Berlin

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

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.

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 ).

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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Log−log scatterplot, Loss−ALAE

log(LOSS)

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Copula type scatterplot, Loss−ALAE

Probability level LOSSP

robabili

Figure 1: Loss-ALAE, scatterplots (log-log and copula type).

Figure 2: Loss-ALAE, histogram of copula type transformation.

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

(x−Xi

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)

Exponential random variables

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Exponential random variables

−1 0 1 2 3 4 5 6

Figure 3: Density estimation of an exponential density, 100 points.

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Kernel based estimation of the uniform density on [0,1]

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Kernel based estimation of the uniform density on [0,1]

Figure 4: Density estimation of an uniform density on [0, 1].

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).

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

(x−Xi

(x + Xi

Remark: It works well when the density f has a null derivative in 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.

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).

−3 −2 −1 0 1 2 3

Density estimation transformed kernel

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Density estimation transformed kernel

Figure 6: The transform kernel principle (with a Φ−1-transformation).

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

1− u

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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Gaussian Kernel approach

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Beta Kernel approach

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Density estimation using beta kernels

Figure 7: The beta-kernel estimate.

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

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]

1(Xi ∈ Ii:m), where x ∈ Ii:m.

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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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Figure 11: Average of histograms, m = 10 and m = 20.

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Gaussian Kernel approach

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Beta Kernel approach

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Average of histograms

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Average of histograms

Figure 12: Shape of the induced kernel based on uniform averaging.

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)]

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)]

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)]

ω2K(ω)dω · h2 + o(h2).

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

K(ω)2dω

· 1nh2

on the interior of the borders (e.g. u = 0 and v ∈ (0, 1)),

V ar(c(0, v, h)) = c(0, v)(∫ 1

K(ω)2dω

)(∫ 1

K(ω)2dω

)· 1nh2

and in the interior ((u, v) ∈ (0, 1)× (0, 1)),

V ar(c(u, v, h)) = c(u, v)(∫ 1

K(ω)2dω

d · 1nh2

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Estimation of Frank copula

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Figure 13: Theoretical density of Frank copula.

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Figure 14: Estimated density of Frank copula, using standard Gaussian (inde-pendent) kernels, h = h∗.

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

(x−Xi

(y − Yi

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)

one gets, substituting f in (3)

c(u, v) =1

nh · g(G−1(u)) · g(G−1(v))

(G−1(u)−G−1(Ui)

h,G−1(v)−G−1(Vi)

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

g(G−1(u))2g(G−1(v))2o

Asymptotic normality and also be derived.

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Figure 15: Estimated density of Frank copula, using a Gaussian kernel, after aGaussian normalization.

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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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Figure 17: Estimated density of Frank copula, using a Gaussian kernel, after aStudent normalization, with 3 degrees of freedom.

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

(u− Ui

(v − Vi

(u + Ui

(v − Vi

(u− Ui

(v + Vi

(u + Ui

(v + Vi

(u− Ui

(v − 2 + Vi

(u + Ui

(v − 2 + Vi

(u− 2 + Ui

(v − Vi

(u− 2 + Ui

(v + Vi

(u− 2 + Ui

(v − 2 + Vi

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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−1.0

−0.5

0.0 0.2 0.4 0.6 0.8 1.00.0

0.81.0

Estimation of the copula density

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−1.0

−0.5

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0.81.0

Estimation of the copula density

Figure 18: Copula density estimation - mirror reflection (n = 100, 1000).

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−1.0

−0.5

0.2 0.4 0.6 0.8

Estimation of the copula density (mirror reflection)

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−1.0

−0.5

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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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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

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

1− u

1− v

where K(·, α, β) denotes the density of the Beta distribution with parametersα and β.

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

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.

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)],

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)

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.

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Estimation of the copula density (Beta kernel, b=0.1) Estimation of the copula density (Beta kernel, b=0.1)

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Figure 22: Estimated density of Frank copula, Beta kernels, b = 0.1

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Estimation of the copula density (Beta kernel, b=0.05) Estimation of the copula density (Beta kernel, b=0.05)

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Figure 23: Estimated density of Frank copula, Beta kernels, b = 0.05

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4Standard Gaussian kernel estimator, n=100

Estimation of the density on the diagonal

Density o

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Standard Gaussian kernel estimator, n=1000

Density o

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Standard Gaussian kernel estimator, n=10000

Density o

Figure 24: Density estimation on the diagonal, standard kernel.

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4Transformed kernel estimator (Gaussian), n=100

Density o

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Transformed kernel estimator (Gaussian), n=1000

Density o

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Density o

Figure 25: Density estimation on the diagonal, transformed kernel.

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4Beta kernel estimator, b=0.05, n=100

Density o

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Beta kernel estimator, b=0.02, n=1000

Density o

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Density o

Figure 26: Density estimation on the diagonal, Beta kernel.

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.

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0.00.2

0.40.6

0.81.0

Copula, positive area

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0.00.2

0.40.6

0.81.0

Semicopula, positive area

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Histogram in dimension 2

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Histogram in dimension 2

Figure 27: Partitioning the unit square.

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Copula density as average of randomized histograms, m=2 Copula density as average of randomized histograms, m=2

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0.00.2

0.40.6

0.81.0

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0.20.4

0.60.8

Figure 28: Copula density using average of histograms.

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0.00.2

0.40.6

0.81.0

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0.0 0.2 0.4 0.6 0.8 1.00.0

0.20.4

0.60.8

Figure 29: Copula density using average of histograms.

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4Average of histograms, n=1000, m=5

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Average of histograms, n=1000, m=10

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Average of histograms, n=1000, m=15

Figure 30: Density estimation on the diagonal, average of histograms.

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

(u− FX(Xi)

h,v − FY (Yi)

), (u, v) ∈ [0, 1].

“this estimator is not consistent in the points on the boundary of the unitsquare.”

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)).

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Scatterplot of observations (Xi,Yi)

Value of the Xis

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Scatterplot of pseudo−observations (Ui,Vi)

Value of the Uis (ranks)V

Figure 31: Observations and pseudo-observation, 500 simulated observationsfrom Frank copula (Xi, Yi) and the associate pseudo-sample (FX(Xi), FY (Yi)).

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).

0 1 2 3 4

0.00.5

1.01.5

Impact of pseudo−observations (n=100)

Distribution of the estimation of the density c(u,v)

ity of

Figure 32: The impact of estimating from pseudo-observations.

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

The nonparametric estimator of the copula density at point (u, v) can beobtained using product beta kernels, on

(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}.

2 4 6 8 10 12 14

Log(LOSS)

Figure 33: The LOSS-ALAE dataset and censoring.

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.

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Figure 34: Estimated density of Frank copula, using Beta kernels, n = 1000,b = 0.050 and no-censoring.

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Figure 35: Estimated density of Frank copula, using Beta kernels, n = 1000,b = 0.050 and 1.7% censored data.

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Figure 36: Estimated density of Frank copula, using Beta kernels, n = 1000,b = 0.050 and 7.8% censored data.

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Figure 37: Estimated density of Frank copula, using Beta kernels, n = 1000,b = 0.050 and 24% censored data.

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Beta kernel estimator, b=0.01, n=1000, no censoring

Density o

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Beta kernel estimator, b=0.01, n=1000, 5% censoring

Density o

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Beta kernel estimator, b=0.01, n=1000, 25% censoring

Density o

Figure 38: Estimation of the density on the diagonal, with censored data .

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

c (u, v)∗,b − c (u, v)

where c (u)∗,b is the density obtained from the bth bootstrap sample.

0 1 2 3 4 5 6

Standard Beta kernel estimator

Estimation of the density near the lower corner

Figure 39: c(12/13, 12/13) using Beta kernels, with censored data (dotted line).

0.0 0.2 0.4 0.6 0.8 1.0

Estimation of the density on the diagonal, 25% censoring

trap b

0.0 0.2 0.4 0.6 0.8 1.0

Estimation of the density on the diagonal, 25% censoringD

trap b

Figure 40: Censoring bias correction using bootstrap techniques.

Figure 40 shows the the evolution of{

u 7→ 1B

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.

0.0 0.2 0.4 0.6 0.8 1.0

Estimation of the density on the diagonal, LOSS−ALAE

0.75 0.80 0.85 0.90 0.95 1.00

Estimation of the density on the diagonal, LOSS−ALAED

Figure 41: Beta kernel estimator for Loss-ALAE, compare with Gumbel copula.

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