Multivariate t-distribution

Multivariate t

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

In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t-distribution is distinct and makes particular use of the matrix structure.

Definition

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

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

and is said to be distributed as a multivariate t-distribution with parameters ${\boldsymbol {\Sigma }},{\boldsymbol {\mu }},\nu$. Note that $\mathbf {\Sigma }$ is not the covariance matrix since the covariance is given by $\nu /(\nu -2)\mathbf {\Sigma }$ (for $\nu >2$).

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

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

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

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

and one approach is to write down a corresponding function of several variables. This is the basic idea of elliptical distribution theory, where one writes down a corresponding function of $p$ variables $t_{i}$ that replaces $t^{2}$ by a quadratic function of all the $t_{i}$. It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom $\nu$. With $\mathbf {A} ={\boldsymbol {\Sigma }}^{-1}$, one has a simple choice of multivariate density function

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

which is the standard but not the only choice.

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

$f(t_{1},t_{2})={\frac {\left|\mathbf {A} \right|^{1/2}}{2\pi }}\left(1+\sum _{i,j=1}^{2,2}A_{ij}t_{i}t_{j}/\nu \right)^{-(\nu +2)/2}$

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

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

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

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

Cumulative distribution function

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

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

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

Conditional Distribution

This was demonstrated by Muirhead though previously derived using the simpler ratio representation above, by Cornish. Let vector $X$ follow the multivariate t distribution and partition into two subvectors of $p_{1},p_{2}$ elements:

$X_{p}={\begin{bmatrix}X_{1}\\X_{2}\end{bmatrix}}\sim t_{p}\left(\mu _{p},\Sigma _{p\times p},\nu \right)$

where $p_{1}+p_{2}=p$, the known mean vector is $\mu _{p}={\begin{bmatrix}\mu _{1}\\\mu _{2}\end{bmatrix}}$ and the scale matrix is $\Sigma _{p\times p}={\begin{bmatrix}\Sigma _{11}&\Sigma _{12}\\\Sigma _{21}&\Sigma _{22}\end{bmatrix}}$.

Then

$p(X_{2}|X_{1})\sim t_{p_{2}}\left(\mu _{2|1},{\frac {\nu +d_{1}}{\nu +p_{1}}}\Sigma _{22|1},\nu +p_{1}\right)$

where

$\mu _{2|1}=\mu _{2}+\Sigma _{21}\Sigma _{11}^{-1}\left(X_{1}-\mu _{1}\right)$ is the conditional mean where it exists or median otherwise.

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

$d_{1}=(X_{1}-\mu _{1})^{T}\Sigma _{11}^{-1}(X_{1}-\mu _{1})$ is the squared Mahalanobis distance of $X_{1}$ from $\mu _{1}$ with scale matrix $\Sigma _{11}$

See for a simple proof of the above conditional distribution.

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 and in the simplest centralised case with spherical symmetry and without scaling, $\Sigma =\operatorname {I} \,$, the multivariate t PDF takes the form

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

where $X=(x_{1},\cdots ,x_{p})^{T}{\text{ is a sampled }}p{\text{-vector}}$ and $\nu$ = degrees of freedom. Muirhead (section 1.5) refers to this as a multivariate Cauchy distribution. The expected covariance of $X$ is

$\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f_{X}(x_{1},\dots ,x_{p})XX^{T}\,dx_{1}\dots dx_{p}={\frac {\nu }{\nu -2}}\operatorname {E} (XX^{T})$

The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder, in a tutorial-style paper, define radial measure $r_{2}=R^{2}={\frac {X^{T}X}{p}}$ such that

$\operatorname {E} [r_{2}]=\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f_{X}(x_{1},\dots ,x_{p}){\frac {X^{T}X}{p}}\,dx_{1}\dots dx_{p}$

which is equivalent to the expected variance of $p$-element vector $X$ treated as a univariate zero-mean random sequence. They note that $r_{2}$ follows the Fisher-Snedecor or $F$ distribution:

$r_{2}\sim F_{F}(p,\nu )=B{\bigg (}{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}^{-1}{\bigg (}{\frac {p}{\nu }}{\bigg )}^{p/2}r_{2}^{p/2-1}{\bigg (}1+{\frac {p}{\nu }}r_{2}{\bigg )}^{-(p+\nu )/2}$

having mean value $\operatorname {E} [r_{2}]={\frac {\nu }{\nu -2}}$.

By a change of random variable to $y={\frac {p}{\nu }}r_{2}={\frac {X^{T}X}{\nu }}$ in the equation above, retaining $p$-vector $X$, we have $\operatorname {E} [y]=\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f_{X}(X){\frac {X^{T}X}{\nu }}\,dx_{1}\dots dx_{p}={\frac {p}{\nu -2}}$ and probability distribution

${\begin{aligned}f_{Y}(y|\,p,\nu )&={\frac {\nu }{p}}B{\bigg (}{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}^{-1}{\big (}{\frac {p}{\nu }}{\big )}^{\,p/2}{\big (}{\frac {p}{\nu }}{\big )}^{-p/2-1}y^{\,p/2-1}{\big (}1+y{\big )}^{-(p+\nu )/2}\\\\&=B{\bigg (}{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}^{-1}y^{\,p/2-1}(1+y)^{-(\nu +p)/2}\end{aligned}}$

which is a regular Beta-prime distribution $y\sim \beta \,'{\bigg (}y;{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}$ having mean value ${\frac {{\frac {1}{2}}p}{{\frac {1}{2}}\nu -1}}={\frac {p}{\nu -2}}$. The cumulative distribution function of $y$ is thus known to be

$F_{Y}(y)\sim I\,{\bigg (}{\frac {y}{1+y}};\,{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}$

where $I$ is the incomplete Beta function.

These results can be derived by straightforward transformation of coordinates from cartesian to spherical. A constant radius surface at $R=(X^{T}X)^{1/2}$ with PDF $p_{X}(X)\propto {\bigg (}1+\nu ^{-1}R^{2}{\bigg )}^{-(\nu +p)/2}$ is an iso-density surface. The quantum of probability in a surface shell of area $A_{R}$ and thickness $\delta R$ at $R$ is $\delta P=p_{X}(R)\,A_{R}\delta R$.

The enclosed sphere in $p$ dimensions has surface area $A_{R}={\frac {2\pi ^{p/2}R^{\,p-1}}{\Gamma (p/2)}}$ and substitution into $\delta P$ shows that the shell has element of probability $\delta P=p_{X}(R){\frac {2\pi ^{p/2}R^{p-1}}{\Gamma (p/2)}}\delta R$. This is equivalent to a radial density function

$f_{R}(R)={\frac {\Gamma {\big (}{\frac {1}{2}}(\nu +p)\,{\big )}}{\nu ^{\,p/2}\pi ^{\,p/2}\Gamma {\big (}{\frac {1}{2}}\nu {\big )}}}{\frac {2\pi ^{p/2}R^{p-1}}{\Gamma (p/2)}}{\bigg (}1+{\frac {R^{2}}{\nu }}{\bigg )}^{-(\nu +p)/2}$

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

Changing the radial variable to $y=R^{2}/\nu$ returns the previous Beta Prime distribution $f_{Y}(y)={\frac {1}{B{\big (}{\frac {1}{2}}p,{\frac {1}{2}}\nu {\big )}}}y^{\,p/2-1}{\bigg (}1+y{\bigg )}^{-(\nu +p)/2}$

To scale the radial variables without changing the radial shape function, define scale matrix $\Sigma =\alpha \operatorname {I}$ , yielding a 3-parameter Cartesian density function, ie. the probability $\Delta _{P}$ in volume element $dx_{1}\dots dx_{p}$ is

$\Delta _{P}{\big (}f_{X}(X\,|\alpha ,p,\nu ){\big )}={\frac {\Gamma {\big (}{\frac {1}{2}}(\nu +p)\,{\big )}}{(\nu \pi )^{\,p/2}\alpha ^{\,p/2}\Gamma {\big (}{\frac {1}{2}}\nu {\big )}}}{\bigg (}1+{\frac {X^{T}X}{\alpha \nu }}{\bigg )}^{-(\nu +p)/2}\;dx_{1}\dots dx_{p}$

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

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

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

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

The moments of $r_{2}=\nu \,y$ are

$\operatorname {E} (r_{2}^{m})=\nu ^{m}\operatorname {E} (y^{m})$

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

$\operatorname {E} (r_{2}^{m}|\alpha )=\alpha ^{m}\nu ^{m}\operatorname {E} (y^{m})$

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

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

Linear Combinations and Affine Transformation

Following section 3.3 of Kibria et.al. let $Z$ be a $p$-vector sampled from a central spherical multivariate t distribution with $\nu$ degrees of freedom: $Z_{p}\sim t_{p}(0,\operatorname {I} ,\nu )$. $X$ is derived from $Z$ via a linear transformation:

$X=\mu +\Sigma ^{1/2}Z$

where $\Sigma$ has full rank, then

$X\sim t_{p}(\mu ,\Sigma ,\nu )$

That is $\operatorname {E} (X)=\mu$ and the covariance of $X$ is $\operatorname {E} {\big [}(X-\mu )(X-\mu )^{T}{\big ]}={\frac {\nu }{\nu -2}}\Sigma$

Furthermore, if $A$ is a non-singular matrix then

$Y=AX+b$ $\sim t_{p}(A\mu +b,A\Sigma A^{T},\nu )$

with mean $\operatorname {E} (Y)=A\mu +b$ and covariance $\operatorname {E} {\big [}(Y-A\mu -b)(Y-A\mu -b)^{T}{\big ]}={\frac {\nu }{\nu -2}}A\Sigma A^{T}$.

Roth (reference below) notes that if $A$ is a $s\times p$ squat matrix with $s<p$ then $Y$ has distribution $Y_{s}\sim t_{s}(A\mu +b,A\Sigma A^{T},\nu )$.

If $A$ takes the form $Y_{s}={\begin{bmatrix}\operatorname {I_{s\times s}} &0_{s\times (p-s)}\end{bmatrix}}X_{p}$ then the PDF of $Y_{s}$ is the marginal distribution of the leading $s$ elements of $X_{p}$.

In the above, the degrees of freedom parameter $\nu$ remains invariant throughout and all vectors must ultimately derive from one initial isotropic spherical vector $Z$ whose elements are not statistically independent. Adding two sample multivariate t vectors generated with independent Chi-squared samples and different $\nu$ values: ${\textstyle {1}/{\sqrt {u_{1}/\nu _{1}}},\;\;{1}/{\sqrt {u_{2}/\nu _{2}}}}$ , as defined in the leading paragraph, will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.

Related concepts

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

See also

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

External links

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