Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Adaptive estimation of the copula correlation
matrix for semiparametric elliptical copulas
Marten Wegkamp
Department of Mathematics
Department of Statistical Science
Cornell University
London, January 7, 2016
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Joint work with
Yue Zhao
Marten Wegkamp
Supported by
NSF Grant DMS 1310119
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Copula
Let the continuous random vector X = (X1 , . . ., Xd )0 ∈Rd
have marginal distribution functions Fj (x) = P Xj ≤ x .
The copula function C(u1 , . . . , ud ) is the joint cumulative
distribution function of the transformation
U = (F1 (X1 ), . . . , Fd (Xd ) )0 ,
i.e., for u = (u1 , . . . , ud )0 ∈ [0, 1]d ,
C(u1 , . . . , ud ) = P {F1 (X1 ) ≤ u1 , . . . , Fd (Xd ) ≤ ud } .
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Copula
The copula C always describes a distribution on the unit
square.
Copula is invariant under strictly increasing transformations
Tj of the marginals Xj — allows separate specifications of
the marginals and the dependence structure.
In this talk we mostly study semiparametric copula models.
The dependence structure is given by a parametric copula,
while the marginals are left unspecified (nonparametric).
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Elliptical distribution
A random vector X ∈ Rd has an elliptical distribution if its
characteristic function E(exp(it 0 X )) can be written as
0
eit µ Ψ(t 0 Σt)
with parameters µ ∈ Rd , covariance matrix Σ ∈ Rd×d , and
characteristic generator Ψ : [0, ∞) → R.
For instance, when Ψ(t) = e−t/2 , X ∼ N(µ, Σ).
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Semiparametric elliptical distributions and the copula
correlation matrix
Elliptical copulas are the copulas of elliptical distributions.
Elliptical copula is characterized by copula parameters: Ψ
and the copula correlation matrix Σ, with
Σij
Σij = q
.
Σii Σjj
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Semiparametric Gaussian graphical copula model
Let X ∼ N(0, Σ), and Ω = Σ−1 be the precision matrix. If
[Ω]k ` = 0, then
Xk ⊥ X` |X\{k ,`} .
Accurate estimator of Ω can be obtained by feeding the
b into the graphical lasso,
sample covariance matrix Σ
CLIME, graphical Dantzig selector, etc.
Assume instead that we merely have f (X ) ∼ N(0, Σ), for
some strictly increasing, unknown function f (i.e., X has a
Gaussian copula with copula correlation matrix Σ), and
again we let Ω = Σ−1 . Now, if [Ω]k ` = 0, then
f (Xk ) ⊥ f (X` )|f (X )\{k ,`} =⇒ Xk ⊥ X` |X\{k ,`} .
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Semiparametric Gaussian graphical copula model
b rank-based (using Kendall’s
Using a rank-based estimator Σ
tau or Spearman’s rho) of Σ, Liu et al 2012, Xue & Zou
2012 recovers the same optimal parametric rate (obtained
under the Gaussian model) under the semiparametric
model.
b rank-based − Σkmax at
This crucially
p depends on estimating kΣ
the rate of log(d)/n (w.h.p.), which is in turn based on
Hoeffding’s classical bound for scalar U-statistic.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Kendall’s tau
e ∈ Rd be an independent copy of X . Kendall’s tau
Let X
between the k th and `th coordinates is
h
i
ek )(X` − X
e` )
τk ` = E sgn (Xk − X
Let T be the matrix of Kendall’s tau:
[T ]k ` = τk ` .
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Kendall’s tau
Kendall’s tau statistic is, for samples X 1 , . . . , X n that are
independent copies of X ,
τbk ` =
XX
2
sgn (Xki − Xkj )(X`i − X`j ) .
n(n − 1)
1≤i<j≤d
b be the matrix of Kendall’s tau statistics:
Let T
b ]k ` = τbk ` .
[T
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Kendall’s tau matrix as plug-in estimator
For elliptical copulas, independent of the characteristic
generator,
π Σ = sin
T
2
with the sine function acting component-wise (Kruskal
1958, Kendall & Gibbons 1990, Fang, Fang & Kotz 2002).
Hence, the natural plug-in estimator for Σ is
b .
b = sin π T
Σ
2
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
For the rest of the talk, we assume
X ∈ Rd follows a semiparametric elliptical distribution,
The copula correlation matrix of X is Σ ∈ Rd×d ,
b =Σ
b n of Σ is constructed from n
The plug-in estimator Σ
independent copies of X , (e.g., Demarta & McNeil 2004,
Klüppelberg & Kuhn 2009)
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Some recent advanced in elliptical/Gaussian copulas
b in the
Klüppelberg & Kuhn 2009 study the property of Σ
asymptotic regime, i.e., d fixed, n → ∞.
Liu et al 2012, Xue & Zou 2012 study estimation of the
precision matrix Σ−1 of Gaussian copulas in high
dimensions.
b − Σkmax is a key step of the study.
Sharp bound on kΣ
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Proposed Research: Part I
First, we establish a sharp bound of the operator norm
b − Σk2 . (See also Han & Liu 2013.)
kΣ
It has often been observed (e.g., Demarta & McNeil 2004,
b is
Klüppelberg & Kuhn 2009) that the plug-in estimator Σ
not always positive semidefinite.
b − Σk2 quantifies the extent to which the
A bound on kΣ
non-positive semidefinite problem may happen.
b is not positive semidefinite, we can project it onto the
If Σ
set of correlation matrices in some specific way to obtain
b + . Furthermore,
Σ
b + − Σk2 ≤ 2kΣ
b − Σk2 .
kΣ
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
Copula, elliptical copula, copula correlation matrix
Proposed research
Proposed Research: Part II
Later, we study a factor model for Σ.
The factor model assumes that Σ = Θ + V for a low-rank
(or nearly low-rank) matrix Θ and a diagonal matrix V .
X and [L, V 1/2 ]Y have the same copula for some elliptical
distribution with Σ = Ir +d . Here Θ = LL0 for some L ∈ Rd×r .
Within this context, the effective number of parameters is
O(r · d), for r = rank(Θ), instead of O(d 2 ) under the
generic model.
e for Σ using a penalized
We propose a refined estimator Σ
least square procedure.
b − Σk2 serves to scale the penalty term.
The bound on kΣ
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
b − Tk
Bounding kT
2
b − Σk
Bounding kΣ
2
b − Σk2
Outline for bounding kΣ
b − T k2 (matrix counterpart of Hoeffding 1963’
Bounding kT
classical bound for scalar U-statistic).
b − T k2 (matrix
b − Σk2 in terms of kT
Bounding kΣ
counterpart of the Lipschitz property).
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
b − Tk
Bounding kT
2
b − Σk
Bounding kΣ
2
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
b − Tk
Bounding kT
2
b − Σk
Bounding kΣ
2
Matrix U-statistic
Theorem
With probability at least 1 − α, we have
np
o
2
b − T k2 < max
kT k2 fn,d,α , fn,d,α
kT
q
2
2
b k2 f 2
≤ kT
n,d,α + fn,d,α /4 + fn,d,α /2
o
np
2
2
+ fn,d,α
kT k2 fn,d,α , fn,d,α
< max
with
s
fn,d,α =
16 d · log(2α−1 d)
·
.
3
n
The above bounds hold for all n, d, α.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
b − Tk
Bounding kT
2
b − Σk
Bounding kΣ
2
Matrix U-statistic: proof
We need a Bernstein-type inequality,
e.g.,oTropp (2012), to
n
b
bound the tail probability P kT − T k2 ≥ t .
That theorem, as many other results on bounding the tail
probability of the operator norm of a sum of random
matrices, requires that the summands be independent.
b does not satisfy this condition.
The U-statistics T
Many results on tail bound rely on the Laplace transform
technique to convert the tail probability into an expectation
b − T.
of some convex function of T
A technique pointed out by Hoeffding (1963) allows us to
b − T k2 into a problem
convert the problem of bounding kT
involving a sum of independent random matrices.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
b − Tk
Bounding kT
2
b − Σk
Bounding kΣ
2
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
b − Tk
Bounding kT
2
b − Σk
Bounding kΣ
2
Matrix Lipschitz property
Theorem
We have, with probability at least 1 − α, that
b − T k2 +
b − Σk2 ≤ πkT
kΣ
3π 2 2
·f
.
16 n,d,α
b − T k2 , we have, with
Combining earlier bound for kT
probability at least 1 − 2α, that
b − Σk2 < π · max
kΣ
o 3π 2
np
2
kT k2 fn,d,α , fn,d,α
+
· f2 .
16 n,d,α
In addition,
2
kΣk2 ≤ kT k2 ≤ kΣk2 .
π
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
b − Tk
Bounding kT
2
b − Σk
Bounding kΣ
2
Potential improvement
We can obtain from the corollary that, for n & d,
r
b − Σk2 . kΣk2 · d · log(d) .
EkΣ
n
b its sample covariance
In contrast, for Gaussian X and Σ
matrix, Koltchinskii & Lounici 2015 shows that, for n & d,
r
b − Σk2 kΣk2 · d .
EkΣ
n
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
A factor model for Σ
From now on, we assume that the copula correlation matrix Σ
satisfies a factor model:
Σ=Θ+V
for some low-rank (or nearly low-rank), positive semidefinite,
symmetric matrix Θ and diagonal matrix V .
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Prelude: an elementary factor model for Σ
In an elementary factor model for Σ = Θ + V , we assume that
rank(Θ) = r < d, and
V = σ 2 Id .
b have the eigen-decomposition
Let the plug-in estimator Σ
d
X
bk u
bk u
bk0 ,
λ
b1 ≥ · · · ≥ λ
bd .
λ
k =1
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Prelude: an elementary factor model for Σ
Construct the following closed-form estimators for r , σ 2 , Θ
and Σ, based on some threshold µ:
b
r =
d
n
o
X
1 λbk ≥ λbd + µ ,
k =1
σ
b2 =
1 Xb
λj ,
d −b
r
j>b
r
b =
Θ
b
r
X
bk − σ
bk u
bk0
(λ
b2 )u
k =1
e = Θ
b − diag(Θ)
b + I.
Σ
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Prelude: an elementary factor model for Σ, continued
Theorem
n
o
b − Σk2 < µ , we have
Assume λr (Θ) ≥ 2µ. On the event 2kΣ
b
r = r,
e − Σk2 ≤ kΘ
b − Θk2 ≤ 8r kΣ
b − Σk2 ,
kΣ
F
F
2
2
2
b
|b
σ − σ | ≤ kΣ − Σk2 .
Here, k · kF denotes the Frobenius norm.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Prelude: an elementary factor model for Σ, continued
Corollary
Set
q
2
4
b k2 f 2
µ = 8 kT
n,d,α + fn,d,α /4 + 8fn,d,α
p
2
.
µ̄ = 8 kT k2 fn,d,α + 12fn,d,α
2
Assume λr (Θ) ≥ 2µ̄, and kT k2 ≥ fn,d,α
. Then,
e − Σk2 ≤ 2r µ̄2 ,
kΣ
F
hold with probability exceeding 1 − 2α.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
e of Σ = Θ + V
The refined estimator Σ
In the factor model we can write
Σ = Θ − diag(Θ) + Id := Θo + Id .
e as the
This motivates us to estimate the low-rank matrix Θ by Θ
solution to the convex program
1
0
0
2
e = argmin
b o k + µkΘ k∗ .
Θ
kΘ − Σ
F
2 o
Θ0 ∈Rd×d
Here, k · k∗ denotes the nuclear norm.
e of the copula correlation
Finally, we let the refined estimator Σ
matrix Σ be
e =Θ
e o + Id .
Σ
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
e with existing estimators
Comparison of Σ
Saunderson et al 2012 study a factor model for Σ in the
noiseless setting using minimum trace factor analysis:

0

Σ = Θ + V
(Θ, V ) = argmin tr(Θ0 ) subject to Θ0 % 0


V diagonal
If Θ has column space U such that its coherence
√
max kPU ei k2 < 1/ 2,
1≤i≤d
then the above algorithm recovers Θ, V exactly.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
e with existing estimators, continued
Comparison of Σ
For Θ with effective low-rank and without sparse
corruption, Lounici 2013 proposes
1 0 b 2
0
e
Θ = argmin
kΘ − ΘkF + µkΘ k∗
2
Θ0 ∈Rd×d
b of Θ (which we lack).
for an unbiased initial estimator Θ
For Σ = Θ + S with low-rank matrix Θ and generic sparse
matrix S, Chandrasekaran et al 2012, Hsu et al 2011
propose
e = argmin 1 kΘ0 + S − Σk
e S)
b 2 + µkΘ0 k∗ + λkSk` .
(Θ,
F
1
2
Θ0 ,S∈Rd×d
This formulation requires λ > 0.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
e with existing estimators, continued
Comparison of Σ
We are in between the above scenarios. We will extend the
result of the former and relax the condition of the latter with
more precise analysis.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
e
Oracle inequality for the refined estimator Σ
Theorem
Let Θr be best rank-r approximation to Θ, with reduced SVD
Θr = Ur Dr Ur0 . Set
q
2
4
b k2 f 2
µ = 10 kT
n,d,α + fn,d,α /4 + 10fn,d,α ,
i
hp
2
2
kT k2 fn,d,α , fn,d,α
+ 15fn,d,α
.
µ̄ = 10 max
Then, with probability at least 1 − 2α,
X
e − Σk2 ≤
kΣ
λ2j (Θ) + 18r µ̄2
F
j>r
for all r with γr = kUr Ur0 kmax ≤ 1/9.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Outline
1
Introduction
Copula, elliptical copula, copula correlation matrix
Proposed research
2
b −Σ
Bounding the operator norm of Σ
b
Bounding kT − T k2
b − Σk2
Bounding kΣ
3
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Simulation result
b − Σk2
kΣ
F
, which our
e − Σk2
kΣ
F
theory predicts to be roughly proportional to d/r as n → ∞.
The object of interest is the ratio
We (somewhat arbitrarily) set C = 0.1, α = 0.1, and forgo
the log(d) factor when choosing the regularization
parameter.
We estimate this ratio by simulation.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Simulation setup
kΣk2 = 2.
10 3
(Larger is better)
e ! 'k2
b ! 'k2 =j'
k'
F
F
10 2
d=128, r=1
10 1
d=32, r=1
10 0
d=128, r=4
d=8, r=1
d=32, r=4
10 -1
d=8, r=4
10 -2
10 2
10 3
Sample size
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Summary
We have obtained a sharp bound on the operator norm of
b − Σ, for the plug-in estimator Σ
b of the copula correlation
Σ
matrix Σ.
We have applied the above bound to the setting of a factor
e a sharp oracle
model of Σ to obtain a refined estimator Σ;
e
inequality for Σ is established.
Marten Wegkamp
Copula correlation matrix for elliptical copulas
Introduction
b−Σ
Bounding the operator norm of Σ
Analyzing a factor model for the copula correlation matrix
The factor model for Σ
Study of an elementary factor model
Analysis of the refined estimator
Simulations
Thanks!
Marten Wegkamp
Copula correlation matrix for elliptical copulas