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Eigenvectors of some large sample covariance matrix ensembles Random Matrix Workshop, Tlcom ParisTech Monday, October 11 th 2010 Olivier Ledoit Sandrine P ech e oledoit@iew.uzh.ch - sandrine.peche@ujf-grenoble.fr University of Z

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Eigenvectors of some large sample covariance matrix ensembles

Random Matrix Workshop, Télécom ParisTech Monday, October 11th 2010

Olivier Ledoit – Sandrine P´ ech´ e

  • ledoit@iew.uzh.ch - sandrine.peche@ujf-grenoble.fr

University of Z¨ urich – Universit´ e Grenoble 1

Eigenvectors of some large sample covariance matrix ensembles – p. 1/23

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Sample Covariance Matrix

Eigenvectors of some large sample covariance matrix ensembles – p. 2/23

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Sample Covariance Matrix

SN = 1 pΣ1/2

N XNX∗ NΣ1/2 N

Eigenvectors of some large sample covariance matrix ensembles – p. 2/23

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Sample Covariance Matrix

SN = 1 pΣ1/2

N XNX∗ NΣ1/2 N

N = number of variables

Eigenvectors of some large sample covariance matrix ensembles – p. 2/23

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Sample Covariance Matrix

SN = 1 pΣ1/2

N XNX∗ NΣ1/2 N

N = number of variables p = sample size

Eigenvectors of some large sample covariance matrix ensembles – p. 2/23

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Sample Covariance Matrix

SN = 1 pΣ1/2

N XNX∗ NΣ1/2 N

N = number of variables p = sample size N and p go to infinity together with N/p → c ∈ (0, +∞)

Eigenvectors of some large sample covariance matrix ensembles – p. 2/23

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Sample Covariance Matrix

SN = 1 pΣ1/2

N XNX∗ NΣ1/2 N

N = number of variables p = sample size N and p go to infinity together with N/p → c ∈ (0, +∞) XN = real or complex iid random variables mean 0, variance 1, bounded 12th moment

Eigenvectors of some large sample covariance matrix ensembles – p. 2/23

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Sample Covariance Matrix

SN = 1 pΣ1/2

N XNX∗ NΣ1/2 N

N = number of variables p = sample size N and p go to infinity together with N/p → c ∈ (0, +∞) XN = real or complex iid random variables mean 0, variance 1, bounded 12th moment ΣN = population covariance matrix

Eigenvectors of some large sample covariance matrix ensembles – p. 2/23

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Population Covariance Matrix ΣN

Eigenvectors of some large sample covariance matrix ensembles – p. 3/23

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Population Covariance Matrix ΣN

Hermitian positive definite matrix

Eigenvectors of some large sample covariance matrix ensembles – p. 3/23

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Population Covariance Matrix ΣN

Hermitian positive definite matrix Independent of XN

Eigenvectors of some large sample covariance matrix ensembles – p. 3/23

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Population Covariance Matrix ΣN

Hermitian positive definite matrix Independent of XN Eigenvalues: τ1 ≤ . . . ≤ τN

Eigenvectors of some large sample covariance matrix ensembles – p. 3/23

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Population Covariance Matrix ΣN

Hermitian positive definite matrix Independent of XN Eigenvalues: τ1 ≤ . . . ≤ τN Eigenvectors: v1, . . . , vN

Eigenvectors of some large sample covariance matrix ensembles – p. 3/23

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Population Covariance Matrix ΣN

Hermitian positive definite matrix Independent of XN Eigenvalues: τ1 ≤ . . . ≤ τN Eigenvectors: v1, . . . , vN Empirical Spectral Distribution (e.s.d.): HN(τ) = 1

N

N

i=1 1[τi,+∞)(τ)

Eigenvectors of some large sample covariance matrix ensembles – p. 3/23

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Population Covariance Matrix ΣN

Hermitian positive definite matrix Independent of XN Eigenvalues: τ1 ≤ . . . ≤ τN Eigenvectors: v1, . . . , vN Empirical Spectral Distribution (e.s.d.): HN(τ) = 1

N

N

i=1 1[τi,+∞)(τ)

HN(τ) → H(τ) at all points of continuity of H

Eigenvectors of some large sample covariance matrix ensembles – p. 3/23

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Population Covariance Matrix ΣN

Hermitian positive definite matrix Independent of XN Eigenvalues: τ1 ≤ . . . ≤ τN Eigenvectors: v1, . . . , vN Empirical Spectral Distribution (e.s.d.): HN(τ) = 1

N

N

i=1 1[τi,+∞)(τ)

HN(τ) → H(τ) at all points of continuity of H Supp(H) bounded away from 0 and +∞

Eigenvectors of some large sample covariance matrix ensembles – p. 3/23

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Spectral Decomposition of SN

Eigenvectors of some large sample covariance matrix ensembles – p. 4/23

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Spectral Decomposition of SN

eigenvalues: λ1 ≤ . . . ≤ λN

Eigenvectors of some large sample covariance matrix ensembles – p. 4/23

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Spectral Decomposition of SN

eigenvalues: λ1 ≤ . . . ≤ λN eigenvectors: u1, . . . , uN

Eigenvectors of some large sample covariance matrix ensembles – p. 4/23

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Spectral Decomposition of SN

eigenvalues: λ1 ≤ . . . ≤ λN eigenvectors: u1, . . . , uN e.s.d: FN(λ) = 1

N

N

i=1 1[λi,+∞)(λ)

Eigenvectors of some large sample covariance matrix ensembles – p. 4/23

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Spectral Decomposition of SN

eigenvalues: λ1 ≤ . . . ≤ λN eigenvectors: u1, . . . , uN e.s.d: FN(λ) = 1

N

N

i=1 1[λi,+∞)(λ)

Marˇ cenko and Pastur (1967), Silverstein (1995): ∃F s.t. FN(λ)

a.s.

− → F(λ) at all points of continuity of F

Eigenvectors of some large sample covariance matrix ensembles – p. 4/23

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Stieltjes Transform

Eigenvectors of some large sample covariance matrix ensembles – p. 5/23

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Stieltjes Transform

∀z ∈ C+ mF(z) = +∞

−∞

1 λ − zdF(λ)

Eigenvectors of some large sample covariance matrix ensembles – p. 5/23

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Stieltjes Transform

∀z ∈ C+ mF(z) = +∞

−∞

1 λ − zdF(λ) mFN(z) = 1 N

N

  • i=1

1 λi − z = 1 N Tr

  • (SN − zI)−1

Eigenvectors of some large sample covariance matrix ensembles – p. 5/23

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Stieltjes Transform

∀z ∈ C+ mF(z) = +∞

−∞

1 λ − zdF(λ) mFN(z) = 1 N

N

  • i=1

1 λi − z = 1 N Tr

  • (SN − zI)−1

Inversion formula: if F is continuous at a and b: F(b) − F(a) = lim

η→0+

1 π b

a

Im [mF(ξ + iη)] dξ

Eigenvectors of some large sample covariance matrix ensembles – p. 5/23

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MP67/Silverstein (1995) Equation

Eigenvectors of some large sample covariance matrix ensembles – p. 6/23

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MP67/Silverstein (1995) Equation

∀z ∈ C+, m = mF(z) is the unique solution in {m ∈ C : c−1

z + cm ∈ C+} to

m = +∞

−∞

1 τ (1 − c − czm) − zdH(τ)

Eigenvectors of some large sample covariance matrix ensembles – p. 6/23

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Extension to the Real Line

Eigenvectors of some large sample covariance matrix ensembles – p. 7/23

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Extension to the Real Line

Silverstein and Choi (1995):

Eigenvectors of some large sample covariance matrix ensembles – p. 7/23

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Extension to the Real Line

Silverstein and Choi (1995): ∀λ ∈ R − {0}, limz∈C+→λ mF(z) ≡ ˘ mF(λ) exists

Eigenvectors of some large sample covariance matrix ensembles – p. 7/23

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Extension to the Real Line

Silverstein and Choi (1995): ∀λ ∈ R − {0}, limz∈C+→λ mF(z) ≡ ˘ mF(λ) exists F has continuous derivative F ′ = 1

πIm [ ˘

mF] on R − {0}

Eigenvectors of some large sample covariance matrix ensembles – p. 7/23

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Intuition for MP67/S95 Equation

Eigenvectors of some large sample covariance matrix ensembles – p. 8/23

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Intuition for MP67/S95 Equation

Sample eigenvalues are a reflection of population eigenvalues

Eigenvectors of some large sample covariance matrix ensembles – p. 8/23

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Intuition for MP67/S95 Equation

Sample eigenvalues are a reflection of population eigenvalues They are noisier, more diffuse, like spreading butter

Eigenvectors of some large sample covariance matrix ensembles – p. 8/23

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Intuition for MP67/S95 Equation

Sample eigenvalues are a reflection of population eigenvalues They are noisier, more diffuse, like spreading butter The higher the c, the more spreading there is

Eigenvectors of some large sample covariance matrix ensembles – p. 8/23

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Intuition for MP67/S95 Equation

Sample eigenvalues are a reflection of population eigenvalues They are noisier, more diffuse, like spreading butter The higher the c, the more spreading there is (Not obvious) Large eigenvalues get more spread out than small ones

Eigenvectors of some large sample covariance matrix ensembles – p. 8/23

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Generalization of MP67/S95 Equation

Eigenvectors of some large sample covariance matrix ensembles – p. 9/23

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Generalization of MP67/S95 Equation

g: piecewise continuous function

Eigenvectors of some large sample covariance matrix ensembles – p. 9/23

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Generalization of MP67/S95 Equation

g: piecewise continuous function mFN(z) = 1 N

N

  • i=1

1 λi − z

N

  • j=1

|u∗

ivj|2 × 1

Eigenvectors of some large sample covariance matrix ensembles – p. 9/23

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Generalization of MP67/S95 Equation

g: piecewise continuous function mFN(z) = 1 N

N

  • i=1

1 λi − z

N

  • j=1

|u∗

ivj|2 × 1

Θg

N(z) =

1 N

N

  • i=1

1 λi − z

N

  • j=1

|u∗

ivj|2 × g(τj)

Eigenvectors of some large sample covariance matrix ensembles – p. 9/23

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Generalization of MP67/S95 Equation

g: piecewise continuous function mFN(z) = 1 N

N

  • i=1

1 λi − z

N

  • j=1

|u∗

ivj|2 × 1

Θg

N(z) =

1 N

N

  • i=1

1 λi − z

N

  • j=1

|u∗

ivj|2 × g(τj)

g(τ) ≡ 1 ⇐ ⇒ Θg

N = mFN

Eigenvectors of some large sample covariance matrix ensembles – p. 9/23

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Generalization of MP67/S95 Equation

Eigenvectors of some large sample covariance matrix ensembles – p. 10/23

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Generalization of MP67/S95 Equation

mFN(z) = 1 N Tr

  • (SN − zI)−1

Eigenvectors of some large sample covariance matrix ensembles – p. 10/23

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Generalization of MP67/S95 Equation

mFN(z) = 1 N Tr

  • (SN − zI)−1

Θg

N(z) =

1 N Tr

  • (SN − zI)−1 g(ΣN)
  • Eigenvectors of some large sample covariance matrix ensembles – p. 10/23
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Generalization of MP67/S95 Equation

Eigenvectors of some large sample covariance matrix ensembles – p. 11/23

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Generalization of MP67/S95 Equation

∃Θg : ∀z ∈ C+ Θg

N(z) a.s.

− → Θg(z)

Eigenvectors of some large sample covariance matrix ensembles – p. 11/23

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Generalization of MP67/S95 Equation

∃Θg : ∀z ∈ C+ Θg

N(z) a.s.

− → Θg(z) Θg(z) = +∞

−∞

g(τ)× 1 τ [1 − c − czmF(z)] − zdH(τ)

Eigenvectors of some large sample covariance matrix ensembles – p. 11/23

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Generalization of MP67/S95 Equation

∃Θg : ∀z ∈ C+ Θg

N(z) a.s.

− → Θg(z) Θg(z) = +∞

−∞

g(τ)× 1 τ [1 − c − czmF(z)] − zdH(τ) Same integration kernel!

Eigenvectors of some large sample covariance matrix ensembles – p. 11/23

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Extension to the Real Line

Eigenvectors of some large sample covariance matrix ensembles – p. 12/23

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Extension to the Real Line

Θg

N(z)

= 1 N

N

  • i=1

1 λi − z

N

  • j=1

|u∗

ivj|2 × g(τj)

Eigenvectors of some large sample covariance matrix ensembles – p. 12/23

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Extension to the Real Line

Θg

N(z)

= 1 N

N

  • i=1

1 λi − z

N

  • j=1

|u∗

ivj|2 × g(τj)

Ωg

N(λ)

= 1 N

N

  • i=1

1[λi,+∞)(λ)

N

  • j=1

|u∗

ivj|2 × g(τj)

Eigenvectors of some large sample covariance matrix ensembles – p. 12/23

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Extension to the Real Line

Θg

N(z)

= 1 N

N

  • i=1

1 λi − z

N

  • j=1

|u∗

ivj|2 × g(τj)

Ωg

N(λ)

= 1 N

N

  • i=1

1[λi,+∞)(λ)

N

  • j=1

|u∗

ivj|2 × g(τj)

Ωg

N(λ) a.s.

→ Ωg(λ) = lim

η→0+

1 π λ

−∞

Im [Θg (l + iη)] dl wherever Ωg is continuous

Eigenvectors of some large sample covariance matrix ensembles – p. 12/23

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Sample Eigenvectors

Eigenvectors of some large sample covariance matrix ensembles – p. 13/23

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Sample Eigenvectors

Fix τ and take g = 1(−∞,τ)

Eigenvectors of some large sample covariance matrix ensembles – p. 13/23

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Sample Eigenvectors

Fix τ and take g = 1(−∞,τ) Ωg

N(λ) = 1

N

N

  • i=1

N

  • j=1

|u∗

ivj|2 1[λi,+∞)(λ) × 1[τj,+∞)(τ)

→ λ

−∞

τ

−∞

clt |t [1 − c − clmF(l)] − l|2dH(t)dF(l)

Eigenvectors of some large sample covariance matrix ensembles – p. 13/23

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Sample Eigenvectors

Fix τ and take g = 1(−∞,τ) Ωg

N(λ) = 1

N

N

  • i=1

N

  • j=1

|u∗

ivj|2 1[λi,+∞)(λ) × 1[τj,+∞)(τ)

→ λ

−∞

τ

−∞

clt |t [1 − c − clmF(l)] − l|2dH(t)dF(l) N|u∗

ivj|2 ≈

cλiτj |τj [1 − c − cλi ˘ mF(λi)] − λi|2

Eigenvectors of some large sample covariance matrix ensembles – p. 13/23

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Eigenvalues with Multiplicity

Eigenvectors of some large sample covariance matrix ensembles – p. 14/23

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Eigenvalues with Multiplicity

ΣN has K distinct eigenvalues t1, . . . , tK with multiplicities n1, . . . , nK

Eigenvectors of some large sample covariance matrix ensembles – p. 14/23

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Eigenvalues with Multiplicity

ΣN has K distinct eigenvalues t1, . . . , tK with multiplicities n1, . . . , nK Pk = projection onto kth eigenspace

Eigenvectors of some large sample covariance matrix ensembles – p. 14/23

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Eigenvalues with Multiplicity

ΣN has K distinct eigenvalues t1, . . . , tK with multiplicities n1, . . . , nK Pk = projection onto kth eigenspace |Pkui|2 ≈ nkcλitk N |tk [1 − c − cλi ˘ mF(λi)] − λi|2

Eigenvectors of some large sample covariance matrix ensembles – p. 14/23

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Estimating the Covariance Matrix (1)

Eigenvectors of some large sample covariance matrix ensembles – p. 15/23

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Estimating the Covariance Matrix (1)

Frobenius norm: A =

  • Tr(AA∗)

Eigenvectors of some large sample covariance matrix ensembles – p. 15/23

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Estimating the Covariance Matrix (1)

Frobenius norm: A =

  • Tr(AA∗)

UN: matrix of eigenvectors of SN

Eigenvectors of some large sample covariance matrix ensembles – p. 15/23

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Estimating the Covariance Matrix (1)

Frobenius norm: A =

  • Tr(AA∗)

UN: matrix of eigenvectors of SN Find matrix closest to ΣN among those that have eigenvectors UN

Eigenvectors of some large sample covariance matrix ensembles – p. 15/23

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Estimating the Covariance Matrix (1)

Frobenius norm: A =

  • Tr(AA∗)

UN: matrix of eigenvectors of SN Find matrix closest to ΣN among those that have eigenvectors UN min

DN diagonal

UNDNU ∗

N − ΣN

Eigenvectors of some large sample covariance matrix ensembles – p. 15/23

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Estimating the Covariance Matrix (1)

Frobenius norm: A =

  • Tr(AA∗)

UN: matrix of eigenvectors of SN Find matrix closest to ΣN among those that have eigenvectors UN min

DN diagonal

UNDNU ∗

N − ΣN

Solution:

  • DN = Diag(

d1, . . . , dN) where

  • di = u∗

i ΣN ui

Eigenvectors of some large sample covariance matrix ensembles – p. 15/23

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Estimating the Covariance Matrix (2)

Eigenvectors of some large sample covariance matrix ensembles – p. 16/23

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Estimating the Covariance Matrix (2)

Take g(τ) = τ

Eigenvectors of some large sample covariance matrix ensembles – p. 16/23

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Estimating the Covariance Matrix (2)

Take g(τ) = τ Ωg

N(λ) = 1

N

N

  • i=1

u∗

iΣNui 1[λi,+∞)(λ)

→ λ

−∞

l |1 − c − clmF(l)|2dF(l)

Eigenvectors of some large sample covariance matrix ensembles – p. 16/23

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Estimating the Covariance Matrix (2)

Take g(τ) = τ Ωg

N(λ) = 1

N

N

  • i=1

u∗

iΣNui 1[λi,+∞)(λ)

→ λ

−∞

l |1 − c − clmF(l)|2dF(l) u∗

iΣNui ≈

λi |1 − c − cλi ˘ mF(λi)|2

Eigenvectors of some large sample covariance matrix ensembles – p. 16/23

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Oracle Estimator

Eigenvectors of some large sample covariance matrix ensembles – p. 17/23

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Oracle Estimator

Keep same eigenvectors as those of Sn,

Eigenvectors of some large sample covariance matrix ensembles – p. 17/23

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Oracle Estimator

Keep same eigenvectors as those of Sn, divide ith sample eigenvalue by |1 − c − cλi ˘ mF(λi)|2

Eigenvectors of some large sample covariance matrix ensembles – p. 17/23

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Oracle Estimator

Keep same eigenvectors as those of Sn, divide ith sample eigenvalue by |1 − c − cλi ˘ mF(λi)|2 − → oracle estimator SN

Eigenvectors of some large sample covariance matrix ensembles – p. 17/23

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Oracle Estimator

Keep same eigenvectors as those of Sn, divide ith sample eigenvalue by |1 − c − cλi ˘ mF(λi)|2 − → oracle estimator SN Percentage Relative Improvement in Average Loss: PRIAL = 100 ×   1 − E

  • SN − UN

DNU ∗

N

  • 2

E

  • SN − UN

DNU ∗

N

  • 2

  

Eigenvectors of some large sample covariance matrix ensembles – p. 17/23

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Monte-Carlo Simulations

Eigenvectors of some large sample covariance matrix ensembles – p. 18/23

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Monte-Carlo Simulations

10,000 simulations

Eigenvectors of some large sample covariance matrix ensembles – p. 18/23

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SLIDE 78

Monte-Carlo Simulations

10,000 simulations c=1/2

Eigenvectors of some large sample covariance matrix ensembles – p. 18/23

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SLIDE 79

Monte-Carlo Simulations

10,000 simulations c=1/2 Population eigenvalues: 20% equal to 1 40% equal to 3 40% equal to 10

Eigenvectors of some large sample covariance matrix ensembles – p. 18/23

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Monte-Carlo Simulations

10,000 simulations c=1/2 Population eigenvalues: 20% equal to 1 40% equal to 3 40% equal to 10 Compare with Ledoit-Wolf (2004) linear shrinkage estimator

Eigenvectors of some large sample covariance matrix ensembles – p. 18/23

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Simulation Results

Eigenvectors of some large sample covariance matrix ensembles – p. 19/23

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SLIDE 82

Simulation Results

10 20 40 60 80 100 120 140 160 180 200 50% 60% 70% 80% 90% 100% Relative Improvement in Average Loss Sample Size γ=2 Optimal Nonlinear Shrinkage Optimal Linear Shrinkage

Eigenvectors of some large sample covariance matrix ensembles – p. 19/23

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SLIDE 83

Inverse of the Covariance Matrix (1)

Eigenvectors of some large sample covariance matrix ensembles – p. 20/23

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SLIDE 84

Inverse of the Covariance Matrix (1)

Find matrix closest to Σ−1

N among those that have

eigenvectors UN

Eigenvectors of some large sample covariance matrix ensembles – p. 20/23

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SLIDE 85

Inverse of the Covariance Matrix (1)

Find matrix closest to Σ−1

N among those that have

eigenvectors UN min

∆N diagonal

UN∆NU ∗

N − Σ−1 N

Eigenvectors of some large sample covariance matrix ensembles – p. 20/23

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SLIDE 86

Inverse of the Covariance Matrix (1)

Find matrix closest to Σ−1

N among those that have

eigenvectors UN min

∆N diagonal

UN∆NU ∗

N − Σ−1 N

Solution:

  • ∆N = Diag(

δ1, . . . , δN) where

  • δi = u∗

i Σ−1 N ui

Eigenvectors of some large sample covariance matrix ensembles – p. 20/23

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SLIDE 87

Inverse of the Covariance Matrix (1)

Find matrix closest to Σ−1

N among those that have

eigenvectors UN min

∆N diagonal

UN∆NU ∗

N − Σ−1 N

Solution:

  • ∆N = Diag(

δ1, . . . , δN) where

  • δi = u∗

i Σ−1 N ui

u∗

i Σ−1 N ui ≥ (u∗ i ΣN ui)−1

Eigenvectors of some large sample covariance matrix ensembles – p. 20/23

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SLIDE 88

Inverse of the Covariance Matrix (2)

Eigenvectors of some large sample covariance matrix ensembles – p. 21/23

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SLIDE 89

Inverse of the Covariance Matrix (2)

Take g(τ) = 1

τ

Eigenvectors of some large sample covariance matrix ensembles – p. 21/23

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SLIDE 90

Inverse of the Covariance Matrix (2)

Take g(τ) = 1

τ

Ωg

N(λ) = 1

N

N

  • i=1

u∗

iΣ−1 N ui 1[λi,+∞)(λ)

→ λ

−∞

1 − c − 2clRe[mF(l)] l dF(l)

Eigenvectors of some large sample covariance matrix ensembles – p. 21/23

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SLIDE 91

Inverse of the Covariance Matrix (2)

Take g(τ) = 1

τ

Ωg

N(λ) = 1

N

N

  • i=1

u∗

iΣ−1 N ui 1[λi,+∞)(λ)

→ λ

−∞

1 − c − 2clRe[mF(l)] l dF(l) u∗

iΣ−1 N ui ≈ 1 − c − 2cλiRe[ ˘

mF(λi)] λi

Eigenvectors of some large sample covariance matrix ensembles – p. 21/23

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SLIDE 92

Conclusion

Eigenvectors of some large sample covariance matrix ensembles – p. 22/23

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SLIDE 93

Conclusion

Generalization of the Marˇ cenko-Pastur (1967)/Silverstein (1995) Equation

Eigenvectors of some large sample covariance matrix ensembles – p. 22/23

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SLIDE 94

Conclusion

Generalization of the Marˇ cenko-Pastur (1967)/Silverstein (1995) Equation Gives location of sample eigenvectors relative to:

Eigenvectors of some large sample covariance matrix ensembles – p. 22/23

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SLIDE 95

Conclusion

Generalization of the Marˇ cenko-Pastur (1967)/Silverstein (1995) Equation Gives location of sample eigenvectors relative to: Population eigenvectors

Eigenvectors of some large sample covariance matrix ensembles – p. 22/23

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SLIDE 96

Conclusion

Generalization of the Marˇ cenko-Pastur (1967)/Silverstein (1995) Equation Gives location of sample eigenvectors relative to: Population eigenvectors Population covariance matrix as a whole

Eigenvectors of some large sample covariance matrix ensembles – p. 22/23

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SLIDE 97

Conclusion

Generalization of the Marˇ cenko-Pastur (1967)/Silverstein (1995) Equation Gives location of sample eigenvectors relative to: Population eigenvectors Population covariance matrix as a whole Inverse of population covariance matrix

Eigenvectors of some large sample covariance matrix ensembles – p. 22/23

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SLIDE 98

Conclusion

Generalization of the Marˇ cenko-Pastur (1967)/Silverstein (1995) Equation Gives location of sample eigenvectors relative to: Population eigenvectors Population covariance matrix as a whole Inverse of population covariance matrix We do for sample eigenvectors what MP67/S95 did for sample eigenvalues

Eigenvectors of some large sample covariance matrix ensembles – p. 22/23

slide-99
SLIDE 99

Directions for Future Research

Eigenvectors of some large sample covariance matrix ensembles – p. 23/23

slide-100
SLIDE 100

Directions for Future Research

  • 1. Construct bona fide nonlinear shrinkage

estimator of the covariance matrix

Eigenvectors of some large sample covariance matrix ensembles – p. 23/23

slide-101
SLIDE 101

Directions for Future Research

  • 1. Construct bona fide nonlinear shrinkage

estimator of the covariance matrix

  • 2. Construct bona fide nonlinear shrinkage

estimator of the inverse of the covariance matrix

Eigenvectors of some large sample covariance matrix ensembles – p. 23/23

slide-102
SLIDE 102

Directions for Future Research

  • 1. Construct bona fide nonlinear shrinkage

estimator of the covariance matrix

  • 2. Construct bona fide nonlinear shrinkage

estimator of the inverse of the covariance matrix

  • 3. Show that N|u∗

ivj|2 is even closer to

cλiτj |τj [1 − c − cλi ˘ mF(λi)] − λi|2 than we have shown in this paper

Eigenvectors of some large sample covariance matrix ensembles – p. 23/23