Multivariate t

Notation	${\displaystyle t_{\nu }({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}$
Parameters	${\displaystyle {\boldsymbol {\mu }}=[\mu _{1},\dots ,\mu _{p}]^{T}}$ location (real ${\displaystyle p\times 1}$ vector) ${\displaystyle {\boldsymbol {\Sigma }}}$ scale matrix (positive-definite real ${\displaystyle p\times p}$ matrix) ${\displaystyle \nu >0}$ (real) represents the degrees of freedom
Support	${\displaystyle \mathbf {x} \in \mathbb {R} ^{p}\!}$
PDF	${\displaystyle {\frac {\Gamma \left[(\nu +p)/2\right]}{\Gamma (\nu /2)\nu ^{p/2}\pi ^{p/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}}\left[1+{\frac {1}{\nu }}({\mathbf {x} }-{\boldsymbol {\mu }})^{\rm {T}}{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right]^{-(\nu +p)/2}}$
CDF	No analytic expression, but see text for approximations
Mean	${\displaystyle {\boldsymbol {\mu }}}$ if ${\displaystyle \nu >1}$; else undefined
Median	${\displaystyle {\boldsymbol {\mu }}}$
Mode	${\displaystyle {\boldsymbol {\mu }}}$
Variance	${\displaystyle {\frac {\nu }{\nu -2}}{\boldsymbol {\Sigma }}}$ if ${\displaystyle \nu >2}$; 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 ${\displaystyle p}$ dimensions, is based on the observation that if ${\displaystyle \mathbf {y} }$ and ${\displaystyle u}$ are independent and distributed as ${\displaystyle N({\mathbf {0} },{\boldsymbol {\Sigma }})}$ and ${\displaystyle \chi _{\nu }^{2}}$ (i.e. multivariate normal and chi-squared distributions) respectively, the matrix ${\displaystyle \mathbf {\Sigma } \,}$ is a p × p matrix, and ${\displaystyle {\boldsymbol {\mu }}}$ is a constant vector then the random variable ${\textstyle {\mathbf {x} }={\mathbf {y} }/{\sqrt {u/\nu }}+{\boldsymbol {\mu }}}$ has the density^[1]

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

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

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

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

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

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

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

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

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

which is the standard but not the only choice.

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

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

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

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

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

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 ${\displaystyle \Sigma }$ 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 ${\displaystyle \mathbf {x} }$ is a real vector):

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

There is no simple formula for ${\displaystyle F(\mathbf {x} )}$, but it can be approximated numerically via Monte Carlo integration.^[2][3][4]

Conditional Distribution

This was demonstrated by Muirhead ^[5] though previously derived using the simpler ratio representation above, by Cornish.^[6] Let vector ${\displaystyle X}$ follow the multivariate t distribution and partition into two subvectors of ${\displaystyle p_{1},p_{2}}$ elements:

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

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

Then

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

where

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

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

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

See ^[7] 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.^[8]

Elliptical Representation

Constructed as an elliptical distribution,^[9] take the simplest centralised case with spherical symmetry and no scaling, ${\displaystyle \Sigma =\operatorname {I} \,}$, then the multivariate t-PDF takes the form

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

where ${\displaystyle X=(x_{1},\cdots ,x_{p})^{T}{\text{ is a }}p{\text{-vector}}}$ and ${\displaystyle \nu }$ = degrees of freedom as defined in Muirhead section 1.5. The covariance of ${\displaystyle X}$ is

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

The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder,^[10] in a tutorial-style paper, define radial measure ${\displaystyle r_{2}=R^{2}={\frac {X^{T}X}{p}}}$ and, noting that the density is dependent only on r₂, we get

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

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

Linear Combinations and Affine Transformation

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: ${\displaystyle f_{X}(X)={\frac {\mathrm {K} }{\left|\Sigma \right|^{1/2}}}\left(1+\nu ^{-1}X^{T}\Sigma ^{-1}X\right)^{-\left(\nu +p\right)/2}}$, where ${\displaystyle \mathrm {K} }$ is a constant and ${\displaystyle \nu }$ is arbitrary but fixed, let ${\displaystyle \Theta \in \mathbb {R} ^{p\times p}}$ be a non-singular matrix and form vector ${\displaystyle Y=\Theta X}$. Then, by a straightforward change of variables we get

${\displaystyle f_{Y}(Y)={\frac {\mathrm {K} }{\left|\Sigma \right|^{1/2}}}\left(1+\nu ^{-1}Y^{T}\Theta ^{-T}\Sigma ^{-1}\Theta ^{-1}Y\right)^{-\left(\nu +p\right)/2}\left|{\frac {\partial Y}{\partial X}}\right|^{-1}}$

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

${\displaystyle f_{Y}(Y)={\frac {\mathrm {K} }{\left|\Sigma \right|^{1/2}\left|\Theta \right|}}\left(1+\nu ^{-1}Y^{T}\Theta ^{-T}\Sigma ^{-1}\Theta ^{-1}Y\right)^{-\left(\nu +p\right)/2}}$

The denominator reduces to

${\displaystyle \left|\Sigma \right|^{1/2}\left|\Theta \right|=\left|\Sigma \right|^{1/2}\left|\Theta \cdot \Theta \right|^{1/2}=\left|\Theta \Sigma \Theta ^{T}\right|^{1/2}}$

where ${\displaystyle \left|\Theta ^{T}\right|=\left|\Theta \right|}$. Finally

${\displaystyle f_{Y}(Y)={\frac {\mathrm {K} }{\left|\Theta \Sigma \Theta ^{T}\right|^{1/2}}}\left(1+\nu ^{-1}Y^{T}\Theta ^{-T}\Sigma ^{-1}\Theta ^{-1}Y\right)^{-\left(\nu +p\right)/2}}$

which is a regular MV-t distribution.

In general if ${\displaystyle X\sim t_{\nu }(\mu ,\Sigma )}$ then ${\displaystyle \Theta X+c\sim t_{\nu }(\Theta \mu +c,\Theta \Sigma \Theta ^{T})}$. Roth shows that the transformation remains valid if ${\displaystyle \Theta }$ is a rectangular matrix ${\displaystyle \Theta \in \mathbb {R} ^{m\times p},m<p}$ which results in dimensionality reduction. The Jacobian ${\displaystyle \left|\Theta \right|}$ is seemingly rectangular but the value ${\displaystyle \left|\Theta \Sigma \Theta ^{T}\right|^{1/2}}$ in the denominator pdf is nevertheless correct and there is a discussion of rectangular matrix product determinants in Aitken.^[11] In extremis, if m = 1 and ${\displaystyle \Theta }$ becomes a row vector, then Y follows a univariate Student-t distribution with ${\displaystyle \nu }$ 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 ${\displaystyle Z}$ 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 ${\displaystyle \nu }$ values: ${\textstyle {1}/{\sqrt {u_{1}/\nu _{1}}},\;\;{1}/{\sqrt {u_{2}/\nu _{2}}}}$ will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.^[12] Taleb compares many examples of elliptical vs non-elliptical multivariate distributions

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 ${\displaystyle \nu \uparrow \infty }$.
• 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

References

Literature

External links

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