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- Multivariate t Distributions and Their Applications. Samuel Kotz and Saralees Nadarajah
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*Measures of multivariate skewness and kurtosis are developed by extending certain studies on robustness of the t statistic.*

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

Either your web browser doesn't support Javascript or it is currently turned off. In the latter case, please turn on Javascript support in your web browser and reload this page. Free to read. 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. The multivariate Student's t distribution is often used as a substitute for the multivariate normal distribution in situations where it is known that the marginal distributions of the individual variables have fatter tails than the normal.

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.

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

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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. The shape of this multivariate t-distribution arises from the observation that the pdf for is given by Equation 1 when is distributed as a multivariate normal distribution with covariance matrix and is distributed as chi-squared. The derivation of the multivariate normal pdf is given in Section 2 to provide background.

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.

*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.*

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