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Multivariate t-distribution

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Multivariate t
Parameters location (real vector)
scale matrix (positive-definite real matrix)
(real) represents 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 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]

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.[3][4][5]

Conditional Distribution[edit]

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:


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

Copulas based on the multivariate t[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

Linear Combinations and Affine Transformation[edit]

Full Rank Transform[edit]

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[edit]

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 .

Rank-Reducing Linear Transform[edit]

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

Related concepts[edit]

  • 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[edit]


  1. ^ 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.
  2. ^ 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.
  3. ^ 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.
  4. ^ 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.
  5. ^ 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.
  6. ^ 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.
  7. ^ 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.
  8. ^ 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.
  9. ^ Demarta, Stefano; McNeil, Alexander (2004). "The t Copula and Related Copulas" (PDF). Risknet.
  10. ^ 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.
  11. ^ 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.
  12. ^ Aitken, A C - (1948). Determinants and Matrices (5th ed.). Edinburgh: Oliver and Boyd. pp. Chapter IV, section 36.
  13. ^ 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.
  14. ^ Okhrin, Y; Schmid, W (2006). "Distributional Properties of Portfolio Weights". Journal of Econometrics. 134: 235–256.
  15. ^ 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.


  • 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[edit]