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]

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

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

which is the standard but not the only choice.
An important special case is the standard bivariate t-distribution, p = 2:

Note that
.
Now, if
is the identity matrix, the density is

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]
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:

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.

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
above then, using the parameters below, the explicit conditional distribution becomes

where
Effective degrees of freedom,
is augmented by the number of disused variables
.
is the conditional mean of 
is the Schur complement of
.
is the squared Mahalanobis distance of
from
with scale matrix 
is the conditional scale matrix for
.
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

where
and
= degrees of freedom as defined in Muirhead [6] section 1.5. The covariance of
is

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

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:

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

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:

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

which further simplifies to
where
is the Beta function.
Changing the radial variable to
returns the previous Beta Prime distribution

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

or, in terms of scalar radial variable
,

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

The moments of
are

while introducing the scale matrix
yields

Moments relating to radial variable
are found by setting
and
whereupon

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

The matrix of partial derivatives is
and the Jacobian becomes
. Thus

The denominator reduces to

In full:

which is a regular MV-t distribution.
In general if
and
has full rank
then

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:

with
, means
, scale matrix 
then
,
such that


If a transformation is constructed in the form

then vector
, as discussed below, has the same distribution as the marginal distribution of
.
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


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