Notation | |||
---|---|---|---|
Parameters | location (real vector) scale matrix (positive-definite real matrix) (real) represents the degrees of freedom | ||
Support | |||
CDF | No analytic expression, but see text for approximations | ||
Mean | if ; else undefined | ||
Median | |||
Mode | |||
Variance | if ; 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.
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:
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.
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]
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):
There is no simple formula for , but it can be approximated numerically via Monte Carlo integration. [3] [4] [5]
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.
Forming first an intermediate distribution , the explicit conditional distribution renders as:
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, , 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.
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 .
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 ,
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
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.
This article includes a list of general references, but it lacks sufficient corresponding inline citations .(May 2012) |
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