HS 67Sampling Distributions1 Chapter 11 Sampling Distributions.
Quant Toolbox - 22. Multivariate distributions - Elliptical distributions
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Quant Toolbox > 22. Multivariate distributions > Elliptical distributions Student t, Cauchy, normal Multivariate normal distribution The random vector X ≡ (X1,...,X¯ n) 0 has a normal distribution X ∼ N (μ, σ 2 ) (22.114) if its pdf reads f N μ,σ 2 (x) = (2π) - ¯ n 2 |σ 2 | - 1 2 e - 1 2 (x-μ) 0 (σ 2 ) -1 (x-μ) (22.115) where μ ∈ R ¯ n , σ 2 0. ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-01-2017 - Last update
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Transcript of Quant Toolbox - 22. Multivariate distributions - Elliptical distributions
Quant Toolbox > 22. Multivariate distributions > Elliptical distributionsStudent t, Cauchy, normal
Multivariate normal distribution
The random vector X ≡ (X1, . . . , Xn̄)′ has a normal distribution
X ∼ N (µ,σ2) (22.114)
if its pdf reads
fNµ,σ2(x) = (2π)−
n̄2 |σ2|−
12 e−
12
(x−µ)′(σ2)−1(x−µ) (22.115)
where µ ∈ Rn̄, σ2 � 0.
The characteristic function reads
ϕNµ,σ2(ω) = eiω
′µe−12ω′σ2ω (22.116)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-01-2017 - Last update
Quant Toolbox > 22. Multivariate distributions > Elliptical distributionsStudent t, Cauchy, normal
Multivariate normal distribution
The random vector X ≡ (X1, . . . , Xn̄)′ has a normal distribution
X ∼ N (µ,σ2) (22.114)
if its pdf reads
fNµ,σ2(x) = (2π)−
n̄2 |σ2|−
12 e−
12
(x−µ)′(σ2)−1(x−µ) (22.115)
where µ ∈ Rn̄, σ2 � 0.
The characteristic function reads
ϕNµ,σ2(ω) = eiω
′µe−12ω′σ2ω (22.116)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-01-2017 - Last update
Quant Toolbox > 22. Multivariate distributions > Elliptical distributionsStudent t, Cauchy, normal
Properties of the multivariate normal distribution
For X ≡ (X1, . . . , Xn̄)′, Z ≡ (Z1, . . . , Zk̄)′ jointly normal random vectors(XZ
)∼ N (
(µXµZ
),
(σ2X σX,Z
σ′X,Z σ2Z
)) (22.117)
the conditional variable is normal
X|z ∼ N (µX +σX,Z(σ2Z)−1(z −µZ),σ2
X −σX,Z(σ2Z)−1σ′X,Z) (22.118)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-01-2017 - Last update
Quant Toolbox > 22. Multivariate distributions > Elliptical distributionsStudent t, Cauchy, normal
Matrix-normal distribution
The n̄× k̄-dimensional random matrix X has a matrix-normal distribution
X ∼ N (µ,σ2,ψ2) (22.122)
with µ ∈ Rn̄×k̄, σ2 � 0 ∈ Rn̄×n̄, ψ2 � 0 ∈ Rk̄×k̄ if
Xd= µ+ σZψ′ (22.123)
where• (σ,ψ) Riccati roots (38.26) of (σ2,ψ2)
• Zn,k ∼ N (0, 1) i.i.d.
Equivalently
X ∼ N (µ,σ2,ψ2) ⇔ vec(X) ∼ N (vec(µ),ψ2 ⊗ σ2) (22..124)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-01-2017 - Last update
