Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
Equation (4.9)
H(x, y) = C(FX(x), FY (y)),
4.4. Focus on the dependence structure
when X, Y take discrete values. The answer is given by the Carley bounds C −
H , C+
H . For all
copulas CH verifying Equation (4.9), we have
C −
H ≤ CH ≤ C +
H ,
cf. Proposition 2 of Genest and Neslehova (2007).
The non-identifiability issue matters, since with discrete copulas the dependence measure,
such as the tau of Kendall or the rho of Spearman, are no longer unique. Furthermore, if X
and Y are independent, it does not imply that the copula of (X, Y ) is the independent copula.
In other words, the copula alone does not characterize the dependence: we need assumptions
on margins.
In the copula literature, efforts have been done to tackle the non-identifiability issue of
dependence measure. The current answer is the interpolated copula C X,Y
, which is a bilinear
interpolation of the distribution function D of the pair (F (X), F (Y )) on the discrete set
Im(FX) × Im(FY ).
Let (ui, vj) ∈ Im(FX) × Im(FY ), i.e. ui = FX(i) and uj = FX(j). For all (u, v) ∈
[ui, ui+1] × [vj, vj+1], the interpolated copula is defined as
C X,Y (u, v) = ¯ λu ¯ λvD(ui, vj)+λu ¯ λvD(ui+1, vj)+ ¯ λuλvD(ui, vj+1)+λuλvD(ui+1, vj+1), (4.12)
where λu = (u−ui)/(ui+1 −ui), λv = (v −vj)/(vj+1 −vj). This copula was already mentioned
in Lemma 2.3.5 of Nelsen (2006) to prove the Sklar theorem. The interpolated copula can
also be interpreted as the copula of (X + U, Y + V ) where U, V are two independent uniform
random variables, see, e.g., Section 4 of Denuit and Lambert (2005). This formulation is useful
when doing random generation.
The properties of the interpolated copula of the pair (X, Y ) are: (i) Kendall’s tau τ(X, Y ) =
τ(C X,Y ) and Spearman’s rho ρ(X, Y ) = ρ(C X,Y ), (ii) C X,Y is absolutely continuous and (iii)
depends on marginals
X ⊥ Y ⇔ C X,Y = Π, the independent copula. Unfortunately, C X,Y
and when FX = FY does not imply C X,Y (u, v) = min(u, v), as well as FX = 1 − FY
C X,Y (u, v) = (u + v − 1)+, see, e.g., Genest and Neslehova (2007).
As the interpolated copula is absolutely continuous, a density can be derived. Indeed, by
differentiating Equation (4.12) with respect to u and v, we get
c X,Y (u, v) = D(ui, vj) − D(ui+1, vj) − D(ui, vj+1) + D(ui+1, vj+1)
, (4.13)
(ui+1 − ui)(vj+1 − vj)
for (u, v) ∈ [ui, ui+1] × [vj, vj+1].
To better see the differences between the copula CY1,Y2 and the interpolated copula C W1,W2 ,
we choose to plot the densities on Figure 4.3 and not the distribution functions. On Figure 4.3,
we plot densities of the continuous copula of the pair (Y1, Y2) and the interpolated copula of
the pair (W1, W2), in the same setting as the previous subsection, i.e. Θ is gamma distributed
Ga(2, 4).
4.4.4 Zero-modified distributions
In the previous Subsections, we worked only with a simple geometric distribution. Following
the model presented in Subsection 4.2.2, we use hereafter a zero-modified distribution
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