Multivariate t-distribution

Multivariate t
Notation	$t_\nu(\boldsymbol\mu,\boldsymbol\Sigma)$
Parameters	$\boldsymbol\mu = [\mu_1, \dots, \mu_p]^T$ location (real $p\times 1$ vector) $\boldsymbol\Sigma$ scale matrix (positive-definite real $p\times p$ matrix) $\nu$ is the degrees of freedom
Support	$\mathbf{x} \in\mathbb{R}^p\!$
PDF	$\frac{\Gamma\left[(\nu+p)/2\right]}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}\left\|{\boldsymbol\Sigma}\right\|^{1/2}\left[1+\frac{1}{\nu}({\mathbf x}-{\boldsymbol\mu})^{\rm T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right]^{(\nu+p)/2}}$
CDF	No analytic expression, but see text for approximations
Mean	$\boldsymbol\mu$ if $\nu > 1$ ; else undefined
Median	$\boldsymbol\mu$
Mode	$\boldsymbol\mu$
Variance	$\frac{\nu}{\nu-2} \boldsymbol\Sigma$ if $\nu > 2$ ; else undefined
Skewness	0

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.

Definition

One common method of construction of a multivariate t-distribution, for the case of $p$ dimensions, is based on the observation that if $\mathbf y$ and $u$ are independent and distributed as ${\mathcal N}({\mathbf 0},{\boldsymbol\Sigma})$ and $\chi^2_\nu$ (i.e. multivariate normal and chi-squared distributions) respectively, the covariance $\mathbf{\Sigma}\,$ is a p × p matrix, and ${\mathbf y}\sqrt{\nu/u}={\mathbf x}-{\boldsymbol\mu}$ , then ${\mathbf x}$ has the density

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

and is said to be distributed as a multivariate t-distribution with parameters ${\boldsymbol\Sigma},{\boldsymbol\mu},\nu$ .

In the special case $\nu=1$ , the distribution is a multivariate Cauchy distribution.

Derivation

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 ( $p=1$ ), with $t=x-\mu$ and $\Sigma=1$ , we have the probability density function

f(t) = \frac{\Gamma[(\nu+1)/2]}{\sqrt{\nu\pi\,}\,\Gamma[\nu/2]} (1+t^2/\nu)^{-(\nu+1)/2}

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 $p$ variables $t_i$ that replaces $t^2$ by a quadratic function of all the $t_i$ . It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom $\nu$ . With $\mathbf{A} = \boldsymbol\Sigma^{-1}$ , one has a simple choice of multivariate density function

f(\mathbf t) = \frac{\Gamma((\nu+p)/2)\left|\mathbf{A}\right|^{1/2}}{\sqrt{\nu^p\pi^p\,}\,\Gamma(\nu/2)} \left(1+\sum_{i,j=1}^{p,p} A_{ij} t_i t_j/\nu\right)^{-(\nu+p)/2}

which is the standard but not the only choice.

An important special case is the standard bivariate t-distribution, p = 2:

f(t_1,t_2) = \frac{\left|\mathbf{A}\right|^{1/2}}{2\pi} \left(1+\sum_{i,j=1}^{2,2} A_{ij} t_i t_j/\nu\right)^{-(\nu+2)/2}

Note that $\frac{\Gamma \left(\frac{\nu +2}{2}\right)}{\pi \ \nu \Gamma \left(\frac{\nu }{2}\right)}= \frac {1} {2\pi}$ .

Now, if $\mathbf{A}$ is the identity matrix, the density is

f(t_1,t_2) = \frac{1}{2\pi} \left(1+(t_1^2 + t_2^2)/\nu\right)^{-(\nu+2)/2}.

The difficulty with the standard representation is revealed by this formula, which does not factorize into the product of the marginal one-dimensional distributions. When $\Sigma$ is diagonal the standard representation can be shown to have zero correlation but the marginal distributions do not agree with statistical independence.

Cumulative distribution function

The definition of the cumulative distribution function (cdf) in one dimension can be extended to multiple dimensions by defining the following probability (here $\mathbf{x}$ is a real vector):

F(\mathbf{x}) = \mathbb{P}(\mathbf{X}\leq \mathbf{x}), \quad \textrm{where}\;\; \mathbf{X}\sim t_\nu(\boldsymbol\mu,\boldsymbol\Sigma).

There is no simple formula for $F(\mathbf{x})$ , but it can be approximated numerically via Monte Carlo integration.^[1]^[2]

Further theory

Many such distributions may be constructed by considering the quotients of normal random variables with the square root of a sample from a chi-squared distribution. These are surveyed in the references and links below.

Copulas based on the multivariate t

The use of such distributions is enjoying renewed interest due to applications in mathematical finance, especially through the use of the Student's t copula.

Related concepts

In univariate statistics, the Student's t-test makes use of Student's t-distribution. Hotelling's T-squared distribution is a distribution that arises in multivariate statistics. The matrix t-distribution is a distribution for random variables arranged in a matrix structure.

References

↑ Botev, Z. I.; L'Ecuyer, P. (6 December 2015). "Efficient probability estimation and simulation of the truncated multivariate student-t distribution". 2015 Winter Simulation Conference (WSC). Huntington Beach, CA, USA: IEEE. pp. 380–391. doi:10.1109/WSC.2015.7408180.
↑ Genz, Alan (2009). Computation of Multivariate Normal and t Probabilities. Springer. ISBN 978-3-642-01689-9.

Literature

Kotz, Samuel; Nadarajah, Saralees (2004). Multivariate t Distributions and Their Applications. Cambridge University Press. ISBN 0521826543.
Cherubini, Umberto; Luciano, Elisa; Vecchiato, Walter (2004). Copula methods in finance. John Wiley & Sons. ISBN 0470863447.

External links

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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