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*Received 29 March ; accepted 14 June ; published 17 June Expressions for the probability density function, for the variances, and for the covariances of the multivariate t-distribution with arbitrary shape parameters for the marginals are given. This expression, which is different in form than the form that is commonly used, allows the shape parameter for each marginal probability density function pdf of the multivariate pdf to be different.*

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In statistics , the multivariate t -distribution or multivariate Student distribution is a multivariate probability distribution. It is a generalization to random vectors of the Student's t -distribution , which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t -distribution is distinct and makes particular use of the matrix structure. There are in fact many candidates for the multivariate generalization of Student's t -distribution. An extensive survey of the field has been given by Kotz and Nadarajah The essential issue is to define a probability density function of several variables that is the appropriate generalization of the formula for the univariate case.

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Multivariate T-Distributions and Their Applications. Samuel Kotz , Saralees Nadarajah. Almost all the results available in the literature on multivariate t-distributions published in the last 50 years are now collected together in this comprehensive reference. Because these distributions are becoming more prominent in many applications, this book is a must for any serious researcher or consultant working in multivariate analysis and statistical distributions. Much of this material has never before appeared in book form. The first part of the book emphasizes theoretical results of a probabilistic nature.

Multivariate t Distributions and Their Applications. Samuel Kotz. George Washington University. Saralees Nadarajah. University of South Florida. SON. QA.

## Multivariate t Distributions and Their Applications. Samuel Kotz and Saralees Nadarajah

Cluster analysis is the automated search for groups of homogeneous observations in a data set. A popular modeling approach for clustering is based on finite normal mixture models, which assume that each cluster is modeled as a multivariate normal distribution. However, the normality assumption that each component is symmetric is often unrealistic. To address these issues, we propose a new class of distributions, multivariate t distributions with the Box-Cox transformation, for mixture modeling.

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Documentation Help Center. The probability density function of the d -dimensional multivariate Student's t distribution is given by. For the singular case, only random number generation is supported. The multivariate Student's t distribution is a generalization of the univariate Student's t to two or more variables. It is a distribution for random vectors of correlated variables, each element of which has a univariate Student's t distribution. In the same way as the univariate Student's t distribution can be constructed by dividing a standard univariate normal random variable by the square root of a univariate chi-square random variable, the multivariate Student's t distribution can be constructed by dividing a multivariate normal random vector having zero mean and unit variances by a univariate chi-square random variable.

Metrics details. In this article, we introduce the multivariate slash and skew-slash t distributions which provide alternative choices in simulating and fitting skewed and heavy tailed data. We study their relationships with other distributions and give the densities, stochastic representations, moments, marginal distributions, distributions of linear combinations and characteristic functions of the random vectors obeying these distributions. We characterize the skew t , the skew-slash normal and the skew-slash t distributions using both the hidden truncation or selective sampling model and the order statistics of the components of a bivariate normal or t variable.

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community. Already on GitHub? Sign in to your account. The multivariate student-t distribution is used extensively within academia, science and finance, primarily for its fatter tails larger kurtosis when compared to the normal distribution.

PDF | This paper makes an attempt to justify a multivariate t -model and provides a modest review of most important results of this model.

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 T 2 test is studied empirically with the help of these measures, and it is found that Hotelling's T 2 test is more sensitive to the measure of skewness than to the measure of kurtosis. Most users should sign in with their email address.

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Contents · 6 - Probability Integrals pp · Access PDF Export citation.