Multivariate t-distribution explained

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 y and u are independent and distributed as N({ 0},{\boldsymbol\Sigma}) and \chi ^{2} _{\nu} (i.e. multivariate normal and chi-squared distributions) respectively, the matrix \Sigma is a p × p matrix, and {\boldsymbol\mu} is a constant vector then the random variable $=/\backslash sqrt\; +$ has the density^[1] (\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]^

and is said to be distributed as a multivariate t-distribution with parameters

{\boldsymbol\Sigma},{\boldsymbol\mu},\nu . Note that \Sigma is not the covariance matrix since the covariance is given by \nu/(\nu -2 ) \Sigma (for \nu> 2 ).

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

1. Generate

u \sim \chi ^{2} _{\nu} and y \sim N( 0 , \boldsymbol{\Sigma}) , independently.

1. Compute

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

p =1 ), with t = x - \mu and \Sigma =1 , we have the probability density functionf(t) = (\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 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 A = \boldsymbol\Sigma ^{-1} , one has a simple choice of multivariate density functionf( t) = (\Gamma((\nu + p)/ 2 )\left| A \right| ^{1/2})/(\sqrt{\nu ^{p\pi} ^{p}) \Gamma(\nu/ 2 )} \left( 1+ \sum ^{p,p} _{i,j =1} A _{ij} t _{i} t ^{- (\nu + p)/ 2} _{j/\nu\right)}

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)} = (\left| A \right| ^{1/2})/(2 \pi) \left( 1+ \sum ^{2 , 2} _{i,j =1} A _{ij} t _{i} t ^{- (\nu +2 )/ 2} _{j/\nu\right)}

Note that

(\Gamma \left( (\nu +2)/(2) \right))/(\pi \nu \Gamma \left( (\nu)/(2) \right)) = (1)/(2 \pi) .

Now, if

A is the identity matrix, the density isf(t _{1,t} _{2)} = (1)/(2 \pi) \left( 1+ (t ^{2} _{1} + t ^{2 )/\nu\right)} _{2} ^{-(\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

x is a real vector):F( x ) = P ( X \leq x ), rm{where} X \sim t _{\nu(\boldsymbol\mu,\boldsymbol\Sigma).} There is no simple formula for F( x ) , but it can be approximated numerically via Monte Carlo integration.^{[2] [3] [4]}

Conditional Distribution

This was developed by Muirhead ^[5] and Cornish.^[6] but later derived using the simpler chi-squared ratio representation above, by Roth and Ding.^[7] Let vector

X follow a multivariate t distribution and partition into two subvectors of p _{1,} p _{2} elements:X _{p} = \begin{bmatrix} X _{1} \\ X _{2} \end{bmatrix} \sim t _{p} \left (\mu _{p,} \Sigma _{p} , \nu \right )

where

p _{1} + p _{2} = p , the known mean vectors are \mu _{p} = \begin{bmatrix} \mu _{1} \\ \mu _{2} \end{bmatrix} and the scale matrix is \Sigma _{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. X _{1|X} _{2} \sim t _{p _{1}} \left( \mu _{1|2} , (\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

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 becomesf(X _{1|X} _{2)} = (\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]^where\tilde \nu = \nu + p _{2} Effective degrees of freedom, \nu is augmented by the number of disused variables p _{2} .\mu _{1|2} = \mu _{1} + \Sigma _{12} \Sigma ^{-1} _{22} \left(X _{2} - \mu _{2} \right ) is the conditional mean of x _{1} \Sigma _{11|2} = \Sigma _{11} - \Sigma _{12} \Sigma _{22} ^{-1} \Sigma _{21} is the Schur complement of \Sigma _{22} in \Sigma ; the conditional covariance.d _{2} = (X _{2} - \mu ^{T} _{2 )} \Sigma ^{-1} _{22} (X _{2} - \mu _{2)} is the squared Mahalanobis distance of X _{2} from \mu _{2} with scale matrix \Sigma _{22} \Psi = (\nu + d _{2})/(\nu + p _{2}) \Sigma _{11|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.^[8]

Elliptical Representation

Constructed as an elliptical distribution,^[9] take the simplest centralised case with spherical symmetry and no scaling,

\Sigma = \operatorname{I} , then the multivariate t-PDF takes the formf _{X(X)=} g(X ^{T} X) = (\Gamma ( (1)/(2) (\nu + p ) ))/(( \nu \pi) ^{p/2} \Gamma ( (1)/(2) \nu )) ( 1 + \nu ^{-1} X ^{T} X ) ^{-(}

where

X = (x _{1,} … ,x _{p} ) ^{Tisa} p -vector and \nu = degrees of freedom as defined in Muirhead section 1.5. The covariance of X is\operatorname{E} \left( XX ^{T} \right) = \int ^{inf ty} _{-inf ty} … \int ^{inf ty} _{-inf ty} f _{X(x} _{1,...,} x _{p)} XX ^{T} dx _{1} ... dx _{p} = (\nu)/(\nu -2) \operatorname{I}

The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder,^[10] define radial measure

r _{2} = R ^{2} = (X ^{TX})/(p) and, noting that the density is dependent only on r₂, we get

\operatorname{E} [ r _{2} ] = \int ^{inf ty} _{-inf ty} … \int ^{inf ty} _{-inf ty} f _{X(x} _{1,...,} x _{p)} (X ^{TX})/(p) dx _{1} ... dx _{p} = (\nu)/(\nu -2)

which is equivalent to the variance of p -element vector X treated as a univariate heavy-tail zero-mean random sequence with uncorrelated, yet statistically dependent, elements.

Radial Distribution

r _{2} = (X ^{TX})/(p) follows the Fisher-Snedecor or F distribution:r _{2} \sim f _{F} ( p,\nu) = B ( (p)/(2) , (\nu)/(2) ) ^{-1} ( (p)/(\nu) ) r ^{p/ 2-1} _{2} ( 1 + (p)/(\nu) r _{2} ) ^{-(p}

having mean value

\operatorname{E} [ r _{2} ] = (\nu)/(\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 = (p)/(\nu) r _{2} = (X ^{T} X)/(\nu) in the equation above, retaining p -vector X , we have \operatorname{E} [ y ] = \int ^{inf ty} _{-inf ty} … \int ^{inf ty} _{-inf ty} f _{X(X)} (X ^{TX})/(\nu) dx _{1} ... dx _{p} = (p)/(\nu -2) and probability distribution \begin{align} f _{Y(y|} p,\nu) & = \left | (p)/(\nu) \right| ^{-1} B ( (p)/(2) , (\nu)/(2) ) ^{-1} ( (p)/(\nu) ) ( (p)/(\nu) ) y ( 1 + y ) ^{-(p} \ \\ & = B ( (p)/(2) , (\nu)/(2) ) ^{-1} y ( 1+ y ) ^{-(\nu} \end{align} y \sim \beta ' ( y; (p)/(2) , (\nu)/(2) ) having mean value ((1)/(2) p)/((1)/(2) \nu - 1) = (p)/(\nu -2) .

Cumulative Radial Distribution

Given the Beta-prime distribution, the radial cumulative distribution function of

y is known:F _{Y(y)} \sim I ( (y)/(1+ y) ; (p)/(2) , (\nu)/(2) ) B ( (p)/(2) , (\nu)/(2) ) ^{-1}

where

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

In the scalar case,

p = 1 , the distribution is equivalent to Student-t with the equivalence 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

R = (X ^{TX)} ^{1/2} with PDF p _{X(X)} \propto ( 1 + \nu ^{-1} R ^{2} ) ^{-(\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 \delta R at R is \delta P = p _{X(R)} A _{R} \delta R .

The enclosed

p -sphere of radius R has surface area A _{R} = (2 \pi ^{p/2} R)/(\Gamma (p/ 2 )) . Substitution into \delta P shows that the shell has element of probability \delta P = p _{X(R)} (2 \pi ^{p/2} R)/(\Gamma (p/ 2 )) \delta R which is equivalent to radial density functionf _{R(R)} = (\Gamma ( (1)/(2) (\nu + p ) ))/(\nu ^{p/ 2} \pi ^{p/2} \Gamma ( (1)/(2) \nu )) (2 \pi ^{p/2} R)/(\Gamma (p/ 2 )) ( 1 + (R ^{2})/(\nu) ) ^{-(} which further simplifies to f _{R(R)} = (2)/(\nu ^{1/2} B ( (1)/(2) p, (1)/(2) \nu )) ( (R ^{2})/(\nu) ) ( 1 + (R ^{2})/(\nu) ) ^{-(} where B(*,*) is the Beta function.

Changing the radial variable to

y = R ^{2} / \nu returns the previous Beta Prime distributionf _{Y(y)} = (1)/(B ( (1)/(2) p, (1)/(2) \nu )) y ( 1 + y ) ^{-(}

To scale the radial variables without changing the radial shape function, define scale matrix

\Sigma = \alpha \operatorname{I} , yielding a 3-parameter Cartesian density function, ie. the probability \Delta _{P} in volume element dx _{1} ... dx _{p} is\Delta _{P} (f _{X(X} |\alpha, p, \nu) ) = (\Gamma ( (1)/(2) (\nu + p ) ))/(( \nu \pi) ^{p/2} \alpha ^{p/2} \Gamma ( (1)/(2) \nu )) ( 1 + (X ^{T} X)/(\alpha \nu) ) ^{-(} dx _{1} ... dx _{p}

or, in terms of scalar radial variable

R ,f _{R(R} |\alpha, p, \nu) = (2)/(\alpha ^{1 / 2} \nu ^{1/2} B ( (1)/(2) p, (1)/(2) \nu )) ( (R ^{2})/(\alpha \nu) ) ( 1 + (R ^{2})/(\alpha \nu) ) ^{-(}

Radial Moments

The moments of all the radial variables, with the spherical distribution assumption, can be derived from the Beta Prime distribution. If

Z \sim \beta'(a,b) then \operatorname{E} (Z ^{m)} = { (B(a + m, b - m))/(B( a ,b )) } , a known result. Thus, for variable y = (p)/(\nu) R ^{2} we have \operatorname{E} (y ^{m)} = { (B( (1)/(p + m, (1)/(2) \nu - m) 2))/(B( (1)/(2) p , (1)/(2) \nu )) } = \frac, \; \nu/2 > m The moments of r _{2} = \nu y are \operatorname{E} (r ^{m)} _{2} = \nu ^{m\operatorname{E}} (y ^{m)} while introducing the scale matrix \alpha \operatorname{I} yields\operatorname{E} (r ^{m} _{2} | \alpha) = \alpha ^{m} \nu ^{m} \operatorname{E} (y ^{m)} Moments relating to radial variable R are found by setting R = (\alpha\nu y) ^{1/2} and M =2 m whereupon\operatorname{E} (R ^{M} ) = \operatorname{E} ( (\alpha \nu y) ^{1/2} ) ^{2} = (\alpha \nu ) ^{M/2} \operatorname{E} (y ^{M/2} ) = (\alpha \nu ) ^{M/2} { (B ( (1)/(2) (p + M), (1)/(2) (\nu - M) ))/(B( (1)/(2) p , (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:

f _{X(X)} = (\Kappa)/(\left|\Sigma \right| ^{1/2}) \left( 1+ \nu ^{-1} X ^{T} \Sigma ^{-1} X \right) , where \Kappa is a constant and \nu is arbitrary but fixed, let \Theta \in R ^{p} be a full-rank matrix and form vector Y = \Theta X . Then, by straightforward change of variablesf _{Y(Y)} = (\Kappa)/(\left|\Sigma \right| ^{1/2}) \left( 1+ \nu ^{-1} Y ^{T} \Theta ^{-T} \Sigma ^{-1} \Theta ^{-1} Y \right) \left| (\partial Y)/(\partial X) \right| ^{-1}

The matrix of partial derivatives is

(\partial Y _{i})/(\partial X _{j}) = \Theta _{i,j} and the Jacobian becomes \left| (\partial Y)/(\partial X) \right| = \left| \Theta \right| . Thusf _{Y(Y)} = (\Kappa)/(\left|\Sigma \right| ^{1/2} \left| \Theta \right|) \left( 1 + \nu ^{-1} Y ^{T} \Theta ^{-T} \Sigma ^{-1} \Theta ^{-1} Y \right)

The denominator reduces to

\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:f _{Y(Y)} = (\Gamma\left[(\nu + p) / 2 \right])/(\Gamma(\nu/ 2 ) (\nu \pi) \left| \Theta \Sigma \Theta ^{T} \right| ^{1/2}) \left( 1 + \nu ^{-1} Y ^{T} \left( \Theta \Sigma \Theta ^{T} \right) ^{-1} Y \right)

which is a regular MV-t distribution.

In general if

X \sim t _{p} ( \mu, \Sigma, \nu ) and \Theta ^{p} has full rank p then\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

X \sim t (p, \mu, \Sigma, \nu ) into two subvectors of p _{1,} p _{2} elements:X _{p} = \begin{bmatrix} X _{1} \\ X _{2} \end{bmatrix} \sim t \left ( p _{1} + p _{2,} \mu _{p,} \Sigma _{p} , \nu \right )

with

p _{1} + p _{2} = p , means \mu _{p} = \begin{bmatrix} \mu _{1} \\ \mu _{2} \end{bmatrix} , scale matrix \Sigma _{p} = \begin{bmatrix} \Sigma _{11} & \Sigma _{12} \\ \Sigma _{21} & \Sigma _{22} \end{bmatrix}

then

X _{1} \sim t \left ( p _{1,} \mu _{1,} \Sigma _{11} , \nu \right ) , X _{2} \sim t \left ( p _{2,} \mu _{2,} \Sigma , \nu \right ) such thatf(X _{1)} = (\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]^f(X _{2)} = (\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]^

If a transformation is constructed in the form

\Theta _{p _{1} x p} = \begin{bmatrix} 1 & … & 0 & … & 0 \\ 0 & \ddots & 0 & … & 0 \\ 0 & … & 1 & … & 0 \end{bmatrix}

then vector

Y = \Theta X , as discussed below, has the same distribution as the marginal distribution of X _{1} .

Rank-Reducing Linear Transform

In the linear transform case, if

\Theta is a rectangular matrix \Theta \in R ^{m} , m < p , of rank m the result is dimensionality reduction. Here, Jacobian \left| \Theta \right| is seemingly rectangular but the value \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.^[11] In general if X \sim t (p, \mu, \Sigma, \nu ) and \Theta ^{m} has full rank m thenY = \Theta X + c \sim t ( m, \Theta \mu + c, \Theta \Sigma \Theta ^{T,} \nu ) f _{Y(Y)} = (\Gamma\left[(\nu + m)/ 2 \right])/(\left[ 1+ \Gamma(\nu/ 2 ) (\nu \pi) \left| \Theta \Sigma \Theta ^{T} \right| ^{1/2}) (1)/(\nu) ( Y - c _{1} ) ^{T} ( \Theta \Sigma \Theta ^{T} ) ^{-1} (Y - c _{1)} \right] ^{-(\nu} , c _{1} = \Theta \mu + c

In extremis, if m = 1 and

\Theta becomes a row vector, then scalar Y follows a univariate double-sided Student-t distribution defined by t ^{2} = Y ^{2} / \sigma ^{2} with the same \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

\nu values: $/\backslash sqrt,\; \backslash ;\; \backslash ;\; /\backslash sqrt$ will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.^[12]

• 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.^{[13] [14] [15]} 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

\nu\uparrow inf ty .

• 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

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

External links

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

Notes and References

1. Web site: Roth . Michael . 17 April 2013 . On the Multivariate t Distribution . live . 1 June 2022 . Automatic Control group. Linköpin University, Sweden . 31 July 2022 . https://web.archive.org/web/20220731142649/http://users.isy.liu.se/en/rt/roth/student.pdf .
2. Book: Botev . Z. . Chen . Y.-L. . 2022 . Botev. Zdravko. Keller. Alexander. Lemieux. Christiane. Tuffin. Bruno. Advances in Modeling and Simulation: Festschrift for Pierre L'Ecuyer . Springer. 65–87 . Chapter 4: Truncated Multivariate Student Computations via Exponential Tilting. . https://doi.org/10.1007/978-3-031-10193-9_4 . 978-3-031-10192-2.
3. Efficient probability estimation and simulation of the truncated multivariate student-t distribution . Botev . Z. I. . L'Ecuyer . P. . 6 December 2015 . IEEE . 2015 Winter Simulation Conference (WSC) . 380–391 . Huntington Beach, CA, USA . 10.1109/WSC.2015.7408180 .
4. Book: Genz, Alan. Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics . 2009. 195 . Springer. 10.1007/978-3-642-01689-9 . 978-3-642-01689-9. 2017-09-05. 2022-08-27. https://web.archive.org/web/20220827214814/https://link.springer.com/book/10.1007/978-3-642-01689-9. live.
5. Book: Muirhead, Robb . Aspects of Multivariate Statistical Theory . Wiley . 1982 . 978-0-47 1-76985-9 . USA . 32–36 Theorem 1.5.4.
6. Cornish . E A . 1954 . The Multivariate t-Distribution Associated with a Set of Normal Sample Deviates. . Australian Journal of Physics . 7 . 531–542 . 10.1071/PH550193. free .
7. Ding . Peng . 2016 . On the Conditional Distribution of the Multivariate t Distribution . The American Statistician . 70 . 3 . 293–295 . 1604.00561 . 10.1080/00031305.2016.1164756 . 55842994.
8. Web site: Demarta . Stefano . McNeil . Alexander . 2004 . The t Copula and Related Copulas . Risknet.
9. Book: Osiewalski . Jacek . Bayesian Analysis in Statistics and Econometrics . Posterior Moments of Scale Parameters in Elliptical Sampling Models . Steele . Mark . Wiley . 1996 . 0-471-11856-7 . 323–335.
10. Kibria . K M G . Joarder . A H . Jan 2006 . A short review of multivariate t distribution . Journal of Statistical Research . 40 . 1 . 59–72. 10.1007/s42979-021-00503-0 . 232163198 .
11. Book: Aitken, A C -. Determinants and Matrices . Oliver and Boyd . 1948 . 5th . Edinburgh . Chapter IV, section 36.
12. Giron . Javier . del Castilo . Carmen . 2010 . The multivariate Behrens–Fisher distribution . Journal of Multivariate Analysis . 101 . 9 . 2091–2102 . 10.1016/j.jmva.2010.04.008 . free .
13. Okhrin . Y . Schmid . W . 2006 . Distributional Properties of Portfolio Weights . Journal of Econometrics . 134 . 235-256.
14. Bodnar . T . Dmytriv . S . Parolya . N . Schmid . W . 2019 . Tests for the Weights of the Global Minimum Variance Portfolio in a High-Dimensional Setting . IEEE Trans. on Signal Processing . 67 . 17 . 4479-4493.
15. Bodnar . T . Okhrin . Y . 2008 . Properties of the Singular, Inverse and Generalized inverse Partitioned Wishart Distribution. . Journal of Multivariate Analysis . 99 . Eqn.20 . 2389-2405.