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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Generalization of gamma distribution to multiple dimensions
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]
It is a family of probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix -valued random variables ). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics . In Bayesian statistics , the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector .[3]
Definition
[ edit ]
Suppose G is a p × n matrix, each column of which is independently drawn from a p -variate normal distribution with zero mean:
G
=
(
g
i
1
,
…
,
g
i
n
)
∼
N
p
(
0
,
V
)
.
{\displaystyle G=(g_{i}^{1},\dots ,g_{i}^{n})\sim {\mathcal {N}}_{p}(0,V).}
Then the Wishart distribution is the probability distribution of the p × p random matrix [4]
S
=
G
G
T
=
∑
i
=
1
n
g
i
g
i
T
{\displaystyle S=GG^{T}=\sum _{i=1}^{n}g_{i}g_{i}^{T}}
known as the scatter matrix . One indicates that S has that probability distribution by writing
S
∼
W
p
(
V
,
n
)
.
{\displaystyle S\sim W_{p}(V,n).}
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.
If p = V = 1 then this distribution is a chi-squared distribution with n degrees of freedom.
Occurrence
[ edit ]
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 fading MIMO wireless channels .[6]
Probability density function
[ edit ]
Spectral density of Wishart-Laguerre ensemble with dimensions (8, 15). A reconstruction of Figure 1 of [7] .
The Wishart distribution can be characterized by its probability density function as follows:
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
f
X
(
X
)
=
1
2
n
p
/
2
|
V
|
n
/
2
Γ
p
(
n
2
)
|
X
|
(
n
−
p
−
1
)
/
2
e
−
1
2
tr
(
V
−
1
X
)
{\displaystyle f_{\mathbf {X} }(\mathbf {X} )={\frac {1}{2^{np/2}\left|{\mathbf {V} }\right|^{n/2}\Gamma _{p}\left({\frac {n}{2}}\right)}}{\left|\mathbf {X} \right|}^{(n-p-1)/2}e^{-{\frac {1}{2}}\operatorname {tr} ({\mathbf {V} }^{-1}\mathbf {X} )}}
where
|
X
|
{\displaystyle \left|{\mathbf {X} }\right|}
is the determinant of
X
{\displaystyle \mathbf {X} }
and Γp is the multivariate gamma function defined as
Γ
p
(
n
2
)
=
π
p
(
p
−
1
)
/
4
∏
j
=
1
p
Γ
(
n
2
−
j
−
1
2
)
.
{\displaystyle \Gamma _{p}\left({\frac {n}{2}}\right)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma \left({\frac {n}{2}}-{\frac {j-1}{2}}\right).}
The density above is not the joint density of all the
p
2
{\displaystyle p^{2}}
elements of the random matrix X (such
p
2
{\displaystyle p^{2}}
-dimensional density does not exist because of the symmetry constrains
X
i
j
=
X
j
i
{\displaystyle X_{ij}=X_{ji}}
), it is rather the joint density of
p
(
p
+
1
)
/
2
{\displaystyle p(p+1)/2}
elements
X
i
j
{\displaystyle X_{ij}}
for
i
≤
j
{\displaystyle i\leq j}
(,[1] page 38). Also, the density formula above applies only to positive definite matrices
x
;
{\displaystyle \mathbf {x} ;}
for other matrices the density is equal to zero.
Spectral density
[ edit ]
The joint-eigenvalue density for the eigenvalues
λ
1
,
…
,
λ
p
≥
0
{\displaystyle \lambda _{1},\dots ,\lambda _{p}\geq 0}
of a random matrix
X
∼
W
p
(
I
,
n
)
{\displaystyle \mathbf {X} \sim W_{p}(\mathbf {I} ,n)}
is,[8] [9]
c
n
,
p
e
−
1
2
∑
i
λ
i
∏
λ
i
(
n
−
p
−
1
)
/
2
∏
i
<
j
|
λ
i
−
λ
j
|
{\displaystyle c_{n,p}e^{-{\frac {1}{2}}\sum _{i}\lambda _{i}}\prod \lambda _{i}^{(n-p-1)/2}\prod _{i<j}|\lambda _{i}-\lambda _{j}|}
where
c
n
,
p
{\displaystyle c_{n,p}}
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]
Use in Bayesian statistics
[ edit ]
In Bayesian statistics , in the context of the multivariate normal distribution , the Wishart distribution is the conjugate prior to the precision matrix Ω = Σ −1 , where Σ is the covariance matrix.[11] : 135 [12]
Choice of parameters
[ edit ]
The least informative, proper Wishart prior is obtained by setting n = p .[citation needed ]
The prior mean of W p (V , n )is n V , suggesting that a reasonable choice for V would be n −1 Σ 0 −1 , where Σ 0 is some prior guess for the covariance matrix.
Properties
[ edit ]
Log-expectation
[ edit ]
The following formula plays a role in variational Bayes derivations for Bayes networks
involving the Wishart distribution. From equation (2.63),[13]
E
[
ln
|
X
|
]
=
ψ
p
(
n
2
)
+
p
ln
(
2
)
+
ln
|
V
|
{\displaystyle \operatorname {E} [\,\ln \left|\mathbf {X} \right|\,]=\psi _{p}\left({\frac {n}{2}}\right)+p\,\ln(2 )+\ln |\mathbf {V} |}
where
ψ
p
{\displaystyle \psi _{p}}
is the multivariate digamma function (the derivative of the log of the multivariate gamma function ).
Log-variance
[ edit ]
The following variance computation could be of help in Bayesian statistics:
Var
[
ln
|
X
|
]
=
∑
i
=
1
p
ψ
1
(
n
+
1
−
i
2
)
{\displaystyle \operatorname {Var} \left[\,\ln \left|\mathbf {X} \right|\,\right]=\sum _{i=1}^{p}\psi _{1}\left({\frac {n+1-i}{2}}\right)}
where
ψ
1
{\displaystyle \psi _{1}}
is the trigamma function. This comes up when computing the Fisher information of the Wishart random variable.
Entropy
[ edit ]
The information entropy of the distribution has the following formula:[11] : 693
H
[
X
]
=
−
ln
(
B
(
V
,
n
)
)
−
n
−
p
−
1
2
E
[
ln
|
X
|
]
+
n
p
2
{\displaystyle \operatorname {H} \left[\,\mathbf {X} \,\right]=-\ln \left(B(\mathbf {V} ,n)\right)-{\frac {n-p-1}{2}}\operatorname {E} \left[\,\ln \left|\mathbf {X} \right|\,\right]+{\frac {np}{2}}}
where B (V , n ) is the normalizing constant of the distribution:
B
(
V
,
n
)
=
1
|
V
|
n
/
2
2
n
p
/
2
Γ
p
(
n
2
)
.
{\displaystyle B(\mathbf {V} ,n)={\frac {1}{\left|\mathbf {V} \right|^{n/2}2^{np/2}\Gamma _{p}\left({\frac {n}{2}}\right)}}.}
This can be expanded as follows:
H
[
X
]
=
n
2
ln
|
V
|
+
n
p
2
ln
2
+
ln
Γ
p
(
n
2
)
−
n
−
p
−
1
2
E
[
ln
|
X
|
]
+
n
p
2
=
n
2
ln
|
V
|
+
n
p
2
ln
2
+
ln
Γ
p
(
n
2
)
−
n
−
p
−
1
2
(
ψ
p
(
n
2
)
+
p
ln
2
+
ln
|
V
|
)
+
n
p
2
=
n
2
ln
|
V
|
+
n
p
2
ln
2
+
ln
Γ
p
(
n
2
)
−
n
−
p
−
1
2
ψ
p
(
n
2
)
−
n
−
p
−
1
2
(
p
ln
2
+
ln
|
V
|
)
+
n
p
2
=
p
+
1
2
ln
|
V
|
+
1
2
p
(
p
+
1
)
ln
2
+
ln
Γ
p
(
n
2
)
−
n
−
p
−
1
2
ψ
p
(
n
2
)
+
n
p
2
{\displaystyle {\begin{aligned}\operatorname {H} \left[\,\mathbf {X} \,\right]&={\frac {n}{2}}\ln \left|\mathbf {V} \right|+{\frac {np}{2}}\ln 2+\ln \Gamma _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\operatorname {E} \left[\,\ln \left|\mathbf {X} \right|\,\right]+{\frac {np}{2}}\\[8pt]&={\frac {n}{2}}\ln \left|\mathbf {V} \right|+{\frac {np}{2}}\ln 2+\ln \Gamma _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\left(\psi _{p}\left({\frac {n}{2}}\right)+p\ln 2+\ln \left|\mathbf {V} \right|\right)+{\frac {np}{2}}\\[8pt]&={\frac {n}{2}}\ln \left|\mathbf {V} \right|+{\frac {np}{2}}\ln 2+\ln \Gamma _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\psi _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\left(p\ln 2+\ln \left|\mathbf {V} \right|\right)+{\frac {np}{2}}\\[8pt]&={\frac {p+1}{2}}\ln \left|\mathbf {V} \right|+{\frac {1}{2}}p(p+1)\ln 2+\ln \Gamma _{p}\left({\frac {n}{2}}\right)-{\frac {n-p-1}{2}}\psi _{p}\left({\frac {n}{2}}\right)+{\frac {np}{2}}\end{aligned}}}
Cross-entropy
[ edit ]
The cross-entropy of two Wishart distributions
p
0
{\displaystyle p_{0}}
with parameters
n
0
,
V
0
{\displaystyle n_{0},V_{0}}
and
p
1
{\displaystyle p_{1}}
with parameters
n
1
,
V
1
{\displaystyle n_{1},V_{1}}
is
H
(
p
0
,
p
1
)
=
E
p
0
[
−
log
p
1
]
=
E
p
0
[
−
log
|
X
|
(
n
1
−
p
1
−
1
)
/
2
e
−
tr
(
V
1
−
1
X
)
/
2
2
n
1
p
1
/
2
|
V
1
|
n
1
/
2
Γ
p
1
(
n
1
2
)
]
=
n
1
p
1
2
log
2
+
n
1
2
log
|
V
1
|
+
log
Γ
p
1
(
n
1
2
)
−
n
1
−
p
1
−
1
2
E
p
0
[
log
|
X
|
]
+
1
2
E
p
0
[
tr
(
V
1
−
1
X
)
]
=
n
1
p
1
2
log
2
+
n
1
2
log
|
V
1
|
+
log
Γ
p
1
(
n
1
2
)
−
n
1
−
p
1
−
1
2
(
ψ
p
0
(
n
0
2
)
+
p
0
log
2
+
log
|
V
0
|
)
+
1
2
tr
(
V
1
−
1
n
0
V
0
)
=
−
n
1
2
log
|
V
1
−
1
V
0
|
+
p
1
+
1
2
log
|
V
0
|
+
n
0
2
tr
(
V
1
−
1
V
0
)
+
log
Γ
p
1
(
n
1
2
)
−
n
1
−
p
1
−
1
2
ψ
p
0
(
n
0
2
)
+
n
1
(
p
1
−
p
0
)
+
p
0
(
p
1
+
1
)
2
log
2
{\displaystyle {\begin{aligned}H(p_{0},p_{1})&=\operatorname {E} _{p_{0}}[\,-\log p_{1}\,]\\[8pt]&=\operatorname {E} _{p_{0}}\left[\,-\log {\frac {\left|\mathbf {X} \right|^{(n_{1}-p_{1}-1)/2}e^{-\operatorname {tr} (\mathbf {V} _{1}^{-1}\mathbf {X} )/2}}{2^{n_{1}p_{1}/2}\left|\mathbf {V} _{1}\right|^{n_{1}/2}\Gamma _{p_{1}}\left({\tfrac {n_{1}}{2}}\right)}}\right]\\[8pt]&={\tfrac {n_{1}p_{1}}{2}}\log 2+{\tfrac {n_{1}}{2}}\log \left|\mathbf {V} _{1}\right|+\log \Gamma _{p_{1}}({\tfrac {n_{1}}{2}})-{\tfrac {n_{1}-p_{1}-1}{2}}\operatorname {E} _{p_{0}}\left[\,\log \left|\mathbf {X} \right|\,\right]+{\tfrac {1}{2}}\operatorname {E} _{p_{0}}\left[\,\operatorname {tr} \left(\,\mathbf {V} _{1}^{-1}\mathbf {X} \,\right)\,\right]\\[8pt]&={\tfrac {n_{1}p_{1}}{2}}\log 2+{\tfrac {n_{1}}{2}}\log \left|\mathbf {V} _{1}\right|+\log \Gamma _{p_{1}}({\tfrac {n_{1}}{2}})-{\tfrac {n_{1}-p_{1}-1}{2}}\left(\psi _{p_{0}}({\tfrac {n_{0}}{2}})+p_{0}\log 2+\log \left|\mathbf {V} _{0}\right|\right)+{\tfrac {1}{2}}\operatorname {tr} \left(\,\mathbf {V} _{1}^{-1}n_{0}\mathbf {V} _{0}\,\right)\\[8pt]&=-{\tfrac {n_{1}}{2}}\log \left|\,\mathbf {V} _{1}^{-1}\mathbf {V} _{0}\,\right|+{\tfrac {p_{1}+1}{2}}\log \left|\mathbf {V} _{0}\right|+{\tfrac {n_{0}}{2}}\operatorname {tr} \left(\,\mathbf {V} _{1}^{-1}\mathbf {V} _{0}\right)+\log \Gamma _{p_{1}}\left({\tfrac {n_{1}}{2}}\right)-{\tfrac {n_{1}-p_{1}-1}{2}}\psi _{p_{0}}({\tfrac {n_{0}}{2}})+{\tfrac {n_{1}(p_{1}-p_{0})+p_{0}(p_{1}+1)}{2}}\log 2\end{aligned}}}
Note that when
p
0
=
p
1
{\displaystyle p_{0}=p_{1}}
and
n
0
=
n
1
{\displaystyle n_{0}=n_{1}}
we recover the entropy.
KL-divergence
[ edit ]
The Kullback–Leibler divergence of
p
1
{\displaystyle p_{1}}
from
p
0
{\displaystyle p_{0}}
is
D
K
L
(
p
0
‖
p
1
)
=
H
(
p
0
,
p
1
)
−
H
(
p
0
)
=
−
n
1
2
log
|
V
1
−
1
V
0
|
+
n
0
2
(
tr
(
V
1
−
1
V
0
)
−
p
)
+
log
Γ
p
(
n
1
2
)
Γ
p
(
n
0
2
)
+
n
0
−
n
1
2
ψ
p
(
n
0
2
)
{\displaystyle {\begin{aligned}D_{KL}(p_{0}\|p_{1})&=H(p_{0},p_{1})-H(p_{0})\\[6pt]&=-{\frac {n_{1}}{2}}\log |\mathbf {V} _{1}^{-1}\mathbf {V} _{0}|+{\frac {n_{0}}{2}}(\operatorname {tr} (\mathbf {V} _{1}^{-1}\mathbf {V} _{0})-p)+\log {\frac {\Gamma _{p}\left({\frac {n_{1}}{2}}\right)}{\Gamma _{p}\left({\frac {n_{0}}{2}}\right)}}+{\tfrac {n_{0}-n_{1}}{2}}\psi _{p}\left({\frac {n_{0}}{2}}\right)\end{aligned}}}
Characteristic function
[ edit ]
The characteristic function of the Wishart distribution is
Θ
↦
E
[
exp
(
i
tr
(
X
Θ
)
)
]
=
|
1
−
2
i
Θ
V
|
−
n
/
2
{\displaystyle \Theta \mapsto \operatorname {E} \left[\,\exp \left(\,i\operatorname {tr} \left(\,\mathbf {X} {\mathbf {\Theta } }\,\right)\,\right)\,\right]=\left|\,1-2i\,{\mathbf {\Theta } }\,{\mathbf {V} }\,\right|^{-n/2}}
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
[ edit ]
If a p × p random matrix X has a Wishart distribution with m degrees of freedom and variance matrix V — write
X
∼
W
p
(
V
,
m
)
{\displaystyle \mathbf {X} \sim {\mathcal {W}}_{p}({\mathbf {V} },m)}
— and C is a q × p matrix of rank q , then [15]
C
X
C
T
∼
W
q
(
C
V
C
T
,
m
)
.
{\displaystyle \mathbf {C} \mathbf {X} {\mathbf {C} }^{T}\sim {\mathcal {W}}_{q}\left({\mathbf {C} }{\mathbf {V} }{\mathbf {C} }^{T},m\right).}
Corollary 1
[ edit ]
If z is a nonzero p × 1 constant vector, then:[15]
σ
z
−
2
z
T
X
z
∼
χ
m
2
.
{\displaystyle \sigma _{z}^{-2}\,{\mathbf {z} }^{T}\mathbf {X} {\mathbf {z} }\sim \chi _{m}^{2}.}
In this case,
χ
m
2
{\displaystyle \chi _{m}^{2}}
is the chi-squared distribution and
σ
z
2
=
z
T
V
z
{\displaystyle \sigma _{z}^{2}={\mathbf {z} }^{T}{\mathbf {V} }{\mathbf {z} }}
(note that
σ
z
2
{\displaystyle \sigma _{z}^{2}}
is a constant; it is positive because V is positive definite).
Corollary 2
[ edit ]
Consider the case where z T = (0, ..., 0, 1, 0, ..., 0) (that is, the j -th element is one and all others zero). Then corollary 1 above shows that
σ
j
j
−
1
w
j
j
∼
χ
m
2
{\displaystyle \sigma _{jj}^{-1}\,w_{jj}\sim \chi _{m}^{2}}
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]
Estimator of the multivariate normal distribution
[ edit ]
The Wishart distribution is the sampling distribution of the maximum-likelihood estimator (MLE) of the covariance matrix of a multivariate normal distribution .[17] A derivation of the MLE uses the spectral theorem .
Bartlett decomposition
[ edit ]
The Bartlett decomposition of a matrix X from a p -variate Wishart distribution with scale matrix V and n degrees of freedom is the factorization:
X
=
L
A
A
T
L
T
,
{\displaystyle \mathbf {X} ={\textbf {L}}{\textbf {A}}{\textbf {A}}^{T}{\textbf {L}}^{T},}
where L is the Cholesky factor of V , and:
A
=
(
c
1
0
0
⋯
0
n
21
c
2
0
⋯
0
n
31
n
32
c
3
⋯
0
⋮
⋮
⋮
⋱
⋮
n
p
1
n
p
2
n
p
3
⋯
c
p
)
{\displaystyle \mathbf {A} ={\begin{pmatrix}c_{1}&0&0&\cdots &0\\n_{21}&c_{2}&0&\cdots &0\\n_{31}&n_{32}&c_{3}&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\n_{p1}&n_{p2}&n_{p3}&\cdots &c_{p}\end{pmatrix}}}
where
c
i
2
∼
χ
n
−
i
+
1
2
{\displaystyle c_{i}^{2}\sim \chi _{n-i+1}^{2}}
and n ij ~ N (0, 1) independently.[18] This provides a useful method for obtaining random samples from a Wishart distribution.[19]
Marginal distribution of matrix elements
[ edit ]
Let V be a 2 × 2 variance matrix characterized by correlation coefficient −1 < ρ <1 and L its lower Cholesky factor:
V
=
(
σ
1
2
ρ
σ
1
σ
2
ρ
σ
1
σ
2
σ
2
2
)
,
L
=
(
σ
1
0
ρ
σ
2
1
−
ρ
2
σ
2
)
{\displaystyle \mathbf {V} ={\begin{pmatrix}\sigma _{1}^{2}&\rho \sigma _{1}\sigma _{2}\\\rho \sigma _{1}\sigma _{2}&\sigma _{2}^{2}\end{pmatrix}},\qquad \mathbf {L} ={\begin{pmatrix}\sigma _{1}&0\\\rho \sigma _{2}&{\sqrt {1-\rho ^{2}}}\sigma _{2}\end{pmatrix}}}
Multiplying through the Bartlett decomposition above, we find that a random sample from the 2 × 2 Wishart distribution is
X
=
(
σ
1
2
c
1
2
σ
1
σ
2
(
ρ
c
1
2
+
1
−
ρ
2
c
1
n
21
)
σ
1
σ
2
(
ρ
c
1
2
+
1
−
ρ
2
c
1
n
21
)
σ
2
2
(
(
1
−
ρ
2
)
c
2
2
+
(
1
−
ρ
2
n
21
+
ρ
c
1
)
2
)
)
{\displaystyle \mathbf {X} ={\begin{pmatrix}\sigma _{1}^{2}c_{1}^{2}&\sigma _{1}\sigma _{2}\left(\rho c_{1}^{2}+{\sqrt {1-\rho ^{2}}}c_{1}n_{21}\right)\\\sigma _{1}\sigma _{2}\left(\rho c_{1}^{2}+{\sqrt {1-\rho ^{2}}}c_{1}n_{21}\right)&\sigma _{2}^{2}\left(\left(1-\rho ^{2}\right)c_{2}^{2}+\left({\sqrt {1-\rho ^{2}}}n_{21}+\rho c_{1}\right)^{2}\right)\end{pmatrix}}}
The diagonal elements, most evidently in the first element, follow the χ 2 distribution with n degrees of freedom (scaled by σ 2 ) 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 χ 2 distribution. The corresponding marginal probability density for the off-diagonal element is therefore the variance-gamma distribution
f
(
x
12
)
=
|
x
12
|
n
−
1
2
Γ
(
n
2
)
2
n
−
1
π
(
1
−
ρ
2
)
(
σ
1
σ
2
)
n
+
1
⋅
K
n
−
1
2
(
|
x
12
|
σ
1
σ
2
(
1
−
ρ
2
)
)
exp
(
ρ
x
12
σ
1
σ
2
(
1
−
ρ
2
)
)
{\displaystyle f(x_{12})={\frac {\left|x_{12}\right|^{\frac {n-1}{2}}}{\Gamma \left({\frac {n}{2}}\right){\sqrt {2^{n-1}\pi \left(1-\rho ^{2}\right)\left(\sigma _{1}\sigma _{2}\right)^{n+1}}}}}\cdot K_{\frac {n-1}{2}}\left({\frac {\left|x_{12}\right|}{\sigma _{1}\sigma _{2}\left(1-\rho ^{2}\right)}}\right)\exp {\left({\frac {\rho x_{12}}{\sigma _{1}\sigma _{2}(1-\rho ^{2})}}\right)}}
where K ν (z ) is the modified Bessel function of the second kind .[20] Similar results may be found for higher dimensions. In general, if
X
{\displaystyle X}
follows a Wishart distribution with parameters,
Σ
,
n
{\displaystyle \Sigma ,n}
, then for
i
≠
j
{\displaystyle i\neq j}
, the off-diagonal elements
X
i
j
∼
VG
(
n
,
Σ
i
j
,
(
Σ
i
i
Σ
j
j
−
Σ
i
j
2
)
1
/
2
,
0
)
{\displaystyle X_{ij}\sim {\text{VG}}(n,\Sigma _{ij},(\Sigma _{ii}\Sigma _{jj}-\Sigma _{ij}^{2})^{1/2},0)}
. [21]
It is also possible to write down the moment-generating function even in the noncentral case (essentially the n th power of Craig (1936)[22] equation 10) although the probability density becomes an infinite sum of Bessel functions.
The range of the shape parameter
[ edit ]
It can be shown [23] that the Wishart distribution can be defined if and only if the shape parameter n belongs to the set
Λ
p
:=
{
0
,
…
,
p
−
1
}
∪
(
p
−
1
,
∞
)
.
{\displaystyle \Lambda _{p}:=\{0,\ldots ,p-1\}\cup \left(p-1,\infty \right).}
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,
Λ
p
∗
:=
{
0
,
…
,
p
−
1
}
,
{\displaystyle \Lambda _{p}^{*}:=\{0,\ldots ,p-1\},}
the corresponding Wishart distribution has no Lebesgue density.
Relationships to other distributions
[ edit ]
The Wishart distribution is related to the inverse-Wishart distribution , denoted by
W
p
−
1
{\displaystyle W_{p}^{-1}}
, as follows: If X ~ W p (V , n ) and if we do the change of variables C = X −1 , then
C
∼
W
p
−
1
(
V
−
1
,
n
)
{\displaystyle \mathbf {C} \sim W_{p}^{-1}(\mathbf {V} ^{-1},n)}
. 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]
In Bayesian statistics , the Wishart distribution is a conjugate prior for the precision parameter of the multivariate normal distribution , when the mean parameter is known.[11]
A generalization is the multivariate gamma distribution .
A different type of generalization is the normal-Wishart distribution , essentially the product of a multivariate normal distribution with a Wishart distribution.
See also
[ edit ]
Complex Wishart distribution
F-distribution
Gamma distribution
Hotelling's T-squared distribution
Inverse-Wishart distribution
Multivariate gamma distribution
Student's t-distribution
Wilks' lambda distribution
References
[ edit ]
^ Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (2018), Livan, Giacomo; Novaes, Marcel; Vivo, Pierpaolo (eds.), "Classical Ensembles: Wishart-Laguerre" , Introduction to Random Matrices: Theory and Practice , SpringerBriefs in Mathematical Physics, Cham: Springer International Publishing, pp. 89–95, doi :10.1007/978-3-319-70885-0_13 , ISBN 978-3-319-70885-0 , retrieved 2023-05-17
^ Koop, Gary; Korobilis, Dimitris (2010). "Bayesian Multivariate Time Series Methods for Empirical Macroeconomics" . Foundations and Trends in Econometrics . 3 (4 ): 267–358. doi :10.1561/0800000013 .
^ Gupta, A. K.; Nagar, D. K. (2000). Matrix Variate Distributions . Chapman & Hall /CRC. ISBN 1584880465 .
^ Gelman, Andrew (2003). Bayesian Data Analysis (2nd ed.). Boca Raton, Fla.: Chapman & Hall. p. 582. ISBN 158488388X . Retrieved 3 June 2015 .
^ Zanella, A.; Chiani, M.; Win, M.Z. (April 2009). "On the marginal distribution of the eigenvalues of wishart matrices" (PDF) . IEEE Transactions on Communications . 57 (4 ): 1050–1060. doi :10.1109/TCOMM.2009.04.070143 . hdl :1721.1/66900 . S2CID 12437386 .
^ 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 . ISSN 0587-4254 . S2CID 119599157 .
^ Muirhead, Robb J. (2005). Aspects of Multivariate Statistical Theory (2nd ed.). Wiley Interscience. ISBN 0471769851 .
^ a b Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Hoboken, N. J.: Wiley Interscience . p. 259. ISBN 0-471-36091-0 .
^ Uhlig, H. (1994). "On Singular Wishart and Singular Multivariate Beta Distributions" . The Annals of Statistics . 22 : 395–405. doi :10.1214/aos/1176325375 .
^ a b c Bishop, C. M. (2006). Pattern Recognition and Machine Learning . Springer.
^ Hoff, Peter D. (2009). A First Course in Bayesian Statistical Methods . New York: Springer. pp. 109–111. ISBN 978-0-387-92299-7 .
^ Nguyen, Duy. "AN IN DEPTH INTRODUCTION TO VARIATIONAL BAYES NOTE" . Retrieved 15 August 2023 .
^ Mayerhofer, Eberhard (2019-01-27). "Reforming the Wishart characteristic function". arXiv :1901.09347 [math.PR ].
^ a b Rao, C. R. (1965). Linear Statistical Inference and its Applications . Wiley. p. 535.
^ Seber, George A. F. (2004). Multivariate Observations . Wiley . ISBN 978-0471691211 .
^ Chatfield, C.; Collins, A. J. (1980). Introduction to Multivariate Analysis . London: Chapman and Hall. pp. 103–108 . ISBN 0-412-16030-7 .
^ Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Hoboken, N. J.: Wiley Interscience . p. 257. ISBN 0-471-36091-0 .
^ Smith, W. B.; Hocking, R. R. (1972). "Algorithm AS 53: Wishart Variate Generator". Journal of the Royal Statistical Society, Series C . 21 (3 ): 341–345. JSTOR 2346290 .
^ Pearson, Karl ; Jeffery, G. B. ; Elderton, Ethel M. (December 1929). "On the Distribution of the First Product Moment-Coefficient, in Samples Drawn from an Indefinitely Large Normal Population". Biometrika . 21 (1/4). Biometrika Trust: 164–201. doi :10.2307/2332556 . JSTOR 2332556 .
^ Fischer, Adrian; Gaunt, Robert E.; Andrey, Sarantsev. "The Variance-Gamma Distribution: A Review" . ArXiv . Retrieved 28 June 2024 .
^ Craig, Cecil C. (1936). "On the Frequency Function of xy" . Ann. Math. Statist . 7 : 1–15. doi :10.1214/aoms/1177732541 .
^ Peddada and Richards, Shyamal Das; Richards, Donald St. P. (1991). "Proof of a Conjecture of M. L. Eaton on the Characteristic Function of the Wishart Distribution" . Annals of Probability . 19 (2 ): 868–874. doi :10.1214/aop/1176990455 .
^ Gindikin, S.G. (1975). "Invariant generalized functions in homogeneous domains". Funct. Anal. Appl. 9 (1 ): 50–52. doi :10.1007/BF01078179 . S2CID 123288172 .
^ Dwyer, Paul S. (1967). "Some Applications of Matrix Derivatives in Multivariate Analysis". J. Amer. Statist. Assoc. 62 (318): 607–625. doi :10.1080/01621459.1967.10482934 . JSTOR 2283988 .
External links
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Wishart_distribution&oldid=1231793170 "
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