Multivariate t-distribution

Multivariate t

Notation	$t_{\nu }({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})$
Parameters	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol\mu = [\mu_1, \dots, \mu_p]^T} location (real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\times 1} vector) ${\boldsymbol {\Sigma }}$ scale matrix (positive-definite real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\times p} matrix) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu > 0} (real) represents the degrees of freedom
Support	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} \in\mathbb{R}^p\!}
PDF	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol\mu} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu > 1} ; else undefined
Median	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol\mu}
Mode	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol\mu}
Variance	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\nu}{\nu-2} \boldsymbol\Sigma} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} dimensions, is based on the observation that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} are independent and distributed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N({\mathbf 0},{\boldsymbol\Sigma})} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2_\nu} (i.e. multivariate normal and chi-squared distributions) respectively, the matrix $\mathbf {\Sigma } \,$ is a p × p matrix, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\boldsymbol\mu}} is a constant vector then the random variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle {\mathbf x}={\mathbf y}/\sqrt{u/\nu} +{\boldsymbol\mu}} has the density

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\boldsymbol\Sigma},{\boldsymbol\mu},\nu} . Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf\Sigma} is not the covariance matrix since the covariance is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu/(\nu-2)\mathbf\Sigma} (for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu>2} ).

The constructive definition of a multivariate t-distribution simultaneously serves as a sampling algorithm:

1. Generate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \sim \chi^2_\nu} and $\mathbf {y} \sim N(\mathbf {0} ,{\boldsymbol {\Sigma }})$, independently.
2. Compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} \gets \sqrt{\nu/u}\mathbf{y}+ \boldsymbol{\mu}} .

This formulation gives rise to the hierarchical representation of a multivariate t-distribution as a scale-mixture of normals: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \sim \mathrm{Ga}(\nu/2,\nu/2)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Ga}(a,b)} indicates a gamma distribution with density proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{a-1}e^{-bx}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}\mid u} conditionally follows Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(\boldsymbol{\mu},u^{-1}\boldsymbol{\Sigma})} .

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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=1} ), with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=x-\mu} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma=1} , we have the probability density function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 use a corresponding function of several variables. This is the basic idea of elliptical distribution theory, where one writes down a corresponding function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_i} that replaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2} by a quadratic function of all the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_i} . It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} . With Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{A} = \boldsymbol\Sigma^{-1}} , one has a simple choice of multivariate density function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 are not statistically independent.

A notable spontaneous occurrence of the elliptical multivariate distribution is its formal mathematical appearance when least squares methods are applied to multivariate normal data such as the classical Markowitz minimum variance econometric solution for asset portfolios.

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.

Conditional Distribution

This was developed by Muirhead and Cornish. but later derived using the simpler chi-squared ratio representation above, by Roth and Ding. Let vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } follow a multivariate t distribution and partition into two subvectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1, p_2 } elements:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_p = \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim t_p \left (\mu_p, \Sigma_{p \times p}, \nu \right ) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1 + p_2 = p } , the known mean vectors are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_p = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}} and the scale matrix is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma_{p \times p} = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} } .

Roth and Ding find the conditional distribution $p(X_{1}|X_{2})$ to be a new t-distribution with modified parameters.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1|X_2 \sim t_{ p_1 }\left( \mu_{1|2},\frac{\nu + d_2}{\nu + p_2} \Sigma_{11|2}, \nu + p_2 \right)}

An equivalent expression in Kotz et. al. is somewhat less concise.

Thus the conditional distribution is most easily represented as a two-step procedure. Form first the intermediate distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1|X_2 \sim t_{ p_1 }\left( \mu_{1|2}, \Psi ,\tilde{ \nu } \right)} above then, using the parameters below, the explicit conditional distribution becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X_1|X_2) =\frac{\Gamma\left[(\tilde \nu +p_1)/2\right]}{\Gamma(\tilde \nu /2) ( \pi \,\tilde \nu )^{p_1/2}\left|{\boldsymbol\Psi}\right|^{1/2}}\left[1+\frac{1}{\tilde \nu}( X_1 - \mu_{1|2} )^T{\boldsymbol\Psi}^{-1}(X_1- \mu_{1|2} )\right]^{-(\tilde \nu + p_1)/2}}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde \nu = \nu + p_2 } Effective degrees of freedom, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } is augmented by the number of disused variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_2 } .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_{1|2} = \mu_1 + \Sigma_{12} \Sigma_{22}^{-1} \left(X_2 - \mu_2 \right ) } is the conditional mean of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma_{11|2} = \Sigma_{11} - \Sigma_{12} \Sigma_{22} ^{-1} \Sigma_{21} } is the Schur complement of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma_{22} \text{ in } \Sigma } .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_2 = (X_2 - \mu_2)^T \Sigma_{22}^{-1} (X_2 - \mu_2) } is the squared Mahalanobis distance of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_2 } from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_2 } with scale matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma_{22} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi = \frac{\nu + d_2}{\nu + p_2} \Sigma_{11|2} } is the conditional covariance for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{\nu} > 2} .

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.

Elliptical Representation

Constructed as an elliptical distribution, take the simplest centralised case with spherical symmetry and no scaling, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma = \operatorname{I} \, } , then the multivariate t-PDF takes the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_X(X)= g(X^T X) = \frac{\Gamma \big ( \frac{1}{2} (\nu + p ) \, \big )}{ ( \nu \pi)^{\,p/2} \Gamma \big( \frac{1}{2} \nu \big)} \bigg( 1 + \nu^{-1} X^T X \bigg)^{-( \nu + p )/2 } }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X =(x_1, \cdots ,x_p )^T\text { is a } p\text{-vector} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } = degrees of freedom as defined in Muirhead section 1.5. The covariance of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} \left( XX^T \right) = \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f_X(x_1,\dots, x_p) XX^T \, dx_1 \dots dx_p = \frac{ \nu }{ \nu - 2 } \operatorname{I} }

The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder, define radial measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_2 = R^2 = \frac{X^TX}{p} } and, noting that the density is dependent only on r₂, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} [ r_2 ] = \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f_X(x_1,\dots, x_p) \frac {X^TX}{p}\, dx_1 \dots dx_p = \frac{\nu}{ \nu -2} }

which is equivalent to the variance of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } -element vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} treated as a univariate heavy-tail zero-mean random sequence with uncorrelated, yet statistically dependent, elements.

Radial Distribution

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_2 = \frac{X^TX}{p}} follows the Fisher-Snedecor or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } distribution:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_2 \sim f_{F}( p,\nu) = B \bigg( \frac {p}{2}, \frac {\nu}{2} \bigg ) ^{-1} \bigg (\frac{p}{\nu} \bigg )^{ p/2 } r_2^ { p/2 -1 } \bigg( 1 + \frac{p}{\nu} r_2 \bigg) ^{-(p + \nu)/2 }}

having mean value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} [ r_2 ] = \frac { \nu }{ \nu - 2 } } . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } -distributions arise naturally in tests of sums of squares of sampled data after normalization by the sample standard deviation.

By a change of random variable to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac{p}{\nu} r_2 = \frac {X^T X}{\nu} } in the equation above, retaining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } -vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} [ y ] = \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f_X(X) \frac {X^TX}{ \nu}\, dx_1 \dots dx_p = \frac { p }{ \nu - 2 }} and probability distribution

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_Y(y| \,p,\nu) & = \left | \frac {p}{\nu} \right|^{-1} B \bigg( \frac {p}{2}, \frac {\nu}{2} \bigg )^{-1} \big (\frac{p}{\nu} \big )^{ \,p/2 } \big (\frac{p}{\nu} \big )^{ -p/2 -1} y^ {\, p/2 -1 } \big( 1 + y \big) ^{-(p + \nu)/2 } \\ \\ & = B \bigg ( \frac {p}{2}, \frac {\nu}{2} \bigg )^{-1} y^{ \,p/2 -1 }(1+ y )^{-(\nu + p)/2} \end{align} }

which is a regular Beta-prime distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \sim \beta \, ' \bigg(y; \frac {p}{2}, \frac {\nu}{2} \bigg ) } having mean value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac { \frac{1}{2} p }{ \frac{1}{2}\nu - 1 } = \frac { p }{ \nu - 2 }} .

Cumulative Radial Distribution

Given the Beta-prime distribution, the radial cumulative distribution function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is known:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_Y(y) \sim I \, \bigg(\frac {y}{1+y}; \, \frac {p}{2}, \frac {\nu}{2} \bigg ) B\bigg( \frac {p}{2}, \frac {\nu}{2} \bigg )^{-1} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is the incomplete Beta function and applies with a spherical Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma } assumption.

In the scalar case, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = 1} , the distribution is equivalent to Student-t with the equivalence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2 = y^2 \sigma^{-1} } , the variable t having double-sided tails for CDF purposes, i.e. the "two-tail-t-test".

The radial distribution can also be derived via a straightforward coordinate transformation from Cartesian to spherical. A constant radius surface at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = (X^TX)^{1/2} } with PDF Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_X(X) \propto \bigg( 1 + \nu^{-1} R^2 \bigg)^{-(\nu+p)/2} } is an iso-density surface. Given this density value, the quantum of probability on a shell of surface area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_R } and thickness Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta R } at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R } is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta P = p_X(R) \, A_R \delta R } .

The enclosed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } -sphere of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R } has surface area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_R = \frac { 2\pi^{p/2 } R^{ \, p-1 } }{ \Gamma (p/2)} } . Substitution into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta P } shows that the shell has element of probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta P = p_X(R) \frac { 2\pi^{p/2 } R^{ p-1 } }{ \Gamma (p/2)} \delta R } which is equivalent to radial density function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_R(R) = \frac{\Gamma \big ( \frac{1}{2} (\nu + p ) \, \big )}{\nu^{\,p/2} \pi^{\,p/2} \Gamma \big( \frac{1}{2} \nu \big)} \frac { 2 \pi^{p/2 } R^{ p-1 } }{ \Gamma (p/2)} \bigg( 1 + \frac{ R^2 }{\nu} \bigg)^{-( \nu + p )/2 } }

which further simplifies to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_R(R) = \frac { 2}{ \nu ^{1/2} B \big( \frac{1}{2} p, \frac{1}{2} \nu \big)} \bigg( \frac {R^2}{ \nu } \bigg)^{ (p-1)/2 } \bigg( 1 + \frac{ R^2 }{\nu} \bigg)^{-( \nu + p )/2 } } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(*,*) } is the Beta function.

Changing the radial variable to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=R^2 / \nu } returns the previous Beta Prime distribution

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_Y(y) = \frac { 1}{ B \big( \frac{1}{2} p, \frac{1}{2} \nu \big)} y^{\, p/2 - 1 } \bigg( 1 + y \bigg)^{-( \nu + p )/2 } }

To scale the radial variables without changing the radial shape function, define scale matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma = \alpha \operatorname{I} } , yielding a 3-parameter Cartesian density function, ie. the probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta_P } in volume element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx_1 \dots dx_p } is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta_P \big (f_X(X \,|\alpha, p, \nu) \big ) = \frac{\Gamma \big ( \frac{1}{2} (\nu + p ) \, \big )}{ ( \nu \pi)^{\,p/2} \alpha^{\,p/2} \Gamma \big( \frac{1}{2} \nu \big)} \bigg( 1 + \frac{X^T X }{ \alpha \nu} \bigg)^{-( \nu + p )/2 } \; dx_1 \dots dx_p }

or, in terms of scalar radial variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R } ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_R(R \,|\alpha, p, \nu) = \frac { 2}{\alpha^{1/2} \; \nu ^{1/2} B \big( \frac{1}{2} p, \frac{1}{2} \nu \big)} \bigg( \frac {R^2}{ \alpha \, \nu } \bigg)^{ (p-1)/2 } \bigg( 1 + \frac{ R^2 }{ \alpha \, \nu} \bigg)^{-( \nu + p )/2 } }

Radial Moments

The moments of all the radial variables , with the spherical distribution assumption, can be derived from the Beta Prime distribution. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z \sim \beta'(a,b) } then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} (Z^m) = {\frac {B(a + m, b - m)}{B( a ,b )}} } , a known result. Thus, for variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac {p}{\nu} R^2} we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} (y^m) = {\frac {B(\frac{1}{2}p + m, \frac{1}{2} \nu - m)}{B( \frac{1}{2} p ,\frac{1}{2} \nu )}} = \frac{\Gamma \big(\frac{1}{2} p + m \big)\; \Gamma \big(\frac{1}{2} \nu - m \big) }{ \Gamma \big( \frac{1}{2} p \big) \; \Gamma \big( \frac{1}{2} \nu \big) }, \; \nu/2 > m }

The moments of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_2 = \nu \, y } are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} (r_2^m) = \nu^m\operatorname{E} (y^m) }

while introducing the scale matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \operatorname{I} } yields

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} (r_2^m | \alpha) = \alpha^m \nu^m \operatorname{E} (y^m) }

Moments relating to radial variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R } are found by setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R =(\alpha\nu y)^{1/2} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=2m } whereupon

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E} (R^M ) =\operatorname{E} \big((\alpha \nu y)^{1/2} \big)^{2 m } = (\alpha \nu )^{M/2} \operatorname{E} (y^{M/2})= (\alpha \nu )^{M/2} {\frac {B \big(\frac{1}{2} (p + M), \frac{1}{2} (\nu - M) \big )}{B( \frac{1}{2} p ,\frac{1}{2} \nu )}} }

Linear Combinations and Affine Transformation

Full Rank Transform

This closely relates to the multivariate normal method and is described in Kotz and Nadarajah, Kibria and Joarder, Roth, and Cornish. Starting from a somewhat simplified version of the central MV-t pdf: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_X(X) = \frac {\Kappa }{ \left|\Sigma \right|^{1/2} } \left( 1+ \nu^{-1} X^T \Sigma^{-1} X \right) ^ { -\left(\nu + p \right)/2} } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Kappa } is a constant and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } is arbitrary but fixed, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta \in \mathbb{R}^{p \times p}} be a full-rank matrix and form vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y = \Theta X } . Then, by straightforward change of variables

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_Y(Y) = \frac {\Kappa }{ \left|\Sigma \right|^{1/2} } \left( 1+ \nu^{-1}Y^T \Theta^{-T} \Sigma^{-1} \Theta^{-1} Y \right) ^ { -\left(\nu + p \right)/2} \left| \frac{\partial Y }{\partial X} \right| ^{-1} }

The matrix of partial derivatives is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial Y_i }{\partial X_j} = \Theta_{i,j} } and the Jacobian becomes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \frac{\partial Y }{\partial X} \right| = \left| \Theta \right| } . Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_Y(Y) = \frac {\Kappa }{ \left|\Sigma \right|^{1/2} \left| \Theta \right| } \left( 1 + \nu^{-1} Y^T \Theta^{-T} \Sigma^{-1} \Theta^{-1} Y \right) ^ { -\left(\nu + p \right)/2} }

The denominator reduces to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\Sigma \right|^{1/2} \left| \Theta \right| = \left|\Sigma \right|^{1/2} \left| \Theta \right|^{1/2} \left|\Theta^T \right|^{1/2} = \left| \Theta \Sigma \Theta^T \right|^{1/2} }

In full:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_Y(Y) = \frac { \Gamma\left[(\nu+p) / 2\right] }{ \Gamma(\nu/2) \, (\nu \, \pi)^{\, p /2}\left| \Theta \Sigma \Theta^T \right|^{1/2} } \left( 1 + \nu^{-1} Y^T \left( \Theta \Sigma \Theta^T \right) ^{-1} Y \right) ^ { -\left(\nu + p \right)/2} }

which is a regular MV-t distribution.

In general if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim t_p ( \mu, \Sigma, \nu ) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta^{p \times p } } has full rank Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta X + c \sim t_p( \Theta \mu +c, \Theta \Sigma \Theta^T, \nu ) }

Marginal Distributions

This is a special case of the rank-reducing linear transform below. Kotz defines marginal distributions as follows. Partition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim t (p, \mu, \Sigma, \nu ) } into two subvectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1, p_2 } elements:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_p = \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim t \left ( p_1 + p_2, \mu_p, \Sigma_{p \times p}, \nu \right ) }

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1 + p_2 = p } , means Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_p = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}} , scale matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma_{p \times p} = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} }

then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1 \sim t \left ( p_1, \mu_1, \Sigma_{11}, \nu \right ) } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_2 \sim t \left ( p_2, \mu_2, \Sigma_{ 22}, \nu \right ) } such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X_1) = \frac{\Gamma\left[(\nu+p_1)/2\right]}{\Gamma(\nu/2) \, (\nu \,\pi)^ {\, p_1/2}\left|{\boldsymbol\Sigma_{11}}\right|^{1/2}}\left[1+\frac{1}{\nu}({\mathbf X_1}-{\boldsymbol\mu_1})^T{\boldsymbol\Sigma}_{11}^{-1}({\mathbf X_1}-{\boldsymbol\mu_1})\right]^{-(\nu \,+ \, p_1)/2}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X_2) = \frac{\Gamma\left[(\nu+p_2)/2\right]}{\Gamma(\nu/2) \, (\nu \, \pi)^{\, p_2 /2}\left|{\boldsymbol\Sigma_{22}}\right|^{1/2}}\left[1+\frac{1}{\nu}({\mathbf X_2} - {\boldsymbol\mu_2})^T{\boldsymbol\Sigma}_{22}^{-1}({\mathbf X_2}-{\boldsymbol\mu_2})\right]^{-(\nu \,+ \, p_2)/2}}

If a transformation is constructed in the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta_{p_1 \times \, p} = \begin{bmatrix} 1 & \cdots & 0 & \cdots & 0 \\ 0 & \ddots & 0 & \cdots & 0 \\ 0 & \cdots & 1 & \cdots & 0 \end{bmatrix} }

then vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y = \Theta X } , as discussed below, has the same distribution as the marginal distribution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1 } .

Rank-Reducing Linear Transform

In the linear transform case, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta } is a rectangular matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta \in \mathbb{R}^{m \times p}, m < p } , of rank Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m } the result is dimensionality reduction. Here, Jacobian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \Theta \right| } is seemingly rectangular but the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \Theta \Sigma \Theta^T \right|^{1/2} } in the denominator pdf is nevertheless correct. There is a discussion of rectangular matrix product determinants in Aitken. In general if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim t (p, \mu, \Sigma, \nu ) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta^{m \times p } } has full rank Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m } then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y = \Theta X + c \sim t ( m, \Theta \mu + c, \Theta \Sigma \Theta^T, \nu ) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_Y(Y) = \frac{\Gamma\left[(\nu + m)/2\right]}{\Gamma(\nu/2) \, (\nu \,\pi)^{\, m / 2} \left| \Theta \Sigma \Theta^T \right|^{1/2}}\left[1+\frac{1}{\nu}( Y - c_1 )^T ( \Theta \Sigma \Theta^T )^{-1} (Y-c_1) \right]^{-(\nu \,+ \, m)/2}, \; c_1 = \Theta \mu + c}

In extremis, if m = 1 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta } becomes a row vector, then scalar Y follows a univariate double-sided Student-t distribution defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2 = Y^2 / \sigma^2 } with the same Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } degrees of freedom. Kibria et. al. use the affine transformation to find the marginal distributions which are also MV-t.

• During affine transformations of variables with elliptical distributions all vectors must ultimately derive from one initial isotropic spherical vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z } whose elements remain 'entangled' and are not statistically independent.
• A vector of independent student-t samples is not consistent with the multivariate t distribution.
• Adding two sample multivariate t vectors generated with independent Chi-squared samples and different Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } values: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle {1}/\sqrt{u_1/\nu_1}, \; \; {1}/\sqrt{u_2/\nu_2}} will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.
• Taleb compares many examples of fat-tail elliptical vs non-elliptical multivariate distributions

Related concepts

• In univariate statistics, the Student's t-test makes use of Student's t-distribution
• The elliptical multivariate-t distribution arises spontaneously in linearly constrained least squares solutions involving multivariate normal source data, for example the Markowitz global minimum variance solution in financial portfolio analysis. which addresses an ensemble of normal random vectors or a random matrix. It does not arise in ordinary least squares (OLS) or multiple regression with fixed dependent and independent variables which problem tends to produce well-behaved normal error probabilities.
• 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.

See also

• Multivariate normal distribution, which is the limiting case of the multivariate Student's t-distribution when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu\uparrow\infty} .
• Chi distribution, the pdf of the scaling factor in the construction the Student's t-distribution and also the 2-norm (or Euclidean norm) of a multivariate normally distributed vector (centered at zero).
  • Rayleigh distribution#Student's t, random vector length of multivariate t-distribution
• Mahalanobis distance

Literature

• Kotz, Samuel; Nadarajah, Saralees (2004). Multivariate t Distributions and Their Applications. Cambridge University Press. ISBN 978-0521826549.
• Cherubini, Umberto; Luciano, Elisa; Vecchiato, Walter (2004). Copula methods in finance. John Wiley & Sons. ISBN 978-0470863442.
• Taleb, Nassim Nicholas (2023). Statistical Consequences of Fat Tails (1st ed.). Academic Press. ISBN 979-8218248031.

External links

• Copula Methods vs Canonical Multivariate Distributions: the multivariate Student T distribution with general degrees of freedom
• Multivariate Student's t distribution