Multivariable generalization of the Student's t-distribution
Multivariate tNotation | ![{\displaystyle t_{\nu }({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d87588a7e2efded9fe566b64d76ea3bc5b60c45d) |
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Parameters | location (real vector)
scale matrix (positive-definite real matrix) (real) represents the degrees of freedom |
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Support | ![{\displaystyle \mathbf {x} \in \mathbb {R} ^{p}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab5c7c27cb906341f04b8838e0ff485d60b1160) |
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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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8f5dfeaf2441f91617b39b38762315c104ae6f) |
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CDF | No analytic expression, but see text for approximations |
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Mean | if ; else undefined |
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Median | ![{\displaystyle {\boldsymbol {\mu }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1aee7d7b4a36d96dfb35bfee9c7751bba1fdfbe) |
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Mode | ![{\displaystyle {\boldsymbol {\mu }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1aee7d7b4a36d96dfb35bfee9c7751bba1fdfbe) |
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Variance | if ; else undefined |
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Skewness | 0 |
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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
dimensions, is based on the observation that if
and
are independent and distributed as
and
(i.e. multivariate normal and chi-squared distributions) respectively, the matrix
is a p × p matrix, and
is a constant vector then the random variable
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb1eb369a63b9f68ba2684a8d69515100650739)
and is said to be distributed as a multivariate t-distribution with parameters
. Note that
is not the covariance matrix since the covariance is given by
(for
).
The constructive definition of a multivariate t-distribution simultaneously serves as a sampling algorithm:
- Generate
and
, independently. - Compute
.
This formulation gives rise to the hierarchical representation of a multivariate t-distribution as a scale-mixture of normals:
where
indicates a gamma distribution with density proportional to
, and
conditionally follows
.
In the special case
, 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 (
), with
and
, 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78275b166b591d1e5597153b1ef315677393849a)
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
variables
that replaces
by a quadratic function of all the
. It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom
. With
, 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f328519c729a4e62c5407af4398f87391387495)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bab4a8fd2a073b736eecc4ac34774d5b831f435e)
Note that
.
Now, if
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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8d1d36fac11f0835beefc063a2e4f19d7f022e)
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
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.[2]
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
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 }}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3c9514e416a503eb4d18f62a884e55c85f6a99)
There is no simple formula for
, but it can be approximated numerically via Monte Carlo integration.[3][4][5]
Conditional Distribution
This was developed by Muirhead [6] and Cornish.[7] but later derived using the simpler chi-squared ratio representation above, by Roth[1] and Ding.[8] Let vector
follow a multivariate t distribution and partition into two subvectors of
elements:
![{\displaystyle X_{p}={\begin{bmatrix}X_{1}\\X_{2}\end{bmatrix}}\sim t_{p}\left(\mu _{p},\Sigma _{p\times p},\nu \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe27322d9bd897c5b262e09224daa15eec65878c)
where
, the known mean vectors are
and the scale matrix is
.
Roth and Ding find the conditional distribution
to be a new t-distribution with modified parameters.
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9db6c78c15d95f27ef11cf738b4eee570d53992c)
An equivalent expression in Kotz et. al. is somewhat less concise.
Forming first an intermediate distribution
, the explicit conditional distribution renders as:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17ce496645b3409fc646a1a52c5f5e1df3c6bf74)
where
Effective degrees of freedom, augmented by the disused variables.
is the conditional mean of ![{\displaystyle x_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308)
is the Schur complement of
; the conditional covariance.
is the squared Mahalanobis distance of
from
with scale matrix ![{\displaystyle \Sigma _{22}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2e84d4f31847e7354a630b9d202284ad55c95d)
![{\displaystyle \Psi ={\frac {\nu +d_{2}}{\nu +p_{2}}}\Sigma _{11|2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38162820edab8a275dd24595edcace3fca0c4849)
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.[9]
Elliptical Representation
Constructed as an elliptical distribution,[10] take the simplest centralised case with spherical symmetry and no scaling,
, 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19b03a47a289a515a5ceb1dc1f75c9616de84330)
where
and
= degrees of freedom as defined in Muirhead[6] section 1.5. The covariance of
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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e0baf21d57bb53e694957b530bb71a6470cc70)
The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder,[11] define radial measure
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b0f1a7a55f2390e8869f23c325ff539e57b0d3)
which is equivalent to the variance of
-element vector
treated as a univariate heavy-tail zero-mean random sequence with uncorrelated, yet statistically dependent, elements.
Radial Distribution
follows the Fisher-Snedecor or
distribution:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e57f6129a544b152980805d5275d6d7c3fe7a73)
having mean value
.
-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
in the equation above, retaining
-vector
, we have
and probability distribution
![{\displaystyle {\begin{aligned}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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d61567aabc20a36dae361461ab61dfe8cb7b07a0)
which is a regular Beta-prime distribution
having mean value
.
Cumulative Radial Distribution
Given the Beta-prime distribution, the radial cumulative distribution function of
is known:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba258881682ca08aaea4943285e5dd082453d19d)
where
is the incomplete Beta function and applies with a spherical
assumption.
In the scalar case,
, the distribution is equivalent to Student-t with the equivalence
, 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
with PDF
is an iso-density surface. Given this density value, the quantum of probability on a shell of surface area
and thickness
at
is
.
The enclosed
-sphere of radius
has surface area
. Substitution into
shows that the shell has element of probability
which is equivalent to radial density function
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2117380e9621f5541adcc1318f7209eeefa03a)
which further simplifies to
where
is the Beta function.
Changing the radial variable to
returns the previous Beta Prime distribution
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5a65c80fbf85721e885dcf937dbbed3d2a0d22)
To scale the radial variables without changing the radial shape function, define scale matrix
, yielding a 3-parameter Cartesian density function, ie. the probability
in volume element
is
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a803c9d9ef97907476bcbbc75bc8cebd3d614e5d)
or, in terms of scalar radial variable
,
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75003a076d5d58463f763ecbd279fe9325eacdd6)
Radial Moments
The moments of all the radial variables , with the spherical distribution assumption, can be derived from the Beta Prime distribution. If
then
, a known result. Thus, for variable
we have
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21bdc327300e3d5cfe2052e17e27f0974808bd36)
The moments of
are
![{\displaystyle \operatorname {E} (r_{2}^{m})=\nu ^{m}\operatorname {E} (y^{m})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f58c02a6331af3caadba2d37602ba353eaedf77)
while introducing the scale matrix
yields
![{\displaystyle \operatorname {E} (r_{2}^{m}|\alpha )=\alpha ^{m}\nu ^{m}\operatorname {E} (y^{m})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64a4ed7b8c46438b2d023ef33aa6044b28be3872)
Moments relating to radial variable
are found by setting
and
whereupon
![{\displaystyle \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 )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df54e91098a7c2f91139291cc045ce4b45d99b5e)
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:
, where
is a constant and
is arbitrary but fixed, let
be a full-rank matrix and form vector
. Then, by straightforward change of variables
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8cc1ba740d4ecb99dc73f42ef491005c4a9667)
The matrix of partial derivatives is
and the Jacobian becomes
. 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5b9c0308f5e8b8dbe7d66764714d0c29e9db4a)
The denominator reduces to
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6be9803882f3abbe1f905f0b1e6449a8e152adad)
In full:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f800afa3cdce63b91e1575a777a98904085c1891)
which is a regular MV-t distribution.
In general if
and
has full rank
then
![{\displaystyle \Theta X+c\sim t_{p}(\Theta \mu +c,\Theta \Sigma \Theta ^{T},\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/108494a54f5cd16e4cae32a5e4196ff0f0e4d573)
Marginal Distributions
This is a special case of the rank-reducing linear transform below. Kotz defines marginal distributions as follows. Partition
into two subvectors of
elements:
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54057f47e726333db0c7b4578d5ce836a409b39d)
with
, means
, scale matrix
then
,
such that
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4834c6c3ea4b15a17f6f1c7d580aff33c508a08)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40b611bc1816b6e44735b762151d104395d1279f)
If a transformation is constructed in the form
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d37b93f519160c08d78894634d44bbb08a2aa67)
then vector
, as discussed below, has the same distribution as the marginal distribution of
.
Rank-Reducing Linear Transform
In the linear transform case, if
is a rectangular matrix
, of rank
the result is dimensionality reduction. Here, Jacobian
is seemingly rectangular but the value
in the denominator pdf is nevertheless correct. There is a discussion of rectangular matrix product determinants in Aitken.[12] In general if
and
has full rank
then
![{\displaystyle Y=\Theta X+c\sim t(m,\Theta \mu +c,\Theta \Sigma \Theta ^{T},\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2067a0e737203f50ba4706e15b3cbc772414a3d)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38b41b9198b8b10f7649a87c0631e6d43a61475b)
In extremis, if m = 1 and
becomes a row vector, then scalar Y follows a univariate double-sided Student-t distribution defined by
with the same
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
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
values:
will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.[13] - 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.[14][15][2] 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
References
- ^ a b Roth, Michael (17 April 2013). "On the Multivariate t Distribution" (PDF). Automatic Control group. Linköpin University, Sweden. Archived (PDF) from the original on 31 July 2022. Retrieved 1 June 2022.
- ^ a b Bodnar, T; Okhrin, Y (2008). "Properties of the Singular, Inverse and Generalized inverse Partitioned Wishart Distribution" (PDF). Journal of Multivariate Analysis. 99 (Eqn.20): 2389–2405.
- ^ Botev, Z.; Chen, Y.-L. (2022). "Chapter 4: Truncated Multivariate Student Computations via Exponential Tilting.". In Botev, Zdravko; Keller, Alexander; Lemieux, Christiane; Tuffin, Bruno (eds.). Advances in Modeling and Simulation: Festschrift for Pierre L'Ecuyer. Springer. pp. 65–87. ISBN 978-3-031-10192-2.
- ^ Botev, Z. I.; L'Ecuyer, P. (6 December 2015). "Efficient probability estimation and simulation of the truncated multivariate student-t distribution". 2015 Winter Simulation Conference (WSC). Huntington Beach, CA, USA: IEEE. pp. 380–391. doi:10.1109/WSC.2015.7408180.
- ^ Genz, Alan (2009). Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics. Vol. 195. Springer. doi:10.1007/978-3-642-01689-9. ISBN 978-3-642-01689-9. Archived from the original on 2022-08-27. Retrieved 2017-09-05.
- ^ a b Muirhead, Robb (1982). Aspects of Multivariate Statistical Theory. USA: Wiley. pp. 32–36 Theorem 1.5.4. ISBN 978-0-47 1-76985-9.
- ^ Cornish, E A (1954). "The Multivariate t-Distribution Associated with a Set of Normal Sample Deviates". Australian Journal of Physics. 7: 531–542. doi:10.1071/PH550193.
- ^ Ding, Peng (2016). "On the Conditional Distribution of the Multivariate t Distribution". The American Statistician. 70 (3): 293–295. arXiv:1604.00561. doi:10.1080/00031305.2016.1164756. S2CID 55842994.
- ^ Demarta, Stefano; McNeil, Alexander (2004). "The t Copula and Related Copulas" (PDF). Risknet.
- ^ Osiewalski, Jacek; Steele, Mark (1996). "Posterior Moments of Scale Parameters in Elliptical Sampling Models". Bayesian Analysis in Statistics and Econometrics. Wiley. pp. 323–335. ISBN 0-471-11856-7.
- ^ Kibria, K M G; Joarder, A H (Jan 2006). "A short review of multivariate t distribution" (PDF). Journal of Statistical Research. 40 (1): 59–72. doi:10.1007/s42979-021-00503-0. S2CID 232163198.
- ^ Aitken, A C - (1948). Determinants and Matrices (5th ed.). Edinburgh: Oliver and Boyd. pp. Chapter IV, section 36.
- ^ Giron, Javier; del Castilo, Carmen (2010). "The multivariate Behrens–Fisher distribution". Journal of Multivariate Analysis. 101 (9): 2091–2102. doi:10.1016/j.jmva.2010.04.008.
- ^ Okhrin, Y; Schmid, W (2006). "Distributional Properties of Portfolio Weights". Journal of Econometrics. 134: 235–256.
- ^ 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.
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
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