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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Nonparametric estimation of copuladensities
A. Charpentier1, J.D. Fermanian2 & O. Scaillet3
Gemeinsame Jahrestagung der Deutschen Mathematiker-Vereinigung und derGesellschaft für Didaktik der Mathematik,
Multivariate Dependence Modelling using Copulas - Applications in Finance
Humboldt-Universität zu Berlin, March 2007
1ensae-ensai-crest & Katholieke Universiteit Leuven, 2CoopeNeff A.M. & crest et 3HEC Genève & FAME
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
One (short) word on copulasDefinition 1. A 2-dimensional copula is a 2-dimensional cumulativedistribution function restricted to [0, 1]2 with standard uniform margins.
Copula (cumulative distribution function) Level curves of the copula
Copula density Level curves of the copula
If C is twice differentiable, let c denote the density of the copula.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Theorem 2. (Sklar) Let C be a copula, and FX and FY two marginaldistributions, then F (x, y) = C(FX(x), FY (y)) is a bivariate distributionfunction, with F ∈ F(FX , FY ).
Conversely, if F ∈ F(FX , FY ), there exists C such thatF (x, y) = C(FX(x), FY (y). Further, if FX and FY are continuous, then C isunique, and given by
C(u, v) = F (F−1X (u), F−1
Y (v)) for all (u, v) ∈ [0, 1]× [0, 1]
We will then define the copula of F , or the copula of (X, Y ).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Motivation
Example Loss-ALAE: consider the following dataset, were the Xi’s are lossamount (paid to the insured) and the Yi’s are allocated expenses. Denote byRi and Si the respective ranks of Xi and Yi. Set Ui = Ri/n = FX(Xi) andVi = Si/n = FY (Yi).
Figure 1 shows the log-log scatterplot (log Xi, log Yi), and the associate copulabased scatterplot (Ui, Vi).
Figure 2 is simply an histogram of the (Ui, Vi), which is a nonparametricestimation of the copula density.
Note that the histogram suggests strong dependence in upper tails (theinteresting part in an insurance/reinsurance context).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
1 2 3 4 5 6
12
34
5
Log−log scatterplot, Loss−ALAE
log(LOSS)
log(A
LA
E)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Copula type scatterplot, Loss−ALAE
Probability level LOSSP
robabili
ty le
vel A
LA
E
Figure 1: Loss-ALAE, scatterplots (log-log and copula type).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Figure 2: Loss-ALAE, histogram of copula type transformation.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Basic notions on nonparametric estimation of densities
Let {X1, ..., Xn} denote an i.i.d. sample with density f .
The standard kernel estimate of a density is,
f(x) =1
nh
n∑
i=1
K
(x−Xi
h
),
where K is a kernel function (e.g. K(ω) = I(|ω| ≤ 1)/2 for the movinghistogram).
If {X1, ..., Xn} of positive random variable (Xi ∈ [0,∞[). Let K denote asymmetric kernel, then
E(f(0, h) =12f(0) + O(h)
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Exponential random variables
Den
sity
−1 0 1 2 3 4 5 6
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Exponential random variables
Den
sity
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0.0
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1.0
1.2
Figure 3: Density estimation of an exponential density, 100 points.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
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Kernel based estimation of the uniform density on [0,1]
Densi
ty
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Kernel based estimation of the uniform density on [0,1]
D
ensi
ty
Figure 4: Density estimation of an uniform density on [0, 1].
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
How to get a proper estimation on the border
Several techniques have been introduce to get a better estimation on theborder,
• boundary kernel (Müller (1991))
• mirror image modification (Deheuvels & Hominal (1989), Schuster(1985))
• transformed kernel (Devroye & Györfi (1981), Wand, Marron &Ruppert (1991))
In the particular case of densities on [0, 1],
• Beta kernel (Brown & Chen (1999), Chen (1999, 2000)),
• average of histograms inspired by Dersko (1998).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
The mirror image modification
The idea is to fit a density to the enlarged sample −X1, ...,−Xn, X1, ..., Xn
(for a positive distribution) and −X1, ...,−Xn, X1, ..., Xn, 2−X1, ..., 2−Xn
(for a distribution with support [0, 1]).
The estimator of the density is
f(x, h) =1
nh
n∑
i=1
K
(x−Xi
h
)+ K
(x + Xi
h
).
Remark: It works well when the density f has a null derivative in 0.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
−1.0 −0.5 0.0 0.5 1.0 1.5 2.0
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Density estimation using mirror image
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Density estimation using mirror image
Figure 5: The mirror image principle.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
The transformed Kernel estimate
The idea was developed in Devroye & Györfi (1981) for univariatedensities.
Consider a transformation T : R→ [0, 1] strictly increasing, continuouslydifferentiable, one-to-one, with a continuously differentiable inverse.
Set Y = T (X). Then Y has density
fY (y) = fX(T−1(y)) · (T−1)′(y).
If fY is estimated by fY , then fX is estimated by
fX(x) = fY (T (x)) · T ′(x).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
−3 −2 −1 0 1 2 3
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0.1
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Density estimation transformed kernel
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0.0
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Density estimation transformed kernel
Figure 6: The transform kernel principle (with a Φ−1-transformation).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
The Beta Kernel estimate
The Beta-kernel based estimator of a density with support [0, 1] at point x, isobtained using beta kernels, which yields
f(x) =1n
n∑
i=1
K
(Xi,
u
b+ 1,
1− u
b+ 1
)
where K(·, α, β) denotes the density of the Beta distribution with parametersα and β,
K(x, α, β) =Γ(α + β)Γ(α)Γ(β)
xα(1− x)β−11(x ∈ [0, 1]).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
02
46
810
Gaussian Kernel approach
0.0 0.2 0.4 0.6 0.8 1.0
02
46
810
Beta Kernel approach
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0.0
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1.0
1.5
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Density estimation using beta kernels
Figure 7: The beta-kernel estimate.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
The average of histogramsIn a very general context Dersko (1998) introduced the intrinsic estimator ofthe density, as the average of all possible histograms. Consider the case of adensity on the unit interval [0, 1]. Let Jm denote the set of all partitions of[0, 1) into m+1 subintervals. Define fm(·) as the average of all the histograms,
fm(·) =1
#Jm
∑
Im∈Jm
fIm(x), (1)
as in Dersko (1998).
Let Im = (I0:m, I1:m, I2:m, . . . , Im:m) a partition of [0, 1) into m + 1subintervals, I =
⋃i Ii:m. Set Ii:m = [ui−1:m, ui:m), where
u0:m = 0 < u1:m < · · · < uk:m < uk+1:m = 1. The histogram of X1, . . . , Xn
associated to Ik is
fIm(x) =1
n · [ui:m − ui−1:m]
n∑
j=1
1(Xi ∈ Ii:m), where x ∈ Ii:m.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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Average of histograms, m=1
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Average of histograms, m=1
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Average of histograms, m=1
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Average of histograms, m=1
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Density as average of randomized histograms, n=250, m=1
Figure 8: Average of histograms, m = 1.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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Average of histograms, m=2
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Average of histograms, m=2
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Average of histograms, m=2
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Average of histograms, m=2
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Density as average of randomized histograms, n=250, m=2
Figure 9: Average of histograms, m = 2.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
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Average of histograms, m=5
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Average of histograms, m=5
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Average of histograms, m=5
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Average of histograms, m=5
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Density as average of randomized histograms, n=250, m=5
Figure 10: Average of histograms, m = 5.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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Density as average of randomized histograms, n=250, m=10
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Density as average of randomized histograms, n=250, m=20
Figure 11: Average of histograms, m = 10 and m = 20.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
02
46
810
Gaussian Kernel approach
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02
46
810
Beta Kernel approach
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02
46
810
Average of histograms
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10
Average of histograms
Figure 12: Shape of the induced kernel based on uniform averaging.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Copula density estimation: the boundary problem
Let (U1, V1), ..., (Un, Vn) denote a sample with support [0, 1]2, and with densityc(u, v), which is assumed to be twice continuously differentiable on (0, 1)2.
If K denotes a symmetric kernel, with support [−1,+1], then for all(u, v) ∈ [0, 1]× [0, 1], in any corners (e.g. (0, 0))
E(c(0, 0, h)) =14· c(u, v)− 1
2[c1(0, 0) + c2(0, 0)]
∫ 1
0
ωK(ω)dω · h + o(h).
on the interior of the borders (e.g. u = 0 and v ∈ (0, 1)),
E(c(0, v, h)) =12· c(u, v)− [c1(0, v)]
∫ 1
0
ωK(ω)dω · h + o(h).
and in the interior ((u, v) ∈ (0, 1)× (0, 1)),
E(c(u, v, h)) = c(u, v) +12[c1,1(u, v) + c2,2(u, v)]
∫ 1
−1
ω2K(ω)dω · h2 + o(h2).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
On borders, there is a multiplicative bias and the order of the bias is O(h)(while it is O(h2) in the interior).
If K denotes a symmetric kernel, with support [−1,+1], then for all(u, v) ∈ [0, 1]× [0, 1], in any corners (e.g. (0, 0))
V ar(c(0, 0, h)) = c(0, 0)(∫ 1
0
K(ω)2dω
)2
· 1nh2
+ o
(1
nh2
).
on the interior of the borders (e.g. u = 0 and v ∈ (0, 1)),
V ar(c(0, v, h)) = c(0, v)(∫ 1
−1
K(ω)2dω
)(∫ 1
0
K(ω)2dω
)· 1nh2
+ o
(1
nh2
).
and in the interior ((u, v) ∈ (0, 1)× (0, 1)),
V ar(c(u, v, h)) = c(u, v)(∫ 1
−1
K(ω)2dω
)2
d · 1nh2
+ o
(1
nh2
).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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Estimation of Frank copula
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Figure 13: Theoretical density of Frank copula.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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Estimation of Frank copula
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Figure 14: Estimated density of Frank copula, using standard Gaussian (inde-pendent) kernels, h = h∗.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Transformed kernel technique
Consider the kernel estimator of the density of the(Xi, Yi) = (G−1(Ui), G−1(Vi))’s, where G is a strictly increasing distributionfunction, with a differentiable density. Since density f is continuous, twicedifferentiable, and bounded above, for all (x, y) ∈ R2, define
f(x, y) =1
nh2
n∑
i=1
K
(x−Xi
h
)K
(y − Yi
h
),
Sincef(x, y) = g(x)g(y)c[G(x), G(y)]. (2)
can be inverted in
c(u, v) =f(G−1(u), G−1(v))
g(G−1(u))g(G−1(v)), (u, v) ∈ [0, 1]× [0, 1], (3)
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
one gets, substituting f in (3)
c(u, v) =1
nh · g(G−1(u)) · g(G−1(v))
n∑
i=1
K
(G−1(u)−G−1(Ui)
h,G−1(v)−G−1(Vi)
h
),
(4)
Therefore,
E(c(u, v, h)) = c(u, v) +o(h)
g(G−1(u))g(G−1(v)).
Similarly,
V ar(c(u, v, h)) =1
g(G−1(u))g(G−1(v))
[c(u, v)nh2
(∫K(ω)2dω
)2]
+1
g(G−1(u))2g(G−1(v))2o
(1
nh2
).
Asymptotic normality and also be derived.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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Estimation of Frank copula
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Figure 15: Estimated density of Frank copula, using a Gaussian kernel, after aGaussian normalization.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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0
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Estimation of Frank copula
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Figure 16: Estimated density of Frank copula, using a Gaussian kernel, after aStudent normalization, with 5 degrees of freedom.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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0
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Estimation of Frank copula
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Figure 17: Estimated density of Frank copula, using a Gaussian kernel, after aStudent normalization, with 3 degrees of freedom.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Using the mirror reflection principleInstead of using only the (Ui, Vi)’s the mirror image consists in reflecting eachdata point with respect to all edges and corners of the unit square [0, 1]× [0, 1].Hence, additional observations can be considered, i.e. the (±Ui,±Vi)’s, the(±Ui, 2− Vi)’s, the (2− Ui,±Vi)’s and the (2− Ui, 2− Vi)’s. Hence, consider
c(u, v) =1
nh
n∑
i=1
K
(u− Ui
h
)K
(v − Vi
h
)+ K
(u + Ui
h
)K
(v − Vi
h
)+
K
(u− Ui
h
)K
(v + Vi
h
)+ K
(u + Ui
h
)K
(v + Vi
h
)+
K
(u− Ui
h
)K
(v − 2 + Vi
h
)+ K
(u + Ui
h
)K
(v − 2 + Vi
h
)+
K
(u− 2 + Ui
h
)K
(v − Vi
h
)+ K
(u− 2 + Ui
h
)K
(v + Vi
h
)+
K
(u− 2 + Ui
h
)K
(v − 2 + Vi
h
).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
In the case of uniformly continuous copula density c on [0, 1]× [0, 1], if K is abounded symmetric kernel that vanishes off [−1, 1], with ωK(ω) → 0 asω →∞, and if nh2 →∞ as h → 0 and n →∞, then f satisfies standardasymptotic properties, e.g. for every (u, v) ∈ [0, 1]× [0, 1]
E(c(u, v, h)) = c(u, v) + O(h) on the borders,
= c(u, v) + O(h2) in the interior,
and moreover, if σ2 =∫ 1
−1K(ω)2dω
V ar(c(u, v, h)) = 4c(u, v)(nh2)−1σ4 + o((nh2)−1) in corner,
= 2c(u, v)(nh2)−1σ4 + o((nh2)−1) in the interior of borders,
= c(u, v)(nh2)−1σ4 + o((nh2)−1) in the interior.
Furthermore, asymptotic normality can also be obtained,√
nh2 (c(u, v, h)− c(u, v)) L→ N(
0, κc(u, v)∫
K2(ω)dω
),
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
−1.0 −0.5 0.0 0.5 1.0 1.5 2.0
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−0.5
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1.0
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2.0
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Estimation of the copula density
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0.0
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1.0
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Estimation of the copula density
Figure 18: Copula density estimation - mirror reflection (n = 100, 1000).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
−1.0 −0.5 0.0 0.5 1.0 1.5 2.0
−1.0
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Estimation of the copula density (mirror reflection)
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Estimation of the copula density (mirror reflection)
Figure 19: Copula density estimation - mirror reflection (n = 100, 1000).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
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Estimation of the copula density (mirror reflection) Estimation of the copula density (mirror reflection)
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Figure 20: Estimated density of Frank copula, mirror reflection principle
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Bivariate Beta kernels
The Beta-kernel based estimator of the copula density at point (u, v), isobtained using product beta kernels, which yields
c(u, v) =1n
n∑
i=1
K
(Xi,
u
b+ 1,
1− u
b+ 1
)·K
(Yi,
v
b+ 1,
1− v
b+ 1
),
where K(·, α, β) denotes the density of the Beta distribution with parametersα and β.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Beta (independent) bivariate kernel , x=0.0, y=0.0 Beta (independent) bivariate kernel , x=0.2, y=0.0 Beta (independent) bivariate kernel , x=0.5, y=0.0
Beta (independent) bivariate kernel , x=0.0, y=0.2 Beta (independent) bivariate kernel , x=0.2, y=0.2 Beta (independent) bivariate kernel , x=0.5, y=0.2
Beta (independent) bivariate kernel , x=0.0, y=0.5 Beta (independent) bivariate kernel , x=0.2, y=0.5 Beta (independent) bivariate kernel , x=0.5, y=0.5
Figure 21: Shape of bivariate Beta kernels K(·, x/b+1, (1−x)/b+1)×K(·, y/b+1, (1− y)/b + 1) for b = 0.2.
38
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Assume that the copula density c is twice differentiable on [0, 1]× [0, 1]. Let(u, v) ∈ [0, 1]× [0, 1]. The bias of c(u, v) is of order b, i.e.
E(c(u, v)) = c(u, v) +Q(u, v) · b + o(b),
where the bias Q(u, v) is
Q(u, v) = (1−2u)c1(u, v)+(1−2v)c2(u, v)+12
[u(1− u)c1,1(u, v) + v(1− v)c2,2(u, v)] .
The bias here is O(b) (everywhere) while it is O(h2) using standard kernels.
Assume that the copula density c is twice differentiable on [0, 1]× [0, 1]. Let(u, v) ∈ [0, 1]× [0, 1]. The variance of c(u, v) is in corners, e.g. (0, 0),
V ar(c(0, 0)) =1
nb2[c(0, 0) + o(n−1)],
in the interior of borders, e.g. u = 0 and v ∈ (0, 1)
V ar(c(0, v)) =1
2nb3/2√
πv(1− v)[c(0, v) + o(n−1)],
39
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
and in the interior,(u, v) ∈ (0, 1)× (0, 1)
V ar(c(u, v)) =1
4nbπ√
v(1− v)u(1− u)[c(u, v) + o(n−1)].
Remark From those properties, an (asymptotically) optimal bandwidth b canbe deduced, maximizing asymptotic mean squared error,
b∗ ≡(
116πnQ(u, v)2
· 1√v(1− v)u(1− u)
)1/3
.
Note (see Charpentier, Fermanian & Scaillet (2005)) that all those results canbe obtained in dimension d ≥ 2.
Example For n = 100 simulated data, from Frank copula, the optimalbandwidth is b ∼ 0.05.
40
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Estimation of the copula density (Beta kernel, b=0.1) Estimation of the copula density (Beta kernel, b=0.1)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 22: Estimated density of Frank copula, Beta kernels, b = 0.1
41
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Estimation of the copula density (Beta kernel, b=0.05) Estimation of the copula density (Beta kernel, b=0.05)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 23: Estimated density of Frank copula, Beta kernels, b = 0.05
42
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4Standard Gaussian kernel estimator, n=100
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Standard Gaussian kernel estimator, n=1000
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Standard Gaussian kernel estimator, n=10000
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
Figure 24: Density estimation on the diagonal, standard kernel.
43
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4Transformed kernel estimator (Gaussian), n=100
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Transformed kernel estimator (Gaussian), n=1000
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Transformed kernel estimator (Gaussian), n=10000
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
Figure 25: Density estimation on the diagonal, transformed kernel.
44
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4Beta kernel estimator, b=0.05, n=100
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Beta kernel estimator, b=0.02, n=1000
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Beta kernel estimator, b=0.005, n=10000
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
Figure 26: Density estimation on the diagonal, Beta kernel.
45
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Average of histograms and copulas
A natural idea would be to extend the notion of average of histograms inhigher dimension. But note that the notion of partition can be morecomplicated.
Recall that C is a copula if for any u1 ≤ u2 and v1 ≤ v2,
C(u2, v2)− C(u1, v2)− C(u2, v1) + C(u1, v1) ≥ 0,
and C is a semicopula (Durante & Sempi (2004)) if for any u1 ≤ u2 andv1 ≤ v2, C(u2, v2)− C(u1, v2) ≥ 0 and C(u2, v2)− C(u2, v1) ≥ 0.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Copula, positive area
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Semicopula, positive area
46
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Histogram in dimension 2
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Histogram in dimension 2
Figure 27: Partitioning the unit square.
47
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Copula density as average of randomized histograms, m=2 Copula density as average of randomized histograms, m=2
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Copula density as average of randomized histograms, m=5 Copula density as average of randomized histograms, m=5
0.0 0.2 0.4 0.6 0.8 1.00.0
0.20.4
0.60.8
1.0
Figure 28: Copula density using average of histograms.
48
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Copula density as average of randomized histograms, m=10 Copula density as average of randomized histograms, m=10
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0
1
2
3
4
5
Copula density as average of randomized histograms, m=15 Copula density as average of randomized histograms, m=15
0.0 0.2 0.4 0.6 0.8 1.00.0
0.20.4
0.60.8
1.0
Figure 29: Copula density using average of histograms.
49
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4Average of histograms, n=1000, m=5
Estimation of the density on the diagonal
Estim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Average of histograms, n=1000, m=10
Estimation of the density on the diagonal
Estim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Average of histograms, n=1000, m=15
Estimation of the density on the diagonal
Estim
ato
r
Figure 30: Density estimation on the diagonal, average of histograms.
50
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Copula density estimation
Gijbels & Mielniczuk (1990): given an i.i.d. sample, a natural estimate forthe normed density is obtained using the transformed sample(FX(X1), FY (Y1)), ..., (FX(Xn), FY (Yn)), where FX and FY are the empiricaldistribution function of the marginal distribution. The copula density can beconstructed as some density estimate based on this sample (Behnen,Husková & Neuhaus (1985) investigated the kernel method).
The natural kernel type estimator c of c is
c(u, v) =1
nh2
n∑
i=1
K
(u− FX(Xi)
h,v − FY (Yi)
h
), (u, v) ∈ [0, 1].
“this estimator is not consistent in the points on the boundary of the unitsquare.”
51
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Copula density estimation and pseudo-observations
Figure 31 shows scatterplots when margins are known (i.e.(FX(Xi), FY (Yi))’s), and when margins are estimated (i.e.(FX(Xi), FY (Yi)’s). Note that the pseudo sample is more “uniform”, in thesense of a lower discrepancy (as in Quasi Monte Carlo techniques, see e.g.Niederreiter, H. (1992)).
52
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Scatterplot of observations (Xi,Yi)
Value of the Xis
Va
lue
of
the
Yis
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Scatterplot of pseudo−observations (Ui,Vi)
Value of the Uis (ranks)V
alu
e o
f th
e V
is (
ran
ks)
Figure 31: Observations and pseudo-observation, 500 simulated observationsfrom Frank copula (Xi, Yi) and the associate pseudo-sample (FX(Xi), FY (Yi)).
53
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Because samples are more “uniform” using ranks and pseudo-observations, thevariance of the estimator of the density, at some given point(u, v) ∈ (0, 1)× (0, 1) is usually smaller. For instance, Figure 32 shows theimpact of considering pseudo observations, i.e. substituting FX and FY tounknown marginal distributions FX and FY . The dotted line shows thedensity of c(u, v) from a n = 100 sample (Ui, Vi) (from Frank copula), and thestraight line shows the density of c(u, v) from the sample (FU (Ui), FV (Vi))(i.e. ranks of observations).
54
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0 1 2 3 4
0.00.5
1.01.5
2.0
Impact of pseudo−observations (n=100)
Distribution of the estimation of the density c(u,v)
Dens
ity of
the e
stima
tor
Figure 32: The impact of estimating from pseudo-observations.
55
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
The problem of censored dataIn the case of censored data, consider sample of ((Xi, δ
Xi ), (Yi, δ
Yi ))’s, where
δXi = 0 when Xi is censored. Xi = min{X0
i , CXi } and Yi = min{X0
i , CYi }
where the Ci’s are censoring variates. The (Xi, Yi) are observed data, the(X0
i , Y 0i )’s are the variables of interest, that might be unobservable.
Kaplan-Meier estimator can be considered, based on the transformed samplemade of the non-zero elements.
The transformed sample is denoted by (X ′i, Y
′i ), i = 1, ..., n′, and its size is
n′ =∑n
i=1 δXi · δY
i of fully uncensored data.
The marginal Kaplan-Meier of the distribution function is
FKMX (x) = 1−
∏
i:Xi≤x
(n− i
n− i + 1
)δXi
.
The nonparametric estimator of the copula density at point (u, v) can beobtained using product beta kernels, on
56
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
(UKMi , V KM
i ) = (FKMX (X ′
i), FKMY (Y ′
i )), i = 1, ..., n′.
Remark In the LOSS-ALAE dataset, the loss amounts Xi’s are censored,because of a policy limit. Hence, Xi = min{loss, limit}.
57
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
2 4 6 8 10 12 14
46
810
12
Log(LOSS)
Log(A
LA
E)
Figure 33: The LOSS-ALAE dataset and censoring.
58
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Unfortunately, such a procedure suffers from the so-called stock sampling (orselection bias) issue. Actually, the subsample used is far from being an i.i.d.sample drawn from the law of (X, Y ).
The observations in hand are chosen as fully uncensored. Thus, smallobserved (Xi, Yi) are more frequent in the subsample than they should be in ausual i.i.d. sample. Therefore, the previous estimator can be advised onlywhen the proportion of censoring is small. Otherwise the bias induced bystock sampling can become large.
59
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0
1
2
3
4
5
Estimation of Frank copula
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
Figure 34: Estimated density of Frank copula, using Beta kernels, n = 1000,b = 0.050 and no-censoring.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0
1
2
3
4
5
Estimation of Frank copula
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
Figure 35: Estimated density of Frank copula, using Beta kernels, n = 1000,b = 0.050 and 1.7% censored data.
61
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0
1
2
3
4
5
Estimation of Frank copula
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
Figure 36: Estimated density of Frank copula, using Beta kernels, n = 1000,b = 0.050 and 7.8% censored data.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0
1
2
3
4
5
Estimation of Frank copula
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
Figure 37: Estimated density of Frank copula, using Beta kernels, n = 1000,b = 0.050 and 24% censored data.
63
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Beta kernel estimator, b=0.01, n=1000, no censoring
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Beta kernel estimator, b=0.01, n=1000, 5% censoring
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Beta kernel estimator, b=0.01, n=1000, 25% censoring
Estimation of the density on the diagonal
Density o
f th
e e
stim
ato
r
Figure 38: Estimation of the density on the diagonal, with censored data .
64
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Stock sampling can be observe as the bias increases as u goes to 1, and as theproportion of censored data increases.
Figure 39 shows the estimated density c(12/13, 12/13) with no censoring, and25% censored observations.
Following the idea proposed in Efron and Tibshirani (1993), bootstraptechniques can be used to estimate the bias. Consider B bootstrap samplesdrawn independently from the original censored data, with replacement,{(
X∗i,b, Y
∗i,b, δ
∗i,b
)}, i = 1, ..., n and b = 1, ..., B. The bias of the density
estimator at point (u, v) can be estimated from
bias (u, v) =1B
B∑
i=1
c (u, v)∗,b − c (u, v)
where c (u)∗,b is the density obtained from the bth bootstrap sample.
65
Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
Standard Beta kernel estimator
Estimation of the density near the lower corner
Den
sity
of t
he e
stim
ator
Figure 39: c(12/13, 12/13) using Beta kernels, with censored data (dotted line).
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Transformed kernel estimator (Gaussian), n=1000
Estimation of the density on the diagonal, 25% censoring
Densi
ty o
f th
e e
stim
ato
r, b
oots
trap b
ias
corr
ect
ion
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Beta kernel estimator, b=0.01, n=1000
Estimation of the density on the diagonal, 25% censoringD
ensi
ty o
f th
e e
stim
ato
r, b
oots
trap b
ias
corr
ect
ion
Figure 40: Censoring bias correction using bootstrap techniques.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
Figure 40 shows the the evolution of{
u 7→ 1B
B∑
i=1
c (u, u)∗,b}
,
on the diagonal using bootstrap techniques (as in Efron (1981)).
For each bootstrap sample, unobservable value Y 0i,b and censoring value Ci,b
are simulated using Kaplan-Meier estimate of the density, and then a sample(Xi,b, Y i, b, δi, b), i = 1, ..., n is obtained, and a new Kaplan-Meier estimationis obtained, F 0
b , and is used to obtain the estimation of the copula density forthis bootstrap sample.
The application to the Loss-ALAE dataset confirms Gumbel model.
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Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
67
Beta kernel estimator, b=0.01, n=1000
Estimation of the density on the diagonal, LOSS−ALAE
De
nsi
ty o
f th
e e
stim
ato
r, b
oo
tstr
ap
bia
s co
rre
ctio
n
0.75 0.80 0.85 0.90 0.95 1.00
01
23
45
67
Beta kernel estimator, b=0.005, n=1000
Estimation of the density on the diagonal, LOSS−ALAED
en
sity
of th
e e
stim
ato
r, b
oo
tstr
ap
bia
s co
rre
ctio
n
Figure 41: Beta kernel estimator for Loss-ALAE, compare with Gumbel copula.
69