Multivariate t-distribution

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Multivariate t
Parameters location (real vector)
scale matrix (positive-definite real matrix)
is the degrees of freedom
CDF No analytic expression, but see text for approximations
Mean if ; else undefined
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:

  1. Generate and , independently.
  2. 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.


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 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 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 do not agree with statistical independence.

Cumulative distribution function[edit]

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.[2][3]

Conditional Distribution[edit]

This was demonstrated by Muirhead [4] though previously derived using the simpler ratio representation above, by Cornish.[5] Let vector follow the multivariate t distribution and partition into two subvectors of elements:

where , the known mean vector is and the scale matrix is .



is the conditional mean where it exists or median otherwise.
is the Schur complement of
is the squared Mahalanobis distance of from with scale matrix

See [6] for a simple proof of the above conditional distribution.

Copulas based on the multivariate t[edit]

The use of such distributions[7] is enjoying renewed interest due to applications in mathematical finance, especially through the use of the Student's t copula.[citation needed]

Elliptical Representation[edit]

Constructed as an elliptical distribution[8] and in the simplest centralised case with spherical symmetry and without scaling, , the multivariate t PDF takes the form

where and = degrees of freedom. Muirhead (section 1.5) refers to this as a multivariate Cauchy distribution. The expected covariance of is

The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder,[9] in a tutorial-style paper, define radial measure such that

which is equivalent to the expected variance of -element vector treated as a univariate zero-mean random sequence. They note that follows the Fisher-Snedecor or distribution:

having mean value .

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 . The cumulative distribution function of is thus known to be

where is the incomplete Beta function.

These results can be derived by straightforward transformation of coordinates from cartesian to spherical. A constant radius surface at with PDF is an iso-density surface. The quantum of probability in a surface shell of area and thickness at is .

The enclosed sphere in dimensions has surface area and substitution into shows that the shell has element of probability . This is equivalent to a radial density function

which 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 can be derived from the Beta Prime distribution. If then , a known result. Thus, for variable , proportional to , we have

The moments of are

while introducing the scale matrix yields

Moments relating to radial variable are found by setting and whereupon

Linear Combinations and Affine Transformation[edit]

Following section 3.3 of Kibria let be a -vector sampled from a central spherical multivariate t distribution with degrees of freedom: . is derived from via a linear transformation:

where has full rank, then

That is and the covariance of is

Furthermore, if is a non-singular matrix then

with mean and covariance .

Roth (reference below) notes that if is a squat matrix with then has distribution .

If takes the form then the PDF of is the marginal distribution of the leading elements of .

In the above, the degrees of freedom parameter remains invariant throughout and all vectors must ultimately derive from one initial isotropic spherical vector whose elements are not statistically independent. Adding two sample multivariate t vectors generated with independent Chi-squared samples and different values: , as defined in the leading paragraph, will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.[10]

Related concepts[edit]

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[edit]


  1. ^ 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.
  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.
  3. ^ 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.
  4. ^ Muirhead, Robb (1982). Aspects of Multivariate Statistical Theory. USA: Wiley. pp. 32-36 Theorem 1.5.4. ISBN 978-0-47 1-76985-9.
  5. ^ 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.
  6. ^ 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.
  7. ^ Demarta, Stefano; McNeil, Alexander (2004). "The t Copula and Related Copulas" (PDF). Risknet.
  8. ^ Osiewalski, Jacek; Steele, Mark (1996). Bayesian Analysis in Statistics and Econometrics Ch(27): Posterior Moments of Scale Parameters in Elliptical Sampling Models. Wiley. pp. 323–335. ISBN 0-471-11856-7.
  9. ^ 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.
  10. ^ 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.


  • 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.

External links[edit]