On the product of a singular Wishart matrix and a singular Gaussian - - PowerPoint PPT Presentation

On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension. Stanislas Muhinyuza

Department of Mathematics, University of Rwanda Department of Mathematics, Stockholm University

First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm 7–8 March 2017

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My Advisors

Taras Bodnar Marcel R.Ndengo

Main advisor Assistant advisor Stockholm University University of Rwanda

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Recall

Asymptotics standard asymptotics

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Recall

Asymptotics standard asymptotics

fixed dimension k and the sample size n goes to infinity;

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Recall

Asymptotics standard asymptotics

fixed dimension k and the sample size n goes to infinity; classical limit theorems hold

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Recall

Asymptotics standard asymptotics

fixed dimension k and the sample size n goes to infinity; classical limit theorems hold

Large dimensional asymptotics (Asymptotic distribution under double asymptotic regime)

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Recall

Asymptotics standard asymptotics

fixed dimension k and the sample size n goes to infinity; classical limit theorems hold

Large dimensional asymptotics (Asymptotic distribution under double asymptotic regime)

both the dimension k and the sample size n goes to infinity;

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Recall

Asymptotics standard asymptotics

fixed dimension k and the sample size n goes to infinity; classical limit theorems hold

Large dimensional asymptotics (Asymptotic distribution under double asymptotic regime)

both the dimension k and the sample size n goes to infinity; the ratio k/n tends to a positive constant c > 0;

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Recall

Asymptotics standard asymptotics

fixed dimension k and the sample size n goes to infinity; classical limit theorems hold

Large dimensional asymptotics (Asymptotic distribution under double asymptotic regime)

both the dimension k and the sample size n goes to infinity; the ratio k/n tends to a positive constant c > 0; classical limit theorems do not hold anymore.

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Introduction

Let X = (X1, . . . , Xn) be a sample of size n from k-variate normal distribution, That is Xi ∼ Nk(0, Σ) for i = 1, . . . , n.

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Introduction

Let X = (X1, . . . , Xn) be a sample of size n from k-variate normal distribution, That is Xi ∼ Nk(0, Σ) for i = 1, . . . , n. A = XX′ ∼ Wk(n, Σ); k ≤ n, Wishart distribution . The properties of A are detailed in [Muirhead, 1982].

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Introduction

Let X = (X1, . . . , Xn) be a sample of size n from k-variate normal distribution, That is Xi ∼ Nk(0, Σ) for i = 1, . . . , n. A = XX′ ∼ Wk(n, Σ); k ≤ n, Wishart distribution . The properties of A are detailed in [Muirhead, 1982]. [Srivastava, 2003] defined A = XX′ ∼ Wk(n, Σ), k > n as a singular Wishart matrix.

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Introduction

Let X = (X1, . . . , Xn) be a sample of size n from k-variate normal distribution, That is Xi ∼ Nk(0, Σ) for i = 1, . . . , n. A = XX′ ∼ Wk(n, Σ); k ≤ n, Wishart distribution . The properties of A are detailed in [Muirhead, 1982]. [Srivastava, 2003] defined A = XX′ ∼ Wk(n, Σ), k > n as a singular Wishart matrix. Singular Gaussian vector is the normal distributed vector with a singular covariance matrix.

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Introduction

Let X = (X1, . . . , Xn) be a sample of size n from k-variate normal distribution, That is Xi ∼ Nk(0, Σ) for i = 1, . . . , n. A = XX′ ∼ Wk(n, Σ); k ≤ n, Wishart distribution . The properties of A are detailed in [Muirhead, 1982]. [Srivastava, 2003] defined A = XX′ ∼ Wk(n, Σ), k > n as a singular Wishart matrix. Singular Gaussian vector is the normal distributed vector with a singular covariance matrix. Wishart distributed matrix does not usually appear alone, but in combination with a normal random vector, its properties are detailed in [Bodnar et al., 2013].

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Introduction

Example of application of the product The weights of the tangency portfolio are computed as products of the inverse sample covariance matrix multiplied by the sample mean vector using historical asset returns.

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Introduction

Example of application of the product The weights of the tangency portfolio are computed as products of the inverse sample covariance matrix multiplied by the sample mean vector using historical asset returns. Assumptions z ∼ Nk(µ, κΣ), κ > 0 with rank(Σ) = r < k;

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Introduction

Example of application of the product The weights of the tangency portfolio are computed as products of the inverse sample covariance matrix multiplied by the sample mean vector using historical asset returns. Assumptions z ∼ Nk(µ, κΣ), κ > 0 with rank(Σ) = r < k; A ∼ Wk(n, Σ) with rank(Σ) = r < k;

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Introduction

Example of application of the product The weights of the tangency portfolio are computed as products of the inverse sample covariance matrix multiplied by the sample mean vector using historical asset returns. Assumptions z ∼ Nk(µ, κΣ), κ > 0 with rank(Σ) = r < k; A ∼ Wk(n, Σ) with rank(Σ) = r < k; M : p × k matrix of constants with rank(M) = p ≤ r ≤ min{n, k} such that MΣ = 0;

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Introduction

Example of application of the product The weights of the tangency portfolio are computed as products of the inverse sample covariance matrix multiplied by the sample mean vector using historical asset returns. Assumptions z ∼ Nk(µ, κΣ), κ > 0 with rank(Σ) = r < k; A ∼ Wk(n, Σ) with rank(Σ) = r < k; M : p × k matrix of constants with rank(M) = p ≤ r ≤ min{n, k} such that MΣ = 0; A and z are independent.

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Introduction

Example of application of the product The weights of the tangency portfolio are computed as products of the inverse sample covariance matrix multiplied by the sample mean vector using historical asset returns. Assumptions z ∼ Nk(µ, κΣ), κ > 0 with rank(Σ) = r < k; A ∼ Wk(n, Σ) with rank(Σ) = r < k; M : p × k matrix of constants with rank(M) = p ≤ r ≤ min{n, k} such that MΣ = 0; A and z are independent. Aim: Distribution of MAz

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Stochastic representation

Theorem 1[Bodnar et al., 2016]

1 Let P(rank((MT, z)TΣ) = p + 1 ≤ r) = 1 , and let Q = PTP with

P = (MΣMT)−1/2MΣ1/2.

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Stochastic representation

Theorem 1[Bodnar et al., 2016]

1 Let P(rank((MT, z)TΣ) = p + 1 ≤ r) = 1 , and let Q = PTP with

P = (MΣMT)−1/2MΣ1/2.Then MAz

d

= ζMΣ1/2t +

• ζ(MΣMT)1/2

× √ tTtIp − √ tTt −

• tT(Ik − Q)t

tTQt PttTPT

• z0,

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Stochastic representation

Theorem 1[Bodnar et al., 2016]

1 Let P(rank((MT, z)TΣ) = p + 1 ≤ r) = 1 , and let Q = PTP with

P = (MΣMT)−1/2MΣ1/2.Then MAz

d

= ζMΣ1/2t +

• ζ(MΣMT)1/2

× √ tTtIp − √ tTt −

• tT(Ik − Q)t

tTQt PttTPT

• z0,

where ζ ∼ χ2

n, t ∼ Nk(Σ1/2µ, κΣ2), and z0 ∼ Np(0, Ip); ζ, t, and

z0 are mutually independent.

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Stochastic representation

Theorem 1[Bodnar et al., 2016]

1 Let P(rank((MT, z)TΣ) = p + 1 ≤ r) = 1 , and let Q = PTP with

P = (MΣMT)−1/2MΣ1/2.Then MAz

d

= ζMΣ1/2t +

• ζ(MΣMT)1/2

× √ tTtIp − √ tTt −

• tT(Ik − Q)t

tTQt PttTPT

• z0,

where ζ ∼ χ2

n, t ∼ Nk(Σ1/2µ, κΣ2), and z0 ∼ Np(0, Ip); ζ, t, and

z0 are mutually independent.

2 Let m be a k-dimensional vector of constants such that mTΣm > 0

and P(zTΣz = 0) = 0.

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Stochastic representation

Theorem 1[Bodnar et al., 2016]

1 Let P(rank((MT, z)TΣ) = p + 1 ≤ r) = 1 , and let Q = PTP with

P = (MΣMT)−1/2MΣ1/2.Then MAz

d

= ζMΣ1/2t +

• ζ(MΣMT)1/2

× √ tTtIp − √ tTt −

• tT(Ik − Q)t

tTQt PttTPT

• z0,

where ζ ∼ χ2

n, t ∼ Nk(Σ1/2µ, κΣ2), and z0 ∼ Np(0, Ip); ζ, t, and

z0 are mutually independent.

2 Let m be a k-dimensional vector of constants such that mTΣm > 0

and P(zTΣz = 0) = 0. Then mTAz

d

= ζmTΣz +

• ζ
• zTΣz · mTΣm − (mTΣz)21/2 z0,

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Stochastic representation

Theorem 1[Bodnar et al., 2016]

1 Let P(rank((MT, z)TΣ) = p + 1 ≤ r) = 1 , and let Q = PTP with

P = (MΣMT)−1/2MΣ1/2.Then MAz

d

= ζMΣ1/2t +

• ζ(MΣMT)1/2

× √ tTtIp − √ tTt −

• tT(Ik − Q)t

tTQt PttTPT

• z0,

where ζ ∼ χ2

n, t ∼ Nk(Σ1/2µ, κΣ2), and z0 ∼ Np(0, Ip); ζ, t, and

z0 are mutually independent.

2 Let m be a k-dimensional vector of constants such that mTΣm > 0

and P(zTΣz = 0) = 0. Then mTAz

d

= ζmTΣz +

• ζ
• zTΣz · mTΣm − (mTΣz)21/2 z0,

where ζ ∼ χ2

n and z0 ∼ N(0, 1); ζ, z0, and z are mutually indep.

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Characteristic function

Theorem 2[Bodnar et al., 2016] The characteristic function of Az is given by ϕAz(u) = exp

• − κ−1

2 µTRΛ−1RTµ

• κr/2|Λ|1/2

∞ |Ω|−1/2fχ2

n(ζ)

× exp

• iζνTΛRTu − ζ2

2 uTRΛΩ−1ΛRTu + 1 2νTΩν

• dζ,

Ω = Ω(ζ) = κ−1Λ−1 + ζ

• Λ · uTΣu − ΛRTuuTRΛ
• ,

ν = ν(ζ) = κ−1Ω−1Λ−1RTµ, Σ = RΛRT singular value decomposition of Σ, with matrix Λ consisting of all r non-zero eigenvalues of Σ and R is the k × r matrix of the corresponding eigenvectors.

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Asymptotic distribution

(A1) (λi, ui) denote the set of non-zero eigenvalues and the corresponding eigenvectors of Σ. There exist l1 and L1 such that 0 < l1 ≤ λ1 ≤ λ2 ≤ . . . ≤ λr ≤ L1 < ∞ uniformly on k.

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Asymptotic distribution

(A1) (λi, ui) denote the set of non-zero eigenvalues and the corresponding eigenvectors of Σ. There exist l1 and L1 such that 0 < l1 ≤ λ1 ≤ λ2 ≤ . . . ≤ λr ≤ L1 < ∞ uniformly on k. (A2) There exists L2 such that |uT

i µ| ≤ L2 for all i = 1, . . . , r uniformly on k.

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Asymptotic distribution

Theorem 3[Bodnar et al., 2016] Assume r

n = c + o(n−1/2), c ∈ [0, 1) and κr = O(1) as n → ∞.

Let m be a k-dimensional vector of constants s.t mTΣm > 0 and |uT

i m| ≤ L2 for all i = 1, . . . , r uniformly on k. Let

P(zTΣz = 0) = 0.

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Asymptotic distribution

Theorem 3[Bodnar et al., 2016] Assume r

n = c + o(n−1/2), c ∈ [0, 1) and κr = O(1) as n → ∞.

Let m be a k-dimensional vector of constants s.t mTΣm > 0 and |uT

i m| ≤ L2 for all i = 1, . . . , r uniformly on k. Let

P(zTΣz = 0) = 0. Then, under (A1) and (A2) it holds that the asymptotic distribution of mTAz is given by √nσ−1 1 nmTAz − mTΣµ

• d

− → N (0, 1) ,

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Asymptotic distribution

Theorem 3[Bodnar et al., 2016] Assume r

n = c + o(n−1/2), c ∈ [0, 1) and κr = O(1) as n → ∞.

Let m be a k-dimensional vector of constants s.t mTΣm > 0 and |uT

i m| ≤ L2 for all i = 1, . . . , r uniformly on k. Let

P(zTΣz = 0) = 0. Then, under (A1) and (A2) it holds that the asymptotic distribution of mTAz is given by √nσ−1 1 nmTAz − mTΣµ

• d

− → N (0, 1) , where σ2 =

• mTΣµ

2 + mTΣm

• κtr(Σ2) + µTΣµ
• + κ

c mTΣ3m.

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Asymptotic distribution

Theorem 4[Bodnar et al., 2016] Assume r

n = c + o(n−1/2), c ∈ [0, 1) and κr = O(1) as n → ∞.

Let M = (m1, . . . , mp)T : p × k be a matrix of constants of rank p such that rank((MT, z)TΣ) = p + 1 ≤ r with probability one and let |uT

i mj| ≤ L2 for all i = 1, . . . , r and j = 1, . . . , p uniformly on

k.

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Asymptotic distribution

Theorem 4[Bodnar et al., 2016] Assume r

n = c + o(n−1/2), c ∈ [0, 1) and κr = O(1) as n → ∞.

Let M = (m1, . . . , mp)T : p × k be a matrix of constants of rank p such that rank((MT, z)TΣ) = p + 1 ≤ r with probability one and let |uT

i mj| ≤ L2 for all i = 1, . . . , r and j = 1, . . . , p uniformly on

• k. Then under (A1) and (A2) the asymptotic distribution of MAz

under the double asymptotic regime is given by √nΩ−1/2 1 nMAz − MΣz

• d

− → N (0, I)

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Asymptotic distribution

Theorem 4[Bodnar et al., 2016] Assume r

n = c + o(n−1/2), c ∈ [0, 1) and κr = O(1) as n → ∞.

Let M = (m1, . . . , mp)T : p × k be a matrix of constants of rank p such that rank((MT, z)TΣ) = p + 1 ≤ r with probability one and let |uT

i mj| ≤ L2 for all i = 1, . . . , r and j = 1, . . . , p uniformly on

• k. Then under (A1) and (A2) the asymptotic distribution of MAz

under the double asymptotic regime is given by √nΩ−1/2 1 nMAz − MΣz

• d

− → N (0, I) where Ω = MΣµµTΣMT + MΣMT κtr(Σ2) + µTΣµ

• + κ

c MΣ3MT.

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Finite sample performance

−4 −2 2 4 0.0 0.1 0.2 0.3 0.4 0.5 Finite sample Asymptotic

(a) k = 750, c = 0.1

−4 −2 2 4 0.0 0.1 0.2 0.3 0.4 0.5 Finite sample Asymptotic

(b) k = 750, c = 0.5

−4 −2 2 4 0.0 0.1 0.2 0.3 0.4 0.5 Finite sample Asymptotic

(c) k = 750, c = 0.8

−4 −2 2 4 0.0 0.1 0.2 0.3 0.4 0.5 Finite sample Asymptotic

(d) k = 750, c = 0.95

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Reference

Bodnar, T., Mazur, S., Muhinyuza, S., and Parolya, N. (2016). On the product of a singular wishart matrix and a singular gaussian vector in high dimension. arXiv preprint arXiv:1611.03042. Bodnar, T., Mazur, S., and Okhrin, Y. (2013). On exact and approximate distributions of the product of the Wishart matrix and normal vector. Journal of Multivariate Analysis, 122:70–81. Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York. Srivastava, M. S. (2003). Singular wishart and multivariate beta distributions.

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Tack s˚ a mycket! Thank you!

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