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### Berlin

1. 1. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Nonparametric estimation of copula densities A. Charpentier1 , J.D. Fermanian2 & O. Scaillet3 Gemeinsame Jahrestagung der Deutschen Mathematiker-Vereinigung und der Gesellschaft 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 1
2. 2. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules One (short) word on copulas Deﬁnition 1. A 2-dimensional copula is a 2-dimensional cumulative distribution 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. 2
3. 3. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Theorem 2. (Sklar) Let C be a copula, and FX and FY two marginal distributions, then F(x, y) = C(FX(x), FY (y)) is a bivariate distribution function, with F ∈ F(FX, FY ). Conversely, if F ∈ F(FX, FY ), there exists C such that F(x, y) = C(FX(x), FY (y). Further, if FX and FY are continuous, then C is unique, and given by C(u, v) = F(F−1 X (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 ). 3
4. 4. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Motivation Example Loss-ALAE: consider the following dataset, were the Xi’s are loss amount (paid to the insured) and the Yi’s are allocated expenses. Denote by Ri and Si the respective ranks of Xi and Yi. Set Ui = Ri/n = ˆFX(Xi) and Vi = Si/n = ˆFY (Yi). Figure 1 shows the log-log scatterplot (log Xi, log Yi), and the associate copula based scatterplot (Ui, Vi). Figure 2 is simply an histogram of the (Ui, Vi), which is a nonparametric estimation of the copula density. Note that the histogram suggests strong dependence in upper tails (the interesting part in an insurance/reinsurance context). 4
5. 5. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 1 2 3 4 5 6 12345 Log−log scatterplot, Loss−ALAE log(LOSS) log(ALAE) 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Copula type scatterplot, Loss−ALAE Probability level LOSSProbabilitylevelALAE Figure 1: Loss-ALAE, scatterplots (log-log and copula type). 5
6. 6. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Figure 2: Loss-ALAE, histogram of copula type transformation. 6
7. 7. 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 moving histogram). If {X1, ..., Xn} of positive random variable (Xi ∈ [0, ∞[). Let K denote a symmetric kernel, then E(f(0, h) = 1 2 f(0) + O(h) 7
8. 8. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Exponential random variables Density −1 0 1 2 3 4 5 6 0.00.20.40.60.81.01.2 Exponential random variables Density −1 0 1 2 3 4 5 6 0.00.20.40.60.81.01.2Figure 3: Density estimation of an exponential density, 100 points. 8
9. 9. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.01.2 Kernel based estimation of the uniform density on [0,1] Density 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.01.2 Kernel based estimation of the uniform density on [0,1] Density Figure 4: Density estimation of an uniform density on [0, 1]. 9
10. 10. 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 the border, • 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). 10
11. 11. 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. 11
12. 12. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 0.00.51.01.52.0 Density estimation using mirror image 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.0 Density estimation using mirror image Figure 5: The mirror image principle. 12
13. 13. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules The transformed Kernel estimate The idea was developed in Devroye & Györfi (1981) for univariate densities. Consider a transformation T : R → [0, 1] strictly increasing, continuously diﬀerentiable, 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). 13
14. 14. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules −3 −2 −1 0 1 2 3 0.00.10.20.30.40.5 Density estimation transformed kernel 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.0 Density estimation transformed kernel Figure 6: The transform kernel principle (with a Φ−1 -transformation). 14
15. 15. 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, is obtained using beta kernels, which yields f(x) = 1 n n i=1 K Xi, u b + 1, 1 − u b + 1 where K(·, α, β) denotes the density of the Beta distribution with parameters α and β, K(x, α, β) = Γ(α + β) Γ(α)Γ(β) xα (1 − x)β−1 1(x ∈ [0, 1]). 15
16. 16. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0246810 Gaussian Kernel approach 0.0 0.2 0.4 0.6 0.8 1.0 0246810 Beta Kernel approach 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.0 Density estimation using beta kernels Figure 7: The beta-kernel estimate. 16
17. 17. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules The average of histograms In a very general context Dersko (1998) introduced the intrinsic estimator of the density, as the average of all possible histograms. Consider the case of a density 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 + 1 subintervals, 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. 17
18. 18. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=1 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=1 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=1 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=1 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.0 Density as average of randomized histograms, n=250, m=1 Figure 8: Average of histograms, m = 1. 18
19. 19. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=2 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=2 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=2 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=2 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.0 Density as average of randomized histograms, n=250, m=2 Figure 9: Average of histograms, m = 2. 19
20. 20. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=5 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=5 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=5 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.5 Average of histograms, m=5 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.0 Density as average of randomized histograms, n=250, m=5 Figure 10: Average of histograms, m = 5. 20
21. 21. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.0 Density as average of randomized histograms, n=250, m=10 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.0 Density as average of randomized histograms, n=250, m=20 Figure 11: Average of histograms, m = 10 and m = 20. 21
22. 22. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0246810 Gaussian Kernel approach 0.0 0.2 0.4 0.6 0.8 1.0 0246810 Beta Kernel approach 0.0 0.2 0.4 0.6 0.8 1.0 0246810 Average of histograms 0.0 0.2 0.4 0.6 0.8 1.00246810 Average of histograms Figure 12: Shape of the induced kernel based on uniform averaging. 22
23. 23. 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 density c(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)) = 1 4 · 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)) = 1 2 · 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) + 1 2 [c1,1(u, v) + c2,2(u, v)] 1 −1 ω2 K(ω)dω · h2 + o(h2 ). 23
24. 24. 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)2 dω 2 · 1 nh2 + 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)2 dω 1 0 K(ω)2 dω · 1 nh2 + 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)2 dω 2 d · 1 nh2 + o 1 nh2 . 24
25. 25. 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 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Figure 13: Theoretical density of Frank copula. 25
26. 26. 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 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Figure 14: Estimated density of Frank copula, using standard Gaussian (inde- pendent) kernels, h = h∗ . 26
27. 27. 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 distribution function, with a diﬀerentiable density. Since density f is continuous, twice diﬀerentiable, 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 , Since f(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) 27
28. 28. 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)2 dω 2 + 1 g(G−1(u))2g(G−1(v))2 o 1 nh2 . Asymptotic normality and also be derived. 28
29. 29. 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 0.20.40.60.8 Figure 15: Estimated density of Frank copula, using a Gaussian kernel, after a Gaussian normalization. 29
30. 30. 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 0.20.40.60.8 Figure 16: Estimated density of Frank copula, using a Gaussian kernel, after a Student normalization, with 5 degrees of freedom. 30
31. 31. 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 0.20.40.60.8 Figure 17: Estimated density of Frank copula, using a Gaussian kernel, after a Student normalization, with 3 degrees of freedom. 31
32. 32. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Using the mirror reﬂection principle Instead of using only the (Ui, Vi)’s the mirror image consists in reﬂecting each data 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 ˆc(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 . 32
33. 33. 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 a bounded symmetric kernel that vanishes oﬀ [−1, 1], with ωK(ω) → 0 as ω → ∞, and if nh2 → ∞ as h → 0 and n → ∞, then f satisﬁes standard asymptotic 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 −1 K(ω)2 dω V ar(c(u, v, h)) = 4c(u, v)(nh2 )−1 σ4 + o((nh2 )−1 ) in corner, = 2c(u, v)(nh2 )−1 σ4 + o((nh2 )−1 ) in the interior of borders, = c(u, v)(nh2 )−1 σ4 + 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 , 33
34. 34. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 −1.0−0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Estimation of the copula density −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 −1.0−0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Estimation of the copula density Figure 18: Copula density estimation - mirror reﬂection (n = 100, 1000). 34
35. 35. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 −1.0−0.50.00.51.01.52.0 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 Estimation of the copula density (mirror reflection) −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 −1.0−0.50.00.51.01.52.0 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 Estimation of the copula density (mirror reflection) Figure 19: Copula density estimation - mirror reﬂection (n = 100, 1000). 35
36. 36. 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 (mirror reflection) Estimation of the copula density (mirror reflection) 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Figure 20: Estimated density of Frank copula, mirror reﬂection principle 36
37. 37. 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), is obtained using product beta kernels, which yields c(u, v) = 1 n 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 parameters α and β. 37
38. 38. 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
39. 39. 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)+ 1 2 [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
40. 40. 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 can be deduced, maximizing asymptotic mean squared error, b∗ ≡ 1 16πnQ(u, v)2 · 1 v(1 − v)u(1 − u) 1/3 . Note (see Charpentier, Fermanian & Scaillet (2005)) that all those results can be obtained in dimension d ≥ 2. Example For n = 100 simulated data, from Frank copula, the optimal bandwidth is b ∼ 0.05. 40
41. 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=0.1) Estimation of the copula density (Beta kernel, b=0.1) 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Figure 22: Estimated density of Frank copula, Beta kernels, b = 0.1 41
42. 42. 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 0.00.20.40.60.81.0 Figure 23: Estimated density of Frank copula, Beta kernels, b = 0.05 42
43. 43. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 01234 Standard Gaussian kernel estimator, n=100 Estimation of the density on the diagonal Densityoftheestimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Standard Gaussian kernel estimator, n=1000 Estimation of the density on the diagonal Densityoftheestimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Standard Gaussian kernel estimator, n=10000 Estimation of the density on the diagonal Densityoftheestimator Figure 24: Density estimation on the diagonal, standard kernel. 43
44. 44. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 01234 Transformed kernel estimator (Gaussian), n=100 Estimation of the density on the diagonal Densityoftheestimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Transformed kernel estimator (Gaussian), n=1000 Estimation of the density on the diagonal Densityoftheestimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Transformed kernel estimator (Gaussian), n=10000 Estimation of the density on the diagonal Densityoftheestimator Figure 25: Density estimation on the diagonal, transformed kernel. 44
45. 45. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 01234 Beta kernel estimator, b=0.05, n=100 Estimation of the density on the diagonal Densityoftheestimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Beta kernel estimator, b=0.02, n=1000 Estimation of the density on the diagonal Densityoftheestimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Beta kernel estimator, b=0.005, n=10000 Estimation of the density on the diagonal Densityoftheestimator Figure 26: Density estimation on the diagonal, Beta kernel. 45
46. 46. 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 in higher dimension. But note that the notion of partition can be more complicated. 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 and v1 ≤ 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 0.00.20.40.60.81.0 Copula, positive area 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Semicopula, positive area 46
47. 47. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Histogram in dimension 2 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Histogram in dimension 2 Figure 27: Partitioning the unit square. 47
48. 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=2 Copula density as average of randomized histograms, m=2 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 1.0 0.00.20.40.60.81.0 Figure 28: Copula density using average of histograms. 48
49. 49. 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 0.00.20.40.60.81.0 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 1.0 0.00.20.40.60.81.0 Figure 29: Copula density using average of histograms. 49
50. 50. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 01234 Average of histograms, n=1000, m=5 Estimation of the density on the diagonal Estimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Average of histograms, n=1000, m=10 Estimation of the density on the diagonal Estimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Average of histograms, n=1000, m=15 Estimation of the density on the diagonal Estimator Figure 30: Density estimation on the diagonal, average of histograms. 50
51. 51. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Copula density estimation Gijbels & Mielniczuk (1990): given an i.i.d. sample, a natural estimate for the normed density is obtained using the transformed sample (FX(X1), FY (Y1)), ..., (FX(Xn), FY (Yn)), where FX and FY are the empirical distribution function of the marginal distribution. The copula density can be constructed 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 unit square.” 51
52. 52. 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. ( ˆFX(Xi), ˆFY (Yi)’s). Note that the pseudo sample is more “uniform”, in the sense of a lower discrepancy (as in Quasi Monte Carlo techniques, see e.g. Niederreiter, H. (1992)). 52
53. 53. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Scatterplot of observations (Xi,Yi) Value of the Xis ValueoftheYis 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Scatterplot of pseudo−observations (Ui,Vi) Value of the Uis (ranks)ValueoftheVis(ranks) Figure 31: Observations and pseudo-observation, 500 simulated observations from Frank copula (Xi, Yi) and the associate pseudo-sample ( ˆFX(Xi), ˆFY (Yi)). 53
54. 54. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Because samples are more “uniform” using ranks and pseudo-observations, the variance of the estimator of the density, at some given point (u, v) ∈ (0, 1) × (0, 1) is usually smaller. For instance, Figure 32 shows the impact of considering pseudo observations, i.e. substituting ˆFX and ˆFY to unknown marginal distributions FX and FY . The dotted line shows the density of ˆc(u, v) from a n = 100 sample (Ui, Vi) (from Frank copula), and the straight line shows the density of ˆc(u, v) from the sample ( ˆFU (Ui), ˆFV (Vi)) (i.e. ranks of observations). 54
55. 55. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0 1 2 3 4 0.00.51.01.52.0 Impact of pseudo−observations (n=100) Distribution of the estimation of the density c(u,v) Densityoftheestimator Figure 32: The impact of estimating from pseudo-observations. 55
56. 56. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules The problem of censored data In the case of censored data, consider sample of ((Xi, δX i ), (Yi, δY i ))’s, where δX i = 0 when Xi is censored. Xi = min{X0 i , CX i } and Yi = min{X0 i , CY i } where the Ci’s are censoring variates. The (Xi, Yi) are observed data, the (X0 i , Y 0 i )’s are the variables of interest, that might be unobservable. Kaplan-Meier estimator can be considered, based on the transformed sample made of the non-zero elements. The transformed sample is denoted by (Xi, Yi ), i = 1, ..., n , and its size is n = n i=1 δX i · δY i of fully uncensored data. The marginal Kaplan-Meier of the distribution function is ˆFKM X (x) = 1 − i:Xi≤x n − i n − i + 1 δX i . The nonparametric estimator of the copula density at point (u, v) can be obtained using product beta kernels, on 56
57. 57. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules ( ˆUKM i , V KM i ) = (FKM X (Xi), FKM Y (Yi )), 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
58. 58. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 2 4 6 8 10 12 14 4681012 Log(LOSS) Log(ALAE) Figure 33: The LOSS-ALAE dataset and censoring. 58
59. 59. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Unfortunately, such a procedure suﬀers from the so-called stock sampling (or selection 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, small observed (Xi, Yi) are more frequent in the subsample than they should be in a usual i.i.d. sample. Therefore, the previous estimator can be advised only when the proportion of censoring is small. Otherwise the bias induced by stock sampling can become large. 59
60. 60. 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 0.20.40.60.8 Figure 34: Estimated density of Frank copula, using Beta kernels, n = 1000, b = 0.050 and no-censoring. 60
61. 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 0.20.40.60.8 Figure 35: Estimated density of Frank copula, using Beta kernels, n = 1000, b = 0.050 and 1.7% censored data. 61
62. 62. 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 0.20.40.60.8 Figure 36: Estimated density of Frank copula, using Beta kernels, n = 1000, b = 0.050 and 7.8% censored data. 62
63. 63. 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 0.20.40.60.8 Figure 37: Estimated density of Frank copula, using Beta kernels, n = 1000, b = 0.050 and 24% censored data. 63
64. 64. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 01234 Beta kernel estimator, b=0.01, n=1000, no censoring Estimation of the density on the diagonal Densityoftheestimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Beta kernel estimator, b=0.01, n=1000, 5% censoring Estimation of the density on the diagonal Densityoftheestimator 0.0 0.2 0.4 0.6 0.8 1.0 01234 Beta kernel estimator, b=0.01, n=1000, 25% censoring Estimation of the density on the diagonal Densityoftheestimator Figure 38: Estimation of the density on the diagonal, with censored data . 64
65. 65. 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 the proportion of censored data increases. Figure 39 shows the estimated density c(12/13, 12/13) with no censoring, and 25% censored observations. Following the idea proposed in Efron and Tibshirani (1993), bootstrap techniques can be used to estimate the bias. Consider B bootstrap samples drawn 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) = 1 B B i=1 c (u, v) ∗,b − c (u, v) where c (u) ∗,b is the density obtained from the bth bootstrap sample. 65
66. 66. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0 1 2 3 4 5 6 0.00.20.40.60.8 Standard Beta kernel estimator Estimation of the density near the lower corner Densityoftheestimator Figure 39: c(12/13, 12/13) using Beta kernels, with censored data (dotted line). 66
67. 67. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 01234 Transformed kernel estimator (Gaussian), n=1000 Estimation of the density on the diagonal, 25% censoring Densityoftheestimator,bootstrapbiascorrection 0.0 0.2 0.4 0.6 0.8 1.0 01234 Beta kernel estimator, b=0.01, n=1000 Estimation of the density on the diagonal, 25% censoringDensityoftheestimator,bootstrapbiascorrection Figure 40: Censoring bias correction using bootstrap techniques. 67
68. 68. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules Figure 40 shows the the evolution of u → 1 B B i=1 c (u, u) ∗,b , on the diagonal using bootstrap techniques (as in Efron (1981)). For each bootstrap sample, unobservable value Y 0 i,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 estimation is obtained, F0 b , and is used to obtain the estimation of the copula density for this bootstrap sample. The application to the Loss-ALAE dataset conﬁrms Gumbel model. 68
69. 69. Arthur CHARPENTIER - Estimation non-paramétrique des densités de copules 0.0 0.2 0.4 0.6 0.8 1.0 01234567 Beta kernel estimator, b=0.01, n=1000 Estimation of the density on the diagonal, LOSS−ALAE Densityoftheestimator,bootstrapbiascorrection 0.75 0.80 0.85 0.90 0.95 1.00 01234567 Beta kernel estimator, b=0.005, n=1000 Estimation of the density on the diagonal, LOSS−ALAEDensityoftheestimator,bootstrapbiascorrection Figure 41: Beta kernel estimator for Loss-ALAE, compare with Gumbel copula. 69