Multivariate t-distribution

Multivariate t-distribution is a probability distribution that generalizes the Student's t-distribution for multiple variables. It is used in various statistical analyses when the sample size is small, and the underlying population's variance is unknown. This distribution is particularly useful in the field of multivariate analysis, where it helps in making inferences about a population based on multivariate sample data.

Definition

The multivariate t-distribution describes the distribution of a random vector. Let \(X\) be a \(p\)-dimensional random vector. \(X\) follows a multivariate t-distribution if its probability density function (pdf) is given by:

\[f(\mathbf{x}; \nu, \mathbf{\mu}, \mathbf{\Sigma}) = \frac{\Gamma\left(\frac{\nu + p}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\nu^{p/2}\pi^{p/2}|\mathbf{\Sigma}|^{1/2}}\left(1 + \frac{1}{\nu}(\mathbf{x} - \mathbf{\mu})^T\mathbf{\Sigma}^{-1}(\mathbf{x} - \mathbf{\mu})\right)^{-\frac{\nu + p}{2}},\]

where: - \(\mathbf{x}\) is a \(p\)-dimensional vector of observations, - \(\nu\) is the degrees of freedom, - \(\mathbf{\mu}\) is the location parameter (mean vector), - \(\mathbf{\Sigma}\) is the scale parameter (covariance matrix), - \(\Gamma\) is the gamma function.

Properties

The multivariate t-distribution shares several properties with its univariate counterpart, including heavy tails and a parameter \(\nu\) that controls the degree of kurtosis. As \(\nu\) increases, the multivariate t-distribution approaches the multivariate normal distribution. Other important properties include: - **Symmetry:** The distribution is symmetric around its mean vector \(\mathbf{\mu}\). - **Marginal distributions:** Any marginal distribution of a subset of the components of \(X\) is also a t-distribution. - **Conditional distributions:** The conditional distribution of a subset of \(X\), given the others, is a multivariate t-distribution.

Applications

The multivariate t-distribution is widely used in statistics and data analysis for: - Modeling multivariate data with outliers, - Developing robust statistical procedures, - Multivariate hypothesis testing, - Bayesian inference in multivariate settings.

See also

• Student's t-distribution
• Multivariate analysis
• Multivariate normal distribution
• Covariance matrix
• Gamma function