Multivariate t-distribution

One common method of construction of a multivariate t-distribution, for the case of $p$ dimensions, is based on the observation that if $\backslash mathbf\; y$ and $u$ are independent and distributed as $N(\{\backslash mathbf\; 0\},\{\backslash boldsymbol\backslash Sigma\})$ and $\backslash chi^2\_\backslash nu$ (i.e. multivariate normal and chi-squared distributions) respectively, the matrix $\backslash mathbf\{\backslash Sigma\}\backslash ,$ is a p × p matrix, and $\{\backslash boldsymbol\backslash mu\}$ is a constant vector then the random variable $\{\backslash mathbf\; x\}=\{\backslash mathbf\; y\}/\backslash sqrt\{u/\backslash nu\}\; +\{\backslash boldsymbol\backslash mu\}$ has the density{{Cite web |last=Roth |first=Michael |date=17 April 2013 |title=On the Multivariate t Distribution |url=http://users.isy.liu.se/en/rt/roth/student.pdf |url-status=live |access-date=1 June 2022 |website=Automatic Control group. Linköpin University, Sweden |archive-date=31 July 2022 |archive-url=https://web.archive.org/web/20220731142649/http://users.isy.liu.se/en/rt/roth/student.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})^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 $\{\backslash boldsymbol\backslash Sigma\},\{\backslash boldsymbol\backslash mu\},\backslash nu$. Note that $\backslash mathbf\backslash Sigma$ is not the covariance matrix since the covariance is given by $\backslash nu/(\backslash nu-2)\backslash mathbf\backslash Sigma$ (for $\backslash nu>2$).

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

1. Generate $u\; \backslash sim\; \backslash chi^2\_\backslash nu$ and $\backslash mathbf\{y\}\; \backslash sim\; N(\backslash mathbf\{0\},\; \backslash boldsymbol\{\backslash Sigma\})$, independently.
2. Compute $\backslash mathbf\{x\}\; \backslash gets\; \backslash mathbf\{y\}\backslash sqrt\{\backslash nu/u\}+\; \backslash boldsymbol\{\backslash mu\}$.

This formulation gives rise to the hierarchical representation of a multivariate t-distribution as a scale-mixture of normals: $u\; \backslash sim\; \backslash mathrm\{Ga\}(\backslash nu/2,\backslash nu/2)$ where $\backslash mathrm\{Ga\}(a,b)$ indicates a gamma distribution with density proportional to $x^\{a-1\}e^\{-bx\}$, and $\backslash mathbf\{x\}\backslash mid\; u$ conditionally follows $N(\backslash boldsymbol\{\backslash mu\},u^\{-1\}\backslash boldsymbol\{\backslash Sigma\})$.

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

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

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 $p$ 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 $\backslash nu$. With $\backslash mathbf\{A\}\; =\; \backslash boldsymbol\backslash Sigma^\{-1\}$, one has a simple choice of multivariate density function

:$f(\backslash mathbf\; t)\; =\; \backslash frac\{\backslash Gamma((\backslash nu+p)/2)\backslash left|\backslash mathbf\{A\}\backslash right|^\{1/2\}\}\{\backslash sqrt\{\backslash nu^p\backslash pi^p\backslash ,\}\backslash ,\backslash Gamma(\backslash nu/2)\}\; \backslash left(1+\backslash sum\_\{i,j=1\}^\{p,p\}\; A\_\{ij\}\; t\_i\; t\_j/\backslash nu\backslash right)^\{-(\backslash nu+p)/2\}$

which is the standard but not the only choice.

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

:$f(t\_1,t\_2)\; =\; \backslash frac\{\backslash left|\backslash mathbf\{A\}\backslash right|^\{1/2\}\}\{2\backslash pi\}\; \backslash left(1+\backslash sum\_\{i,j=1\}^\{2,2\}\; A\_\{ij\}\; t\_i\; t\_j/\backslash nu\backslash right)^\{-(\backslash nu+2)/2\}$

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

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

:$f(t\_1,t\_2)\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash left(1+(t\_1^2\; +\; t\_2^2)/\backslash nu\backslash right)^\{-(\backslash nu+2)/2\}.$

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 $\backslash Sigma$ 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.

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

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

There is no simple formula for $F(\backslash mathbf\{x\})$, but it can be [http://www.mathworks.com/matlabcentral/fileexchange/53796 approximated numerically] via Monte Carlo integration.{{cite book |last1=Botev |first1=Z. |last2=Chen |first2=Y.-L. |date=2022 |editor-last1=Botev|editor-first1=Zdravko|editor-last2=Keller|editor-first2=Alexander|editor-last3=Lemieux|editor-first3=Christiane| editor-last4=Tuffin|editor-first4=Bruno|title=Advances in Modeling and Simulation: Festschrift for Pierre L'Ecuyer |publisher=Springer|pages=65–87 |chapter=Chapter 4: Truncated Multivariate Student Computations via Exponential Tilting. |doi=10.1007/978-3-031-10193-9_4 |chapter-url=https://doi.org/10.1007/978-3-031-10193-9_4 |isbn=978-3-031-10192-2}}{{cite conference |title=Efficient probability estimation and simulation of the truncated multivariate student-t distribution |last1=Botev |first1=Z. I. |last2=L'Ecuyer |first2=P. |date=6 December 2015 |publisher=IEEE |book-title=2015 Winter Simulation Conference (WSC) |pages=380–391 |location=Huntington Beach, CA, USA |doi=10.1109/WSC.2015.7408180 }}

{{cite book|last=Genz|first=Alan|title=Computation of Multivariate Normal and t Probabilities|series=Lecture Notes in Statistics |date=2009|volume=195 |publisher=Springer|doi=10.1007/978-3-642-01689-9 |isbn=978-3-642-01689-9|url=https://www.springer.com/statistics/computational+statistics/book/978-3-642-01688-2|access-date=2017-09-05|archive-date=2022-08-27|archive-url=https://web.archive.org/web/20220827214814/https://link.springer.com/book/10.1007/978-3-642-01689-9|url-status=live}}

This was developed by Muirhead {{Cite book |last=Muirhead |first=Robb |title=Aspects of Multivariate Statistical Theory |publisher=Wiley |year=1982 |isbn=978-0-47 1-76985-9 |location=USA |pages=32–36 Theorem 1.5.4}} and Cornish.{{Cite journal |last=Cornish |first=E A |date=1954 |title=The Multivariate t-Distribution Associated with a Set of Normal Sample Deviates. |url=https://www.publish.csiro.au/PH/pdf/PH540531 |journal=Australian Journal of Physics |volume=7 |pages=531–542 |doi=10.1071/PH550193|doi-access=free }} but later derived using the simpler chi-squared ratio representation above, by Roth and Ding.{{cite journal |last1=Ding |first1=Peng |year=2016 |title=On the Conditional Distribution of the Multivariate t Distribution |url=https://www.tandfonline.com/doi/full/10.1080/00031305.2016.1164756 |journal=The American Statistician |volume=70 |issue=3 |pages=293–295 |arxiv=1604.00561 |doi=10.1080/00031305.2016.1164756 |s2cid=55842994}} Let vector $X$ follow a multivariate t distribution and partition into two subvectors of $p\_1,\; p\_2$ elements:

:$X\_p\; =\; \backslash begin\{bmatrix\}$

X_1 \\

X_2 \end{bmatrix} \sim t_p \left (\mu_p, \Sigma_{p \times p}, \nu \right )

where $p\_1\; +\; p\_2\; =\; p$, the known mean vectors are $\backslash mu\_p\; =\; \backslash begin\{bmatrix\}$

\mu_1 \\

\mu_2 \end{bmatrix} and the scale matrix is $\backslash Sigma\_\{p\; \backslash times\; p\}\; =\; \backslash begin\{bmatrix\}$

\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.

:$X\_1|X\_2\; \backslash sim\; t\_\{\; p\_1\; \}\backslash left(\; \backslash mu\_\{1|2\},\backslash frac\{\backslash nu\; +\; d\_2\}\{\backslash nu\; +\; p\_2\}\; \backslash Sigma\_\{11|2\},\; \backslash nu\; +\; p\_2\; \backslash right)$

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 $X\_1|X\_2\; \backslash sim\; t\_\{\; p\_1\; \}\backslash left(\; \backslash mu\_\{1|2\},\; \backslash Psi\; ,\backslash tilde\{\; \backslash nu\; \}\; \backslash right)$ above then, using the parameters below, the explicit conditional distribution becomes

:$$

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

: $\backslash tilde\; \backslash nu\; =\; \backslash nu\; +\; p\_2$ Effective degrees of freedom, $\backslash nu$ is augmented by the number of disused variables $p\_2$.

: $\backslash mu\_\{1|2\}\; =\; \backslash mu\_1\; +\; \backslash Sigma\_\{12\}\; \backslash Sigma\_\{22\}^\{-1\}\; \backslash left(X\_2\; -\; \backslash mu\_2\; \backslash right\; )$ is the conditional mean of $x\_1$

: $\backslash Sigma\_\{11|2\}\; =\; \backslash Sigma\_\{11\}\; -\; \backslash Sigma\_\{12\}\; \backslash Sigma\_\{22\}\; ^\{-1\}\; \backslash Sigma\_\{21\}$ is the Schur complement of $\backslash Sigma\_\{22\}\; \backslash text\{\; in\; \}\; \backslash Sigma$.

: $d\_2\; =\; (X\_2\; -\; \backslash mu\_2)^T\; \backslash Sigma\_\{22\}^\{-1\}\; (X\_2\; -\; \backslash mu\_2)$ is the squared Mahalanobis distance of $X\_2$ from $\backslash mu\_2$ with scale matrix $\backslash Sigma\_\{22\}$

: $\backslash Psi\; =\; \backslash frac\{\backslash nu\; +\; d\_2\}\{\backslash nu\; +\; p\_2\}\; \backslash Sigma\_\{11|2\}$ is the conditional scale matrix for $\backslash tilde\{\backslash nu\}\; >\; 2$.

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.{{Cite web |last1=Demarta |first1=Stefano |last2=McNeil |first2=Alexander |date=2004 |title=The t Copula and Related Copulas |url=https://www.risknet.de/uploads/tx_bxelibrary/t-Copula-Demarta-ETH.pdf |website=Risknet}}

Constructed as an elliptical distribution,{{Cite book |last1=Osiewalski |first1=Jacek |title=Bayesian Analysis in Statistics and Econometrics |chapter=Posterior Moments of Scale Parameters in Elliptical Sampling Models |last2=Steele |first2=Mark |publisher=Wiley |year=1996 |isbn=0-471-11856-7 |pages=323–335}} take the simplest centralised case with spherical symmetry and no scaling, $\backslash Sigma\; =\; \backslash operatorname\{I\}\; \backslash ,$, then the multivariate t-PDF takes the form

: $f\_X(X)=\; g(X^T\; X)\; =\; \backslash frac\{\backslash Gamma\; \backslash big\; (\; \backslash frac\{1\}\{2\}\; (\backslash nu\; +\; p\; )\; \backslash ,\; \backslash big\; )\}\{\; (\; \backslash nu\; \backslash pi)^\{\backslash ,p/2\}\; \backslash Gamma\; \backslash big(\; \backslash frac\{1\}\{2\}\; \backslash nu\; \backslash big)\}\; \backslash bigg(\; 1\; +\; \backslash nu^\{-1\}\; X^T\; X\; \backslash bigg)^\{-(\; \backslash nu\; +\; p\; )/2\; \}$

where $X\; =(x\_1,\; \backslash cdots\; ,x\_p\; )^T\backslash text\; \{\; is\; a\; \}\; p\backslash text\{-vector\}$ and $\backslash nu$ = degrees of freedom as defined in Muirhead section 1.5. The covariance of $X$ is

:$\backslash operatorname\{E\}\; \backslash left(\; XX^T\; \backslash right)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash cdots\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f\_X(x\_1,\backslash dots,\; x\_p)\; XX^T\; \backslash ,\; dx\_1\; \backslash dots\; dx\_p\; =\; \backslash frac\{\; \backslash nu\; \}\{\; \backslash nu\; -\; 2\; \}\; \backslash operatorname\{I\}$

The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder,{{Cite journal |last1=Kibria |first1=K M G |last2=Joarder |first2=A H |date=Jan 2006 |title=A short review of multivariate t distribution |url=https://link.springer.com/content/pdf/10.1007/s42979-021-00503-0.pdf |journal=Journal of Statistical Research |volume=40 |issue=1 |pages=59–72|doi=10.1007/s42979-021-00503-0 |s2cid=232163198 }} define radial measure $r\_2\; =\; R^2\; =\; \backslash frac\{X^TX\}\{p\}$ and, noting that the density is dependent only on r₂, we get

$\backslash operatorname\{E\}\; [\; r\_2\; ]\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash cdots\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f\_X(x\_1,\backslash dots,\; x\_p)\; \backslash frac\; \{X^TX\}\{p\}\backslash ,\; dx\_1\; \backslash dots\; dx\_p\; =\; \backslash frac\{\backslash nu\}\{\; \backslash nu\; -2\}$

$r\_2\; =\; \backslash frac\{X^TX\}\{p\}$ follows the Fisher-Snedecor or $F$ distribution:

:$r\_2\; \backslash sim\; f\_\{F\}(\; p,\backslash nu)\; =\; B\; \backslash bigg(\; \backslash frac\; \{p\}\{2\},\; \backslash frac\; \{\backslash nu\}\{2\}\; \backslash bigg\; )\; ^\{-1\}\; \backslash bigg\; (\backslash frac\{p\}\{\backslash nu\}\; \backslash bigg\; )^\{\; p/2\; \}\; r\_2^\; \{\; p/2\; -1\; \}$

\bigg( 1 + \frac{p}{\nu} r_2 \bigg) ^{-(p + \nu)/2 }

having mean value $\backslash operatorname\{E\}\; [\; r\_2\; ]\; =\; \backslash frac\; \{\; \backslash nu\; \}\{\; \backslash nu\; -\; 2\; \}$. $F$-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 $y\; =\; \backslash frac\{p\}\{\backslash nu\}\; r\_2\; =\; \backslash frac\; \{X^T\; X\}\{\backslash nu\}$ in the equation above, retaining $p$-vector $X$, we have $\backslash operatorname\{E\}\; [\; y\; ]\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash cdots\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f\_X(X)\; \backslash frac\; \{X^TX\}\{\; \backslash nu\}\backslash ,\; dx\_1\; \backslash dots\; dx\_p\; =\; \backslash frac\; \{\; p\; \}\{\; \backslash nu\; -\; 2\; \}$ and probability distribution

:

& = 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 $y\; \backslash sim\; \backslash beta\; \backslash ,\; \text{'}\; \backslash bigg(y;\; \backslash frac\; \{p\}\{2\},\; \backslash frac\; \{\backslash nu\}\{2\}\; \backslash bigg\; )$ having mean value $\backslash frac\; \{\; \backslash frac\{1\}\{2\}\; p\; \}\{\; \backslash frac\{1\}\{2\}\backslash nu\; -\; 1\; \}\; =\; \backslash frac\; \{\; p\; \}\{\; \backslash nu\; -\; 2\; \}$.

Given the Beta-prime distribution, the radial cumulative distribution function of $y$ is known:

:$F\_Y(y)\; \backslash sim\; I\; \backslash ,\; \backslash bigg(\backslash frac\; \{y\}\{1+y\};\; \backslash ,\; \backslash frac\; \{p\}\{2\},\; \backslash frac\; \{\backslash nu\}\{2\}\; \backslash bigg\; )\; B\backslash bigg(\; \backslash frac\; \{p\}\{2\},\; \backslash frac\; \{\backslash nu\}\{2\}\; \backslash bigg\; )^\{-1\}$

where $I$ is the incomplete Beta function and applies with a spherical $\backslash Sigma$ assumption.

In the scalar case, $p\; =\; 1$, the distribution is equivalent to Student-t with the equivalence $t^2\; =\; y^2\; \backslash sigma^\{-1\}$, 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 $R\; =\; (X^TX)^\{1/2\}$ with PDF $p\_X(X)\; \backslash propto\; \backslash bigg(\; 1\; +\; \backslash nu^\{-1\}\; R^2\; \backslash bigg)^\{-(\backslash nu+p)/2\}$ is an iso-density surface. Given this density value, the quantum of probability on a shell of surface area $A\_R$ and thickness $\backslash delta\; R$ at $R$ is $\backslash delta\; P\; =\; p\_X(R)\; \backslash ,\; A\_R\; \backslash delta\; R$.

The enclosed $p$-sphere of radius $R$ has surface area $A\_R\; =\; \backslash frac\; \{\; 2\backslash pi^\{p/2\; \}\; R^\{\; \backslash ,\; p-1\; \}\; \}\{\; \backslash Gamma\; (p/2)\}$. Substitution into $\backslash delta\; P$ shows that the shell has element of probability $\backslash delta\; P\; =\; p\_X(R)\; \backslash frac\; \{\; 2\backslash pi^\{p/2\; \}\; R^\{\; p-1\; \}\; \}\{\; \backslash Gamma\; (p/2)\}\; \backslash delta\; R$ which is equivalent to radial density function

:$f\_R(R)\; =\; \backslash frac\{\backslash Gamma\; \backslash big\; (\; \backslash frac\{1\}\{2\}\; (\backslash nu\; +\; p\; )\; \backslash ,\; \backslash big\; )\}\{\backslash nu^\{\backslash ,p/2\}\; \backslash pi^\{\backslash ,p/2\}\; \backslash Gamma\; \backslash big(\; \backslash frac\{1\}\{2\}\; \backslash nu\; \backslash big)\}\; \backslash frac\; \{\; 2\; \backslash pi^\{p/2\; \}\; R^\{\; p-1\; \}\; \}\{\; \backslash Gamma\; (p/2)\}\; \backslash bigg(\; 1\; +\; \backslash frac\{\; R^2\; \}\{\backslash nu\}\; \backslash bigg)^\{-(\; \backslash nu\; +\; p\; )/2\; \}$

which further simplifies to $f\_R(R)\; =\; \backslash frac\; \{\; 2\}\{\; \backslash nu\; ^\{1/2\}\; B\; \backslash big(\; \backslash frac\{1\}\{2\}\; p,\; \backslash frac\{1\}\{2\}\; \backslash nu\; \backslash big)\}\; \backslash bigg(\; \backslash frac\; \{R^2\}\{\; \backslash nu\; \}\; \backslash bigg)^\{\; (p-1)/2\; \}\; \backslash bigg(\; 1\; +\; \backslash frac\{\; R^2\; \}\{\backslash nu\}\; \backslash bigg)^\{-(\; \backslash nu\; +\; p\; )/2\; \}$ where $B(*,*)$ is the Beta function.

Changing the radial variable to $y=R^2\; /\; \backslash nu$ returns the previous Beta Prime distribution

:$f\_Y(y)\; =\; \backslash frac\; \{\; 1\}\{\; B\; \backslash big(\; \backslash frac\{1\}\{2\}\; p,\; \backslash frac\{1\}\{2\}\; \backslash nu\; \backslash big)\}\; y^\{\backslash ,\; p/2\; -\; 1\; \}\; \backslash bigg(\; 1\; +\; y\; \backslash bigg)^\{-(\; \backslash nu\; +\; p\; )/2\; \}$

To scale the radial variables without changing the radial shape function, define scale matrix $\backslash Sigma\; =\; \backslash alpha\; \backslash operatorname\{I\}$ , yielding a 3-parameter Cartesian density function, ie. the probability $\backslash Delta\_P$ in volume element $dx\_1\; \backslash dots\; dx\_p$ is

:$\backslash Delta\_P\; \backslash big\; (f\_X(X\; \backslash ,|\backslash alpha,\; p,\; \backslash nu)\; \backslash big\; )\; =\; \backslash frac\{\backslash Gamma\; \backslash big\; (\; \backslash frac\{1\}\{2\}\; (\backslash nu\; +\; p\; )\; \backslash ,\; \backslash big\; )\}\{\; (\; \backslash nu\; \backslash pi)^\{\backslash ,p/2\}\; \backslash alpha^\{\backslash ,p/2\}\; \backslash Gamma\; \backslash big(\; \backslash frac\{1\}\{2\}\; \backslash nu\; \backslash big)\}\; \backslash bigg(\; 1\; +\; \backslash frac\{X^T\; X\; \}\{\; \backslash alpha\; \backslash nu\}\; \backslash bigg)^\{-(\; \backslash nu\; +\; p\; )/2\; \}\; \backslash ;\; dx\_1\; \backslash dots\; dx\_p$

or, in terms of scalar radial variable $R$,

:$f\_R(R\; \backslash ,|\backslash alpha,\; p,\; \backslash nu)\; =\; \backslash frac\; \{\; 2\}\{\backslash alpha^\{1/2\}\; \backslash ;\; \backslash nu\; ^\{1/2\}\; B\; \backslash big(\; \backslash frac\{1\}\{2\}\; p,\; \backslash frac\{1\}\{2\}\; \backslash nu\; \backslash big)\}\; \backslash bigg(\; \backslash frac\; \{R^2\}\{\; \backslash alpha\; \backslash ,\; \backslash nu\; \}\; \backslash bigg)^\{\; (p-1)/2\; \}\; \backslash bigg(\; 1\; +\; \backslash frac\{\; R^2\; \}\{\; \backslash alpha\; \backslash ,\; \backslash nu\}\; \backslash bigg)^\{-(\; \backslash nu\; +\; p\; )/2\; \}$

The moments of all the radial variables , with the spherical distribution assumption, can be derived from the Beta Prime distribution. If $Z\; \backslash sim\; \backslash beta\text{'}(a,b)$ then $\backslash operatorname\{E\}\; (Z^m)\; =\; \{\backslash frac\; \{B(a\; +\; m,\; b\; -\; m)\}\{B(\; a\; ,b\; )\}\}$, a known result. Thus, for variable $y\; =\; \backslash frac\; \{p\}\{\backslash nu\}\; R^2$ we have

:$\backslash operatorname\{E\}\; (y^m)\; =\; \{\backslash frac\; \{B(\backslash frac\{1\}\{2\}p\; +\; m,\; \backslash frac\{1\}\{2\}\; \backslash nu\; -\; m)\}\{B(\; \backslash frac\{1\}\{2\}\; p\; ,\backslash frac\{1\}\{2\}\; \backslash nu\; )\}\}\; =\; \backslash frac\{\backslash Gamma\; \backslash big(\backslash frac\{1\}\{2\}\; p\; +\; m\; \backslash big)\backslash ;\; \backslash Gamma\; \backslash big(\backslash frac\{1\}\{2\}\; \backslash nu\; -\; m\; \backslash big)\; \}\{\; \backslash Gamma\; \backslash big(\; \backslash frac\{1\}\{2\}\; p\; \backslash big)\; \backslash ;\; \backslash Gamma\; \backslash big(\; \backslash frac\{1\}\{2\}\; \backslash nu\; \backslash big)\; \},\; \backslash ;\; \backslash nu/2\; >\; m$

The moments of $r\_2\; =\; \backslash nu\; \backslash ,\; y$ are

:$\backslash operatorname\{E\}\; (r\_2^m)\; =\; \backslash nu^m\backslash operatorname\{E\}\; (y^m)$

while introducing the scale matrix $\backslash alpha\; \backslash operatorname\{I\}$ yields

:$\backslash operatorname\{E\}\; (r\_2^m\; |\; \backslash alpha)\; =\; \backslash alpha^m\; \backslash nu^m\; \backslash operatorname\{E\}\; (y^m)$

Moments relating to radial variable $R$ are found by setting $R\; =(\backslash alpha\backslash nu\; y)^\{1/2\}$ and $M=2m$ whereupon

:$\backslash operatorname\{E\}\; (R^M\; )\; =\backslash operatorname\{E\}\; \backslash big((\backslash alpha\; \backslash nu\; y)^\{1/2\}\; \backslash big)^\{2\; m\; \}\; =\; (\backslash alpha\; \backslash nu\; )^\{M/2\}\; \backslash operatorname\{E\}\; (y^\{M/2\})=\; (\backslash alpha\; \backslash nu\; )^\{M/2\}\; \{\backslash frac\; \{B\; \backslash big(\backslash frac\{1\}\{2\}\; (p\; +\; M),\; \backslash frac\{1\}\{2\}\; (\backslash nu\; -\; M)\; \backslash big\; )\}\{B(\; \backslash frac\{1\}\{2\}\; p\; ,\backslash frac\{1\}\{2\}\; \backslash nu\; )\}\}$

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: $f\_X(X)\; =\; \backslash frac\; \{\backslash Kappa\; \}\{\; \backslash left|\backslash Sigma\; \backslash right|^\{1/2\}\; \}\; \backslash left(\; 1+\; \backslash nu^\{-1\}\; X^T\; \backslash Sigma^\{-1\}\; X\; \backslash right)\; ^\; \{\; -\backslash left(\backslash nu\; +\; p\; \backslash right)/2\}$, where $\backslash Kappa$ is a constant and $\backslash nu$ is arbitrary but fixed, let $\backslash Theta$

\in \mathbb{R}^{p \times p} be a full-rank matrix and form vector $Y\; =\; \backslash Theta\; X$. Then, by straightforward change of variables

: $f\_Y(Y)\; =\; \backslash frac\; \{\backslash Kappa\; \}\{\; \backslash left|\backslash Sigma\; \backslash right|^\{1/2\}\; \}\; \backslash left(\; 1+\; \backslash nu^\{-1\}Y^T\; \backslash Theta^\{-T\}\; \backslash Sigma^\{-1\}\; \backslash Theta^\{-1\}\; Y\; \backslash right)\; ^\; \{\; -\backslash left(\backslash nu\; +\; p\; \backslash right)/2\}\; \backslash left|\; \backslash frac\{\backslash partial\; Y\; \}\{\backslash partial\; X\}\; \backslash right|\; ^\{-1\}$

The matrix of partial derivatives is $\backslash frac\{\backslash partial\; Y\_i\; \}\{\backslash partial\; X\_j\}\; =\; \backslash Theta\_\{i,j\}$ and the Jacobian becomes $\backslash left|\; \backslash frac\{\backslash partial\; Y\; \}\{\backslash partial\; X\}\; \backslash right|\; =\; \backslash left|\; \backslash Theta\; \backslash right|$. Thus

: $f\_Y(Y)\; =\; \backslash frac\; \{\backslash Kappa\; \}\{\; \backslash left|\backslash Sigma\; \backslash right|^\{1/2\}\; \backslash left|\; \backslash Theta\; \backslash right|\; \}\; \backslash left(\; 1\; +\; \backslash nu^\{-1\}\; Y^T\; \backslash Theta^\{-T\}\; \backslash Sigma^\{-1\}\; \backslash Theta^\{-1\}\; Y\; \backslash right)\; ^\; \{\; -\backslash left(\backslash nu\; +\; p\; \backslash right)/2\}$

The denominator reduces to

:$\backslash left|\backslash Sigma\; \backslash right|^\{1/2\}\; \backslash left|\; \backslash Theta\; \backslash right|\; =\; \backslash left|\backslash Sigma\; \backslash right|^\{1/2\}\; \backslash left|\; \backslash Theta\; \backslash right|^\{1/2\}\; \backslash left|\backslash Theta^T\; \backslash right|^\{1/2\}\; =\; \backslash left|\; \backslash Theta\; \backslash Sigma\; \backslash Theta^T\; \backslash right|^\{1/2\}$

In full:

: $f\_Y(Y)\; =\; \backslash frac\; \{\; \backslash Gamma\backslash left[(\backslash nu+p)\; /\; 2\backslash right]\; \}\{\; \backslash Gamma(\backslash nu/2)\; \backslash ,\; (\backslash nu\; \backslash ,\; \backslash pi)^\{\backslash ,\; p\; /2\}\backslash left|\; \backslash Theta\; \backslash Sigma\; \backslash Theta^T\; \backslash right|^\{1/2\}\; \}\; \backslash left(\; 1\; +\; \backslash nu^\{-1\}\; Y^T\; \backslash left(\; \backslash Theta\; \backslash Sigma\; \backslash Theta^T\; \backslash right)\; ^\{-1\}\; Y\; \backslash right)\; ^\; \{\; -\backslash left(\backslash nu\; +\; p\; \backslash right)/2\}$

which is a regular MV-t distribution.

In general if $X\; \backslash sim\; t\_p\; (\; \backslash mu,\; \backslash Sigma,\; \backslash nu\; )$ and $\backslash Theta^\{p\; \backslash times\; p\; \}$ has full rank $p$ then

: $\backslash Theta\; X\; +\; c\; \backslash sim\; t\_p(\; \backslash Theta\; \backslash mu\; +c,\; \backslash Theta\; \backslash Sigma\; \backslash Theta^T,\; \backslash nu\; )$

This is a special case of the rank-reducing linear transform below. Kotz defines marginal distributions as follows. Partition $X\; \backslash sim\; t\; (p,\; \backslash mu,\; \backslash Sigma,\; \backslash nu\; )$ into two subvectors of $p\_1,\; p\_2$ elements:

:$X\_p\; =\; \backslash begin\{bmatrix\}$

X_1 \\

X_2 \end{bmatrix} \sim t \left ( p_1 + p_2, \mu_p, \Sigma_{p \times p}, \nu \right )

with $p\_1\; +\; p\_2\; =\; p$, means $\backslash mu\_p\; =\; \backslash begin\{bmatrix\}$

\mu_1 \\

\mu_2 \end{bmatrix}, scale matrix $\backslash Sigma\_\{p\; \backslash times\; p\}\; =\; \backslash begin\{bmatrix\}$

\Sigma_{11} & \Sigma_{12} \\

\Sigma_{21} & \Sigma_{22} \end{bmatrix}

then $X\_1\; \backslash sim\; t\; \backslash left\; (\; p\_1,\; \backslash mu\_1,\; \backslash Sigma\_\{11\},\; \backslash nu\; \backslash right\; )$, $X\_2\; \backslash sim\; t\; \backslash left\; (\; p\_2,\; \backslash mu\_2,\; \backslash Sigma\_\{\; 22\},\; \backslash nu\; \backslash right\; )$ such that

:$f(X\_1)\; =$

\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}

:$f(X\_2)\; =$

\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

: $$

\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 $Y\; =\; \backslash Theta\; X$, as discussed below, has the same distribution as the marginal distribution of $X\_1$ .

In the linear transform case, if $\backslash Theta$ is a rectangular matrix , of rank $m$ the result is dimensionality reduction. Here, Jacobian $\backslash left|\; \backslash Theta\; \backslash right|$ is seemingly rectangular but the value $\backslash left|\; \backslash Theta\; \backslash Sigma\; \backslash Theta^T\; \backslash right|^\{1/2\}$ in the denominator pdf is nevertheless correct. There is a discussion of rectangular matrix product determinants in Aitken.{{Cite book |last=Aitken |first=A C -|title=Determinants and Matrices |publisher=Oliver and Boyd |year=1948 |edition=5th |location=Edinburgh |pages=Chapter IV, section 36}} In general if $X\; \backslash sim\; t\; (p,\; \backslash mu,\; \backslash Sigma,\; \backslash nu\; )$ and $\backslash Theta^\{m\; \backslash times\; p\; \}$ has full rank $m$ then

: $Y\; =\; \backslash Theta\; X\; +\; c\; \backslash sim\; t\; (\; m,\; \backslash Theta\; \backslash mu\; +\; c,\; \backslash Theta\; \backslash Sigma\; \backslash Theta^T,\; \backslash nu\; )$

: $f\_Y(Y)\; =\; \backslash frac\{\backslash Gamma\backslash left[(\backslash nu\; +\; m)/2\backslash right]\}\{\backslash Gamma(\backslash nu/2)\; \backslash ,\; (\backslash nu\; \backslash ,\backslash pi)^\{\backslash ,\; m\; /\; 2\}\; \backslash left|\; \backslash Theta\; \backslash Sigma\; \backslash Theta^T\; \backslash right|^\{1/2\}\}\backslash left[1+\backslash frac\{1\}\{\backslash nu\}(\; Y\; -\; c\_1\; )^T\; (\; \backslash Theta\; \backslash Sigma\; \backslash Theta^T\; )^\{-1\}\; (Y-c\_1)\; \backslash right]^\{-(\backslash nu\; \backslash ,+\; \backslash ,\; m)/2\},\; \backslash ;\; c\_1\; =\; \backslash Theta\; \backslash mu\; +\; c$

In extremis, if m = 1 and $\backslash Theta$ becomes a row vector, then scalar Y follows a univariate double-sided Student-t distribution defined by $t^2\; =\; Y^2\; /\; \backslash sigma^2$ with the same $\backslash nu$ 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 $Z$ 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 $\backslash nu$ values: $\{1\}/\backslash sqrt\{u\_1/\backslash nu\_1\},\; \backslash ;\; \backslash ;\; \{1\}/\backslash sqrt\{u\_2/\backslash nu\_2\}$ will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.{{Cite journal |last1=Giron |first1=Javier |last2=del Castilo |first2=Carmen |date=2010 |title=The multivariate Behrens–Fisher distribution |journal=Journal of Multivariate Analysis |volume=101 |issue=9 |pages=2091–2102 |doi=10.1016/j.jmva.2010.04.008 |doi-access=free }}
• Taleb compares many examples of fat-tail elliptical vs non-elliptical multivariate distributions

• 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.{{Cite journal |last1=Okhrin |first1=Y |last2=Schmid |first2=W |date=2006 |title=Distributional Properties of Portfolio Weights |url=https://www.sciencedirect.com/science/article/abs/pii/S0304407605001442 |journal=Journal of Econometrics |volume=134 |pages=235–256|doi=10.1016/j.jeconom.2005.06.022 }}{{Cite journal |last1=Bodnar |first1=T |last2=Dmytriv |first2=S |last3=Parolya |first3=N |last4=Schmid |first4=W |date=2019 |title=Tests for the Weights of the Global Minimum Variance Portfolio in a High-Dimensional Setting |url=https://ieeexplore.ieee.org/document/8767989 |journal=IEEE Trans. On Signal Processing |volume=67 |issue=17 |pages=4479–4493|doi=10.1109/TSP.2019.2929964 |arxiv=1710.09587 |bibcode=2019ITSP...67.4479B }}{{Cite journal |last1=Bodnar |first1=T |last2=Okhrin |first2=Y |date=2008 |title=Properties of the Singular, Inverse and Generalized inverse Partitioned Wishart Distribution. |url=https://core.ac.uk/download/pdf/82469023.pdf |journal=Journal of Multivariate Analysis |volume=99 |issue=Eqn.20 |pages=2389–2405|doi=10.1016/j.jmva.2008.02.024 }} 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.

{{more footnotes|date=May 2012}}

• Multivariate normal distribution, which is the limiting case of the multivariate Student's t-distribution when $\backslash nu\backslash uparrow\backslash infty$.
• 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

{{Reflist|refs=

}}

{{refbegin}}

• {{cite book |title= Multivariate t Distributions and Their Applications |last= Kotz |first= Samuel |author2=Nadarajah, Saralees |year= 2004 |publisher= Cambridge University Press |isbn= 978-0521826549 }}
• {{cite book |title= Copula methods in finance |last= Cherubini |first= Umberto |author2=Luciano, Elisa |author3=Vecchiato, Walter |year= 2004 |publisher= John Wiley & Sons |isbn= 978-0470863442 }}
• {{Cite book |last=Taleb |first=Nassim Nicholas |title=Statistical Consequences of Fat Tails |publisher=Academic Press |year=2023 |isbn=979-8218248031 |edition=1st}}

{{refend}}

• [https://web.archive.org/web/20061202010900/http://www.mth.kcl.ac.uk/~shaww/web_page/papers/MultiStudentc.pdf Copula Methods vs Canonical Multivariate Distributions: the multivariate Student T distribution with general degrees of freedom]
• [http://www.statlect.com/mcdstu1.htm Multivariate Student's t distribution]

{{ProbDistributions|multivariate}}

{{DEFAULTSORT:Multivariate Normal Distribution}}

Category:Continuous distributions

Category:Multivariate continuous distributions