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Efficient goodness-of-fit tests in multi-dimensional vinecopula models
Ulf Schepsmeier
Technische Universitat MunchenLehrstuhl fur Mathematische Statistik
January 5, 2014International Workshop on High-Dimensional Dependence and Copulas:
Theory, Modeling, and Applications, Beijing, China
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 1 / 23
Motivation
Vines: very flexible class of multivariate copulas
We can do estimation and selection for vine copulas
But: Little validation tools for vine copulas are available
So far: likelihood, AIC, BIC, Vuong-test, Clarke-test
Goodness-of-fit tests for bivariate copulas
New: Goodness-of-fit test for vine copulas
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 2 / 23
Motivation
Vines: very flexible class of multivariate copulas
We can do estimation and selection for vine copulas
But: Little validation tools for vine copulas are available
So far: likelihood, AIC, BIC, Vuong-test, Clarke-test
Goodness-of-fit tests for bivariate copulas
New: Goodness-of-fit test for vine copulas
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 2 / 23
Motivation
Vines: very flexible class of multivariate copulas
We can do estimation and selection for vine copulas
But: Little validation tools for vine copulas are available
So far: likelihood, AIC, BIC, Vuong-test, Clarke-test
Goodness-of-fit tests for bivariate copulas
New: Goodness-of-fit test for vine copulas
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 2 / 23
Motivation
Vines: very flexible class of multivariate copulas
We can do estimation and selection for vine copulas
But: Little validation tools for vine copulas are available
So far: likelihood, AIC, BIC, Vuong-test, Clarke-test
Goodness-of-fit tests for bivariate copulas
New: Goodness-of-fit test for vine copulas
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 2 / 23
1 Motivation
2 Short introduction to R-vines
3 Goodness-of-fit tests for R-vines
4 Power study
5 Application
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 3 / 23
Copulas
Consider d random variables X = (X1, ...,Xd) with
density function distribution function
marginal fi (xi ), i = 1, ..., d Fi (xi ), i = 1, ..., d
joint f (x1, ..., xd) F (x1, ..., xd)
conditional fi |j(xi |xj), i 6= j Fi |j(xi |xj), i 6= j
Copula
A d-dimensional copula C is a multivariate distribution on [0, 1]d withuniformly distributed marginals.
Copula density function: c(u1, ..., ud) := ∂d
∂u1...∂udC (u1, ..., ud)
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 4 / 23
Sklar’s theorem
Theorem (Sklar 1959)
F (x1, ..., xd) = C (F1(x1), ...,Fd(xd))
f (x1, ..., xd) = c(F1(x1), ...,Fd(xd))f1(x1)...fd(xd)
for some d-dimensional copula C .
d = 2 :f (x1, x2) = c12(F1(x1),F2(x2))f1(x1)f2(x2)
f2|1(x2|x1) = c12(F1(x1),F2(x2))f2(x2)
cij ;D is bivariate copula pdf associated with (Xi ,Xj) given XD .
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 5 / 23
Pair-copula construction in 3 dimensions
One possible decomposition of f (x1, x2, x3) is:
f (x1, x2, x3) = f3|12(x3|x1, x2)f2|1(x2|x1)f1(x1)
We can represent the density f asa product of copula densities and marginal densities!
f2|1(x2|x1) = c12(F1(x1),F2(x2))f2(x2)
f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2),F3|2(x3|x2))f3|2(x3|x2)
f3|2(x3|x2) = c23(F2(x2),F3(x3))f3(x3)
⇒ f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2),F3|2(x3|x2))c23(F2(x2),F3(x3))f3(x3)
f (x1, x2, x3) = f3(x3)f2(x2)f1(x1) (marginals)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) (unconditional pairs)
× c13;2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 6 / 23
Pair-copula construction in 3 dimensions
One possible decomposition of f (x1, x2, x3) is:
f (x1, x2, x3) = f3|12(x3|x1, x2)f2|1(x2|x1)f1(x1)
We can represent the density f asa product of copula densities and marginal densities!
f2|1(x2|x1) = c12(F1(x1),F2(x2))f2(x2)
f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2),F3|2(x3|x2))f3|2(x3|x2)
f3|2(x3|x2) = c23(F2(x2),F3(x3))f3(x3)
⇒ f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2),F3|2(x3|x2))c23(F2(x2),F3(x3))f3(x3)
f (x1, x2, x3) = f3(x3)f2(x2)f1(x1) (marginals)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) (unconditional pairs)
× c13;2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 6 / 23
Pair-copula construction in 3 dimensions
One possible decomposition of f (x1, x2, x3) is:
f (x1, x2, x3) = f3|12(x3|x1, x2)f2|1(x2|x1)f1(x1)
We can represent the density f asa product of copula densities and marginal densities!
f2|1(x2|x1) = c12(F1(x1),F2(x2))f2(x2)
f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2),F3|2(x3|x2))f3|2(x3|x2)
f3|2(x3|x2) = c23(F2(x2),F3(x3))f3(x3)
⇒ f3|12(x3|x1, x2) = c13;2(F1|2(x1|x2),F3|2(x3|x2))c23(F2(x2),F3(x3))f3(x3)
f (x1, x2, x3) = f3(x3)f2(x2)f1(x1) (marginals)
× c12(F1(x1),F2(x2)) · c23(F2(x2),F3(x3)) (unconditional pairs)
× c13;2(F1|2(x1|x2),F3|2(x3|x2))(conditional pair)
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 6 / 23
R-vine structure (d = 5)
1 3 4
2 5
1,3 3,4
1,2
1,5
T1
1,2 1,3 3,4
1,5
2,3;1 1,4;3
3,5;1 T2
2,3;1 1,4;3 3,5;12,4;1,3 4,5;1,3
T3
2,4;1,3 4,5;1,32,5;1,3,4
T4
Pair-copulas:
1 c12, c13, c34, c15
2 proximity condition If two
nodes are joined by an edge in tree
i + 1, the corresponding edges in tree
i share a node.
3 c23;1, c14;3, c35;1
4 c24;13, c45;13
5 c25;134
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 7 / 23
Goodness-of-fit tests for R-vines
H0 : −H = C versus H1 : −H 6= C H0 : C ∈ C0 versus H1 : C /∈ C0
multivariate PITut ∼ C 7→ yt ∼ C⊥
t = 1, . . . , T
test statistic based onempirical copula process using u
(ECP test)
original aggregationy1, . . . ,yn 7→ s1, . . . , sn
st =∑di=1 Γ(yti), t = 1, . . . , n
ordered aggregationyT(1), . . . ,y
T(d) 7→ s1, . . . , sn
st =∑di=1 Γ(yt(i)), t = 1, . . . , n
test statisticbased on H + C(White test)
test statisticbased on C−1H
(IR test)
Γ(·) = Φ−1(·)2(Breymann tests)
Γ(·) = | · −0.5|(Berg tests)
Γ(·) = (· − 0.5)α
α = 2, 4, . . .(Berg2 tests)
test statistic based onempirical copula process using y
(ECP2 test)mCvM mKS
mCvM mKS
AD CvM KS AD CvM KS AD CvM KS
Legend:
AD univariate Anderson-Darling test CvM univariate Cramer-von Mises test KS univariate Kolmogoroov-Smirnov test
mCvM multivariate Cramer-von Mises test mKS multivariate Kolmogorov-Smirnov test
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 8 / 23
The misspecification test (White 1982)
Let U = (U1, . . . ,Ud)T ∈ [0, 1]d be a d-dimensional random vector withcopula distribution Cθ(u1, . . . , ud). Then
H(θ) = E[∂2θ `(θ|U)
],
C(θ) = E[∂θ`(θ|U)
(∂θ`(θ|U)
)T ]
are the expected Hessian matrix and the expected outer product ofgradient, respectively, and `(θ|u) := ln(cθ(u1, . . . , ud))Under correct model specification (θ = θ0)
−H(θ0) = C(θ0)
⇒ Test problem:
H0 : H(θ0) + C(θ0) = 0 against H1 : H(θ0) + C(θ0) 6= 0,
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 9 / 23
Test statistic
Proposition (Huang and Prokhorov 2013; Schepsmeier 2013b)
Under correct vine copula specification, given margins and suitableregularity conditions (A1-A10 in White 1982) the information matrix teststatistic is
T = n(d (θ)
)TV−1
θnd (θ), (1)
where V−1
θnis a consistent estimate for the inverse asymptotic covariance
matrix. Further, T is asymptotically χ2p(p+1)/2 distributed.
d (θ) := vech(H(θ) + C(θ)), ∇Dθ := E [∂θkd `(θ|ut)]`=1,...,p(p+1)
2,k=1,...,p
,
Vθ0 := E
[(d(θ0|ut)−∇Dθ0H(θ0)
−1∂θ0`(θ0|ut))(d(θ0|ut)−∇Dθ0H(θ0)
−1∂θ0`(θ0|ut))T]
,
where Vθ0 is the asymptotic covariance matrix of√nd (θn)
vech() vectorizes the lower triangular of a matrix (including the diagonal).
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 10 / 23
Information matrix ratio test (IR test)
Zhou et al. (2012) followed a different approach: Information matrix ratio:
Ψ(θ) := −H(θ)−1C(θ)
⇒ Test problem:
H0 : Ψ(θ0) = Ip against H1 : Ψ(θ0) 6= Ip,
where Ip is the p-dimensional identity matrix.Ψ(θ0) corresponds to the information ratio (IR) statistic
IR := tr(Ψ(θ0))/p.
⇒ Test problem:
H0 : IR = 1 against H1 : IR 6= 1,
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 11 / 23
Information matrix ratio test (IR test)
Zhou et al. (2012) followed a different approach: Information matrix ratio:
Ψ(θ) := −H(θ)−1C(θ)
⇒ Test problem:
H0 : Ψ(θ0) = Ip against H1 : Ψ(θ0) 6= Ip,
where Ip is the p-dimensional identity matrix.Ψ(θ0) corresponds to the information ratio (IR) statistic
IR := tr(Ψ(θ0))/p.
⇒ Test problem:
H0 : IR = 1 against H1 : IR 6= 1,
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 11 / 23
Information matrix ratio test (IR test)
Further letW := (W1, . . . ,Wp(p+1))T = (vech(C(θ)), vech(H(θ)))T ∈ Rp(p+1), thenPresnell and Boos (2004) showed that
Σ−1/2W
√n(W − µW )
d→ Np(p+1)(0p(p+1), Ip(p+1)),
where µW is the mean vector and ΣW is the asymptotic covariance matrixof W .Furthermore let
D(θ) :=
(∂IR
∂Wi
)
i=1,...,p(p+1)
∈ Rp(p+1).
the partial derivatives of IR with respect to the components of W .
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 12 / 23
Test statistic
Proposition (Presnell and Boos 2004; Schepsmeier 2013a)
Let U ∼ RV (V,B(V), θ(B(V))) satisfy some regularity conditionsSchepsmeier (2013a). Then the IR test statistic
Zn :=IRn − 1
σIR
d→ N(0, 1) as n→∞,
where σIR is the standard error of the IR test statistic, defined as
σ2IR :=
1
nDTΣWD.
Here ΣW is the asymptotic covariance matrix of W , andD := D(θn)|
θnP→θ0
.
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 13 / 23
Rosenbatt’s transform tests
u11 . . . u1d
......
un1 . . . und
Rosenblatt−−−−−−→
(PIT )
y11 . . . y1d
......
yn1 . . . ynd
Aggregation−−−−−−−−→
st=∑d
j=1 Γ(ytj )
s1
...sn
univariate−−−−−−→
GOF tests
Breymann et al. (2003): Γ(ytj) = Φ−1(ytj)2
Berg and Bakken (2007): Γ(ytj) = |ytj − 0.5| orΓ(ytj) = (ytj − 0.5)α, α = (2, 4, . . .)
Use univariate Cramer-von Mises (CvM), Kolmogorov-Smirnov (KS) orAnderson-Darling (AD) GOF tests in the last step.New:
Probability integral transform (PIT) for Vines (Schepsmeier 2013a)
Extension to vine copulas
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 14 / 23
Rosenbatt’s transform tests
u11 . . . u1d
......
un1 . . . und
Rosenblatt−−−−−−→
(PIT )
y11 . . . y1d
......
yn1 . . . ynd
Aggregation−−−−−−−−→
st=∑d
j=1 Γ(ytj )
s1
...sn
univariate−−−−−−→
GOF tests
Breymann et al. (2003): Γ(ytj) = Φ−1(ytj)2
Berg and Bakken (2007): Γ(ytj) = |ytj − 0.5| orΓ(ytj) = (ytj − 0.5)α, α = (2, 4, . . .)
Use univariate Cramer-von Mises (CvM), Kolmogorov-Smirnov (KS) orAnderson-Darling (AD) GOF tests in the last step.New:
Probability integral transform (PIT) for Vines (Schepsmeier 2013a)
Extension to vine copulas
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 14 / 23
Empirical copula process testsLet Cθn(u) be the copula distribution function with estimated parameter θ
and the empirical copula Cn(v) = 1n+1
∑nt=1 1{ut1≤v1,...,utd≤vd}
ECP-mCvM: nω2ECP := n
∫
[0,1]d(
empirical copula process︷ ︸︸ ︷Cn(u)− Cθn(u) )2dCn(u) and
ECP-mKS: Dn,ECP := supu∈[0,1]d
|Cn(u)− Cθn(u)|.
Problem: Need double bootstrap procedure to estimate p-valuesVariant: use transformed data y = (y1, . . . , yd) of the PIT approach:
ECP2-mCvM: nω2ECP2 := n
∫
[0,1]d(Cn(y)− C⊥(y))2dCn(y) and
ECP2-mKS: Dn,ECP2 := supy∈[0,1]d
|Cn(y)− C⊥(y)|,
Advantage: Calculation of the independence copula C⊥(y) is easy.
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 15 / 23
Empirical copula process testsLet Cθn(u) be the copula distribution function with estimated parameter θ
and the empirical copula Cn(v) = 1n+1
∑nt=1 1{ut1≤v1,...,utd≤vd}
ECP-mCvM: nω2ECP := n
∫
[0,1]d(
empirical copula process︷ ︸︸ ︷Cn(u)− Cθn(u) )2dCn(u) and
ECP-mKS: Dn,ECP := supu∈[0,1]d
|Cn(u)− Cθn(u)|.
Problem: Need double bootstrap procedure to estimate p-valuesVariant: use transformed data y = (y1, . . . , yd) of the PIT approach:
ECP2-mCvM: nω2ECP2 := n
∫
[0,1]d(Cn(y)− C⊥(y))2dCn(y) and
ECP2-mKS: Dn,ECP2 := supy∈[0,1]d
|Cn(y)− C⊥(y)|,
Advantage: Calculation of the independence copula C⊥(y) is easy.
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 15 / 23
Simulation study - Test models
T1
2
1
3
4 5
1,2 1,3
1,4
4,5
T2
1,2 1,3
1,4
4,5
2, 4; 1 3,4;1
1,5;4
T3
2, 4; 1 1, 5; 4
3, 4; 12, 3; 1, 4 3,
5;1,4
T4
2, 3; 1, 4
3, 5; 1, 4
2,5;1,3,4
Mixed Kendall’s τ in a range from 0.1 to 0.75
R-package: VineCopula
Null hypothesis: M1 = R-Vine
Alternatives: M2 = C-vine, D-vine or multivariate Gauss copula
Most interesting properties are size and power.
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 16 / 23
Power study - ResultsAll results are based on bootstrapped p-values!
● WhiteIR
Breymann−ADBreymann−CvMBreymann−KS
Berg−ADBerg−CvMBerg−KS
Berg2−ADBerg2−CvMBerg2−KS
ECP−mCvMECP−mKS
ECP2−mCvMECP2−mKS
●
●
●
●
model
pow
er
R−vine D−vine C−vine Gauss
0.05
0.15
0.30
0.45
0.60
0.75
0.90
(a) d = 5, n = 500
●
●
●
●
model
pow
er
R−vine D−vine C−vine Gauss
0.05
0.15
0.30
0.45
0.60
0.75
0.90
(b) d = 5, n = 750
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 17 / 23
Power study - ResultsAll results are based on bootstrapped p-values!
● WhiteIR
Breymann−ADBreymann−CvMBreymann−KS
Berg−ADBerg−CvMBerg−KS
Berg2−ADBerg2−CvMBerg2−KS
ECP−mCvMECP−mKS
ECP2−mCvMECP2−mKS
●
●
●
●
model
pow
er
R−vine D−vine C−vine Gauss
0.05
0.15
0.30
0.45
0.60
0.75
0.90
(c) d = 5, n = 1000
●
● ● ●
model
pow
er
R−vine D−vine C−vine Gauss
0.05
0.15
0.30
0.45
0.60
0.75
0.90
(d) d = 5, n = 2000
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 18 / 23
Power study - Results
All proposed GOF tests maintain their given size.
Power > size for White, IR, ECP and ECP2 ⇒ good performancein mean; Breymann undecided
Power < size for Berg and Berg2 ⇒ poor performance
↑ number of observations ⇒ ↑ power
Given sufficient data points the White, IR and ECP2 tests areconsistent
Power seems to depend on the alternative
For ECP, ECP2 or Breymann tests CvM performs better than KS.
Not in the plots
↑ Kendall’s τ ⇒ ↑ power
↑ dimension d ⇒ no influence on the top performing GOF tests(White, IR, ECP, ECP2)
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 19 / 23
Power study - Results
All proposed GOF tests maintain their given size.
Power > size for White, IR, ECP and ECP2 ⇒ good performancein mean; Breymann undecided
Power < size for Berg and Berg2 ⇒ poor performance
↑ number of observations ⇒ ↑ power
Given sufficient data points the White, IR and ECP2 tests areconsistent
Power seems to depend on the alternative
For ECP, ECP2 or Breymann tests CvM performs better than KS.
Not in the plots
↑ Kendall’s τ ⇒ ↑ power
↑ dimension d ⇒ no influence on the top performing GOF tests(White, IR, ECP, ECP2)
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 19 / 23
Application: Indices and volatility indices
Daily log returns of 4 major stock indices and their correspondingvolatility indices.• German DAX and VDAX-NEW
• European EuroSTOXX50 and VSTOXX
• US S&P500 and VIX
• Swiss SMI and VSMI
Observed from August, 9th, 2007 until April 30th, 2013 (currentfinancial crisis; 1405 observations).
Time series are filtered using AR(1)-GARCH(1,1) with Student’s tinnovations.
Data set of standardized residuals transformed to [0,1].
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 20 / 23
Application
DAX
0.0 0.4 0.8
xy
x
y
0.0 0.4 0.8
x
y
x
y
0.0 0.4 0.8
x
y
x
y
0.0 0.4 0.8
0.0
0.4
0.8
x
y
0.0
0.4
0.8 0.01
0.0
1775
51
STOXX50
xy
x
y
x
y
x
y
x
y
x
y
0.01 0.01
0.0
1775
51
SMI
xy
x
y
x
y
x
y
0.0
0.4
0.8
x
y
0.0
0.4
0.8 0.01 0.01 0.01
S&P500
xy
x
y
x
y
x
y
0.01
0.01387755
0.01
0.01387755
0.01
0.01387755
0.01 0.01387755 VDAX−NEW
xy
x
y
0.0
0.4
0.8
x
y
0.0
0.4
0.8
0.01
0.01387755
0.01
0.01387755
0.01 0.01 0.01387755
0.01
0.0
1775
51
VSTOXX
xy
x
y
0.01
0.01387755
0.01
0.01387755
0.01
0.01387755
0.01 0.01387755
0.01
0.01387755
0.01
0.01387755
VSMI
0.0
0.4
0.8
xy
0.0 0.4 0.8
0.0
0.4
0.8
0.01
0.01387755
0.01
0.01387755
0.0 0.4 0.8
0.01
0.0177551
0.01
0.01387755
0.0 0.4 0.8
0.01
0.01387755
0.01
0.01387755
0.0 0.4 0.8
0.01
0.01387755
VIX
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 21 / 23
Application - Results
log-lik #par AIC BIC
R-vine 7652 33 -15238 -15065C-vine 7585 42 -15086 -14865D-vine 7654 41 -15226 -15011Gauss 7320 28 -14584 -14445
White ECP ECP2 IR
CvM KS CvM KS
R-vine 0.002 0.18 0.98 0.30 0.67 0.75C-vine 0.14 0.51 0.36 0.01 < 0.01 0.74D-vine 0.41 0.82 0.24 0.55 0.67 0.52Gauss < 0.01 0.60 0.28 < 0.01 < 0.01 < 0.01
Tabelle: Bootstrapped p-values of the White, ECP, ECP2 and IR goodness-of-fittest for the 4 considered (vine) copula models
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 22 / 23
Summary/Outlook
Summary
New GOF tests for vine copulas are introduced
For comparison further GOF tests were extended from the bivariatecase to the vine copula case
Information matrix based GOF tests perform very well
Empirical copula process based GOF tests work well and are fast
Outlook
Extension of further copula GOF tests to the vine copula case, e.g.tests based on Kendall’s process, likelihood ratio based tests,...
Hybrid approach of Zhang et al. (2013)
GOF tests for classes of vines (here the vine structure was fixed)
Thank you for your attention!
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 23 / 23
Summary/Outlook
Summary
New GOF tests for vine copulas are introduced
For comparison further GOF tests were extended from the bivariatecase to the vine copula case
Information matrix based GOF tests perform very well
Empirical copula process based GOF tests work well and are fast
Outlook
Extension of further copula GOF tests to the vine copula case, e.g.tests based on Kendall’s process, likelihood ratio based tests,...
Hybrid approach of Zhang et al. (2013)
GOF tests for classes of vines (here the vine structure was fixed)
Thank you for your attention!
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 23 / 23
Summary/Outlook
Summary
New GOF tests for vine copulas are introduced
For comparison further GOF tests were extended from the bivariatecase to the vine copula case
Information matrix based GOF tests perform very well
Empirical copula process based GOF tests work well and are fast
Outlook
Extension of further copula GOF tests to the vine copula case, e.g.tests based on Kendall’s process, likelihood ratio based tests,...
Hybrid approach of Zhang et al. (2013)
GOF tests for classes of vines (here the vine structure was fixed)
Thank you for your attention!
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 23 / 23
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Schepsmeier, U. (2013b).A goodness-of-fit test for regular vine copula models.preprint, available at: http://arxiv.org/abs/1306.0818.
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Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 23 / 23
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Zhou, Q. M., P. X.-K. Song, and M. E. Thompson (2012).Information ratio test for model misspecification in quasi-likelihood inference.Journal of the American Statistical Association 107(497), 205–213.
Schepsmeier (TUM) Gof-tests for R-vines January 5, 2014, Beijing 23 / 23