aspects of some robustness studies for the t statistic which involve I1 and 32. It should be noted that measures of multivariate dispersion have been available for quite some time (Wilks, 1932, 1960; Hotelling, 1951). We deal with the measure of skewness in ? 2 and with the measure of kurtosis in ? 3. In ? 4 we give two important applications of these measures, namely, a test of multivariate normality and a study of the effect of nonnormality on the size of the one-sample Hotelling's T2 test. Both of these problems have attracted attention recently. The first problem has been treated by Wagle (1968) and Day (1969) and the second by Arnold (1964), but our approach differs from theirs.
3,774 citations
Journal Article•
TL;DR: In this paper, a nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard.
Abstract: On introduit une nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard
2,470 citations
Book•
01 Nov 1989TL;DR: In this article, the authors define marginal distributions, moments and density marginal distributions moments density the relationship between (phi and f) conditional distributions properties of elliptically symmetric distributions mixtures of normal distributions robust statistics and regression model robust statistics regression model log-elliptical and additive logistic elliptical distributions multivariate log elliptical distribution additive logistics elliptical distribution complex elliptical symmetric distribution.
Abstract: Part 1 Preliminaries: construction of symmetric multivariate distributions notation of algebraic entities and characteristics of random quantities the "d" operator groups and invariance dirichlet distribution problems 1. Part 2 Spherically and elliptically symmetric distributions: introduction and definition marginal distributions, moments and density marginal distributions moments density the relationship between (phi) and f conditional distributions properties of elliptically symmetric distributions mixtures of normal distributions robust statistics and regression model robust statistics regression model log-elliptical and additive logistic elliptical distributions multivariate log-elliptical distribution additive logistic elliptical distributions complex elliptically symmetric distributions. Part 3 Some subclasses of elliptical distributions: multiuniform distribution the characteristic function moments marginal distribution conditional distributions uniform distribution in the unit sphere discussion symmetric Kotz type distributions definition distribution of R(2) moments multivariate normal distributions the c.f. of Kotz type distributions symmetric multivariate Pearson type VII distributions definition marginal densities conditional distributions moments conditional distributions moments some examples extended Tn family relationships between Ln and Tn families of distributions order statistics mixtures of exponential distributions independence, robustness and characterizations problems V. Part 6 Multivariate Liouville distributions: definitions and properties examples marginal distributions conditional distribution characterizations scale-invariant statistics survival functions inequalities and applications.
2,106 citations
1,872 citations