Quant Toolbox > 22. Multivariate distributions > Notable distributionsWishart distribution

Wishart distributionAn n̄× n̄ symmetric and positive (semi) definite matrix-valued randomvariable W 2 is said to have a Wishart distribution with ν degrees offreedom and dispersion parameter σ2

W 2 ∼Wishart(ν,σ2) (22.131)

if it satisfiesW 2 ≡X1X

′1 + · · ·+XνX

′ν (22.133)

whereXt ∼ N (0,σ2), t = 1, . . . , ν ≥ n̄ (22.132)

The pdf reads

fWishartν,σ2 (x)

1← |σ2|−ν2 |x|

ν−n̄−12 e−

12tr((σ2)−1x) (22.134)

where tr(·) is the trace (39.4) of a matrix.

Example: bivariate Wishart distribution

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Quant Toolbox > 22. Multivariate distributions > Notable distributionsWishart distribution

Properties of the Wishart distribution

A 2 by 2 positive (semi) definite matrix W 2 must satisfies the constraints

|W 2| ≡ [W 2]1,1[W 2]2,2 − [W 2]21,2 ≥ 0 (22.137)

tr(W 2) ≡ [W 2]1,1 + [W 2]2,2 ≥ 0 (22.138)

Example: Wishart distribution

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Quant Toolbox > 22. Multivariate distributions > Notable distributionsWishart distribution

Properties of the Wishart distribution

• ExpectationE{W 2} = νσ2 (22.140)

• CovarianceCv{W 2} = ν(In̄ + Kn̄,n̄)(σ2 ⊗ σ2) (22.141)

where Kn̄,n̄ is the n̄2 × n̄2 commutation matrix (E.27.203) and ⊗ is theKronecker product.

• Relationships with other distributions

Wishart(ν, σ2) ⇔ Gamma(ν

2, 2σ2) ⇔ σ2χ2

ν (22.144)

• For any generic n̄-dimensional vector a we have

a′W 2a ∼Wishart(ν,a′σ2a) (22.145)

especially[W 2]n,n ∼Wishart(ν, [σ2]n,n) (22.146)

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Quant Toolbox > 22. Multivariate distributions > Notable distributionsWishart distribution

Inverse Wishart distribution

An n̄× n̄ symmetric and positive (semi) definite random variable Σ2 is saidto have a inverse-Wishart distribution with ν degrees of freedom

Σ2 ∼ InvWishart(ν,ψ2) (22.149)

if its inverse has a Wishart distribution (21.131), or

(Σ2)−1 ∼Wishart(ν, (ψ2)−1) (22.150)

The pdf reads

f InvWishartν,ψ (x)

1← |ψ2|ν2 |x|−

ν+n̄+12 e−

12tr(ψ2x−1) (22.151)

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Quant Toolbox > 22. Multivariate distributions > Notable distributionsWishart distribution

Properties of the inverse Wishart distribution

• Expectation

E{Σ2} =1

ν − n̄− 1ψ2 (22.152)

• Covariance

Cv{[Σ2]m,n, [Σ2]p,q}

=2[ψ2]m,n[ψ2]p,q + (ν − n̄− 1)([ψ2]m,p[ψ

2]n,q + [ψ2]m,q[ψ2]n,p)

(ν − n̄)(ν − n̄− 1)2(ν − n̄− 3)(22.153)

Example: inverse Wishart distribution

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