In
statistics, the Wishart distribution is a generalization of the
gamma distribution to multiple dimensions. It is named in honor of
John Wishart, who first formulated the distribution in 1928.[1] Other names include Wishart ensemble (in
random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve
Laguerre polynomials), or LOE, LUE, LSE (in analogy with
GOE, GUE, GSE).[2]
known as the
scatter matrix. One indicates that S has that probability distribution by writing
The positive integer n is the number of degrees of freedom. Sometimes this is written W(V, p, n). For n ≥ p the matrix S is invertible with probability 1 if V is invertible.
The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from a
multivariate normal distribution. It occurs frequently in
likelihood-ratio tests in multivariate statistical analysis. It also arises in the spectral theory of
random matrices[citation needed] and in multidimensional Bayesian analysis.[5] It is also encountered in wireless communications, while analyzing the performance of
Rayleigh fadingMIMO wireless channels .[6]
Probability density function
Spectral density of Wishart-Laguerre ensemble with dimensions (8, 15). A reconstruction of Figure 1 of [7].
Let X be a p × p symmetric matrix of random variables that is
positive semi-definite. Let V be a (fixed) symmetric positive definite matrix of size p × p.
Then, if n ≥ p, X has a Wishart distribution with n degrees of freedom if it has the
probability density function
The density above is not the joint density of all the elements of the random matrix X (such -dimensional density does not exist because of the symmetry constrains ), it is rather the joint density of elements for (,[1] page 38). Also, the density formula above applies only to positive definite matrices for other matrices the density is equal to zero.
Spectral density
The joint-eigenvalue density for the eigenvalues of a random matrix is,[8][9]
where is a constant.
In fact the above definition can be extended to any real n > p − 1. If n ≤ p − 1, then the Wishart no longer has a density—instead it represents a singular distribution that takes values in a lower-dimension subspace of the space of p × p matrices.[10]
where E[⋅] denotes expectation. (Here Θ is any matrix with the same dimensions as V, 1 indicates the identity matrix, and i is a square root of −1).[9] Properly interpreting this formula requires a little care, because noninteger complex powers are
multivalued; when n is noninteger, the correct branch must be determined via
analytic continuation.[14]
Theorem
If a p × p random matrix X has a Wishart distribution with m degrees of freedom and variance matrix V — write — and C is a q × p matrix of
rankq, then [15]
Corollary 1
If z is a nonzero p × 1 constant vector, then:[15]
In this case, is the
chi-squared distribution and (note that is a constant; it is positive because V is positive definite).
Corollary 2
Consider the case where zT = (0, ..., 0, 1, 0, ..., 0) (that is, the j-th element is one and all others zero). Then corollary 1 above shows that
gives the marginal distribution of each of the elements on the matrix's diagonal.
George Seber points out that the Wishart distribution is not called the “multivariate chi-squared distribution” because the marginal distribution of the
off-diagonal elements is not chi-squared. Seber prefers to reserve the term
multivariate for the case when all univariate marginals belong to the same family.[16]
where and nij ~ N(0, 1) independently.[18] This provides a useful method for obtaining random samples from a Wishart distribution.[19]
Marginal distribution of matrix elements
Let V be a 2 × 2 variance matrix characterized by
correlation coefficient−1 < ρ < 1 and L its lower Cholesky factor:
Multiplying through the Bartlett decomposition above, we find that a random sample from the 2 × 2 Wishart distribution is
The diagonal elements, most evidently in the first element, follow the χ2 distribution with n degrees of freedom (scaled by σ2) as expected. The off-diagonal element is less familiar but can be identified as a
normal variance-mean mixture where the mixing density is a χ2 distribution. The corresponding marginal probability density for the off-diagonal element is therefore the
variance-gamma distribution
where Kν(z) is the
modified Bessel function of the second kind.[20] Similar results may be found for higher dimensions. In general, if follows a Wishart distribution with parameters, , then for , the off-diagonal elements
It is also possible to write down the
moment-generating function even in the noncentral case (essentially the nth power of Craig (1936)[22] equation 10) although the probability density becomes an infinite sum of Bessel functions.
The range of the shape parameter
It can be shown [23] that the Wishart distribution can be defined if and only if the shape parameter n belongs to the set
This set is named after Gindikin, who introduced it[24] in the 1970s in the context of gamma distributions on homogeneous cones. However, for the new parameters in the discrete spectrum of the Gindikin ensemble, namely,
the corresponding Wishart distribution has no Lebesgue density.
Relationships to other distributions
The Wishart distribution is related to the
inverse-Wishart distribution, denoted by , as follows: If X ~ Wp(V, n) and if we do the change of variables C = X−1, then . This relationship may be derived by noting that the absolute value of the
Jacobian determinant of this change of variables is |C|p+1, see for example equation (15.15) in.[25]
^Livan, Giacomo; Vivo, Pierpaolo (2011). "Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities". Acta Physica Polonica B. 42 (5): 1081.
arXiv:1103.2638.
doi:
10.5506/APhysPolB.42.1081.
ISSN0587-4254.
S2CID119599157.
^Muirhead, Robb J. (2005). Aspects of Multivariate Statistical Theory (2nd ed.). Wiley Interscience.
ISBN0471769851.