Multivariate t-distribution

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. One common method of construction of a multivariate t-distribution, for the case of dimensions, is based on the observation that if and are independent and distributed as and (i.e. multivariate normal and chi-squared distributions) respectively, the matrix is a p × p matrix, and is a constant vector then the random variable has the density and is said to be distributed as a multivariate t-distribution with parameters . Note that is not the covariance matrix since the covariance is given by (for ). The constructive definition of a multivariate t-distribution simultaneously serves as a sampling algorithm: Generate and , independently. Compute . This formulation gives rise to the hierarchical representation of a multivariate t-distribution as a scale-mixture of normals: where indicates a gamma distribution with density proportional to , and conditionally follows . In the special case , the distribution is a multivariate Cauchy distribution. 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 (2004). 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. In one dimension (), with and , we have the probability density function and one approach is to write down a corresponding function of several variables. This is the basic idea of elliptical distribution theory, where one writes down a corresponding function of variables that replaces by a quadratic function of all the .