One common method of construction of a multivariate t-distribution, for the case of
The constructive definition of a multivariate t-distribution simultaneously serves as a sampling algorithm: This formulation gives rise to the hierarchical representation of a multivariate t-distribution as a scale-mixture of normals:
There are in fact many candidates for the multivariate generalization of Student's t-distribution.
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.
This is the basic idea of elliptical distribution theory, where one writes down a corresponding function of
It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom
An important special case is the standard bivariate t-distribution, p = 2: Note that
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.
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.
, but it can be approximated numerically via Monte Carlo integration.
[7] but later derived using the simpler chi-squared ratio representation above, by Roth[1] and Ding.
Thus the conditional distribution is most easily represented as a two-step procedure.
above then, using the parameters below, the explicit conditional distribution becomes where 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] Constructed as an elliptical distribution,[10] take the simplest centralised case with spherical symmetry and no scaling,
= degrees of freedom as defined in Muirhead[6] section 1.5.
Kibria and Joarder,[11] define radial measure
treated as a univariate heavy-tail zero-mean random sequence with uncorrelated, yet statistically dependent, elements.
, 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.
Given this density value, the quantum of probability on a shell of surface area
returns the previous Beta Prime distribution To scale the radial variables without changing the radial shape function, define scale matrix
yields Moments relating to radial variable
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:
Then, by straightforward change of variables The matrix of partial derivatives is
Thus The denominator reduces to In full: which is a regular MV-t distribution.
There is a discussion of rectangular matrix product determinants in Aitken.
becomes a row vector, then scalar Y follows a univariate double-sided Student-t distribution defined by
al. use the affine transformation to find the marginal distributions which are also MV-t.