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Arthur CHARPENTIER, Distortion in actuarial sciences

Distorting probabilitiesin actuarial sciences

Arthur Charpentier

Université Rennes 1

[email protected]

http ://freakonometrics.blog.free.fr/

Univeristé Laval, Québec, Avril 2011

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http://freakonometrics.blog.free.fr/

Arthur CHARPENTIER, Distortion in actuarial sciences 1 DECISION THEORY AND DISTORTED RISK MEASURES

1 Decision theory and distorted risk measures

Consider a preference ordering among risks, � such that1. � is distribution based, i.e. if X � Y , ∀X L= X Y

L= Y , then X � Y ; hence,we can write FX � FY

2. � is total, reflexive and transitive,3. � is continuous, i.e. ∀FX , FY and FZ such that FX � FY � FZ , ∃λ, µ ∈ (0, 1)

such thatλFX + (1− λ)FZ � FY � µFX + (1− µ)FZ .

4. � satisfies an independence axiom, i.e. ∀FX , FY and FZ , and ∀λ ∈ (0, 1),

FX � FY =⇒ λFX + (1− λ)FZ � λFY + (1− λ)FZ .

5. � satisfies an ordering axiom, ∀X and Y constant (i.e.P(X = x) = P(Y = y) = 1, FX � FY =⇒ x ≤ y.

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Theorem1Ordering � satisfies axioms 1-2-3-4-5 if and only if ∃u : R→ R, continuous, strictlyincreasing and unique (up to an increasing affine transformation) such that ∀FX and FY :

FX � FY ⇔∫Ru(x)dFX(x) ≤

∫Ru(x)dFY (x)

⇔ E[u(X)] ≤ E[u(Y )].

But if we consider an alternative to the independence axiom4’. � satisfies an dual independence axiom, i.e. ∀FX , FY and FZ , and ∀λ ∈ (0, 1),

FX � FY =⇒ [λF−1X + (1− λ)F−1

Z ]−1 � [λF−1Y + (1− λ)F−1

Z ]−1.

we (Yaari (1987)) obtain a dual representation theorem,Theorem2Ordering � satisfies axioms 1-2-3-4’-5 if and only if ∃g : [0, 1]→ R, continuous, strictlyincreasing such that ∀FX and FY :

FX � FY ⇔∫Rg(FX(x))dx ≤

∫Rg(FY (x))dx

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Standard axioms required on risque measures R : X → R,– law invariance, X L= Y =⇒ R(X) = R(Y )– increasing X ≥ Y =⇒ R(X) ≥ R(Y ),– translation invariance ∀k ∈ R, =⇒ R(X + k) = R(X) + k,– homogeneity ∀λ ∈ R+, R(λX) = λ · R(X),– subadditivity R(X + Y ) ≤ R(X) +R(Y ),– convexity ∀β ∈ [0, 1], R(βλX + [1− β]Y ) ≤ β · R(X) + [1− β] · R(Y ).

– additivity for comonotonic risks ∀X and Y comonotonic,R(X + Y ) = R(X) +R(Y ),

– maximal correlation (w.r.t. measure µ) ∀X,

R(X) = sup {E(X · U) where U ∼ µ}

– strong coherence ∀X and Y , sup{R(X + Y )} = R(X) +R(Y ), where X L= X

and Y L= Y .

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Proposition1IfR is a monetary convex fonction, then the three statements are equivalent,– R is strongly coherent,– R is additive for comonotonic risks,– R is a maximal correlation measure.

Proposition2A coherente risk measureR is additive for comonotonic risks if and only if there exists adecreasing positive function φ on [0, 1] such that

R(X) =∫ 1

0φ(t)F−1(1− t)dt

where F (x) = F(X ≤ x).

see Kusuoka (2001), i.e. R is a spectral risk measure.

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Definition1A distortion function is a function g : [0, 1]→ [0, 1] such that g(0) = 0 and g(1) = 1.

For positive risks,Definition1Given distortion function g, Wang’s risk measure, denotedRg , is

Rg (X) =∫ ∞

0g (1− FX(x)) dx =

∫ ∞0

g(FX(x)

)dx (1)

Proposition1Wang’s risk measure can be defined as

Rg (X) =∫ 1

0F−1X (1− α) dg(α) =

∫ 1

0VaR[X; 1− α] dg(α). (2)

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More generally (risks taking value in R)Definition2We call distorted risk measure

R(X) =∫ 1

0F−1(1− u)dg(u)

where g is some distortion function.

Proposition3R(X) can be written

R(X) =∫ +∞

0g(1− F (x))dx−

∫ 0

−∞[1− g(1− F (x))]dx.

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risk measures R distortion function g

VaR g (x) = I[x ≥ p]Tail-VaR g (x) = min {x/p, 1}PH g (x) = xp

Dual Power g (x) = 1− (1− x)1/p

Gini g (x) = (1 + p)x− px2

exponential transform g (x) = (1− px) / (1− p)

Table 1 – Standard risk measures, p ∈ (0, 1).

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Here, it looks like risk measures can be seen as R(X) = Eg◦P(X).Remark1Let Q denote the distorted measure induced by g on P, denoted g ◦ P i.e.

Q([a,+∞)) = g(P([a,+∞))).

Since g is increasing on [0, 1] Q is a capacity.

Example1Consider function g(x) = xk. The PH - proportional hazard - risk measure is

R(X; k) =∫ 1

0F−1(1− u)kuk−1du =

∫ ∞0

[F (x)]kdx

If k is an integer [F (x)]k is the survival distribution of the minimum over k values.

Definition2The Esscher risk measure with parameter h > 0 is Es[X;h], defined as

Es[X;h] = E[X exp(hX)]MX(h) = d

dhlnMX(h).

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Arthur CHARPENTIER, Distortion in actuarial sciences 2 ARCHIMEDEAN COPULAS

2 Archimedean copulas

Definition3Let φ denote a decreasing function (0, 1]→ [0,∞] such that φ(1) = 0, and such thatφ−1 is d-monotone, i.e. for all k = 0, 1, · · · , d, (−1)k[φ−1](k)(t) ≥ 0 for all t. Definethe inverse (or quasi-inverse if φ(0) <∞) as

φ−1(t) =

φ−1(t) for 0 ≤ t ≤ φ(0)0 for φ(0) < t <∞.

The function

C(u1, · · · , un) = φ−1(φ(u1) + · · ·+ φ(ud)), u1, · · · , un ∈ [0, 1],

is a copula, called an Archimedean copula, with generator φ.

Let Φd denote the set of generators in dimension d.Example2The independent copula C⊥ is an Archimedean copula, with generator φ(t) = − log t.

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The upper Fréchet-Hoeffding copula, defined as the minimum componentwise,M(u) = min{u1, · · · , ud}, is not Archimedean (but can be obtained as the limit ofsome Archimedean copulas).

Set λ(t) = exp[−φ(t)] (the multiplicative generator), then

C(u1, ..., ud) = λ−1(λ(u1) · · ·λ(ud)),∀u1, ..., ud ∈ [0, 1],

which can be written

C(u1, ..., ud) = λ−1(C⊥[λ(u1), . . . , λ(ud)]),∀u1, ..., ud ∈ [0, 1],

Note that it is possible to get an interpretation of that distortion of theindependence.

A large subclass of Archimedean copula in dimension d is the class ofArchimedean copulas obtained using the frailty approach.

Consider random variables X1, · · · , Xd conditionally independent, given a latentfactor Θ, a positive random variable, such that P (Xi ≤ xi|Θ) = Gi (x)Θ whereGi denotes a baseline distribution function.

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The joint distribution function of X is given by

FX (x1, · · · , xd) = E (P (X1 ≤ x1, · · · , Xd ≤ Xd|Θ))

= E

(d∏i=1

P (Xi ≤ xi|Θ))

= E

(d∏i=1

Gi (xi)Θ

)

= E

(d∏i=1

exp [−Θ (− logGi (xi))])

= ψ

(−

d∑i=1

logGi (xi)),

where ψ is the Laplace transform of the distribution of Θ, i.e.ψ (t) = E (exp (−tΘ)) . Because the marginal distributions are given respectivelyby

Fi(xi) = P(Xi ≤ xi) = ψ (− logGi (xi)) ,the copula of X is

C (u) = FX

(F−1

1 (u1) , · · · , F−1d (ud)

)= ψ

(ψ−1 (u) + · · ·+ ψ−1 (ud)

)This copula is an Archimedean copula with generator φ = ψ−1 (see e.g. Clayton(1978), Oakes (1989), Bandeen-Roche & Liang (1996) for more details).

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Arthur CHARPENTIER, Distortion in actuarial sciences 3 HIERARCHICAL ARCHIMEDEAN COPULAS

3 Hierarchical Archimedean copulas

It is possible to look at C(u1, · · · , ud) defined as

φ−11 [φ1[φ−1

2 (φ2[· · ·φ−1d−1[φd−1(u1) + φd−1(u2)] + · · ·+ φ2(ud−1))] + φ1(ud)]

where φi are generators. C is a copula if φi ◦ φ−1i−1 is the inverse of a Laplace

transform. This copula is said to be a fully nested Archimedean (FNA) copula.E.g. in dimension d = 5, we get

φ−11 [φ1(φ−1

2 [φ2(φ−13 [φ3(φ−1

4 [φ4(u1) + φ4(u2)]) + φ3(u3)]) + φ2(u4)]) + φ1(u5)].

It is also possible to consider partially nested Archimedean (PNA) copulas, e.g.by coupling (U1, U2, U3), and (U4, U5),

φ−14 [φ4(φ−1

1 [φ1(φ−12 [φ2(u1) + φ2(u2)]) + φ1(u3)]) + φ4(φ−1

3 [φ3(u4) + φ3(u5)])]

Again, it is a copula if φ2 ◦ φ−11 is the inverse of a Laplace transform, as well as

φ4 ◦ φ−11 and φ4 ◦ φ−1

3 .

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U1 U2 U3 U4 U5

φ4

φ3

φ2

φ1

U1 U2 U3 U4 U5

φ2

φ1

φ3

φ4

Figure 1 – fully nested Archimedean copula, and partially nested Archimedeancopula.

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It is also possible to consider

φ−13 [φ3(φ−1

1 [φ1(u1) + φ1(u2) + φ1(u3)]) + φ3(φ−12 [φ2(u4) + φ2(u5)])].

if φ3 ◦ φ−11 and φ3 ◦ φ−1

2 are inverses of Laplace transform. Or

φ−13 [φ3(φ−1

1 [φ1(u1) + φ1(u2)] + φ3(u3) + φ3(φ−12 [φ2(u4) + φ2(u5)])].

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U1 U2 U3 U4 U5

φ1

φ3

φ2

U1 U2 U3 U4 U5

φ1

φ3

φ2

Figure 2 – Copules Archimédiennes hiérarchiques avec deux constructions dif-férentes.

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Example3If φi’s are Gumbel’s generators, with parameter θi, a sufficient condition for C to be aFNA copula is that θi’s increasing. Similarly if φi’s are Clayton’s generators.

Again, an heuristic interpretation can be derived, see Hougaard (2000), with twofrailties Θ1 and Θ2 such that

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Arthur CHARPENTIER, Distortion in actuarial sciences 4 DISTORTING COPULAS

4 Distorting copulas

Genest & Rivest (2001) extended the concept of Archimedean copulasintroducing the multivariate probability integral transformation (Wang, Nelsen &Valdez (2005) called this the distorted copula, while Klement, Mesiar & Pap(2005) or Durante & Sempi (2005) called this the transformed copula). Considera copula C. Let h be a continuous strictly concave increasing function[0, 1]→ [0, 1] satisfying h (0) = 0 and h (1) = 1, such that

Dh (C) (u1, · · · , ud) = h−1 (C (h (u1) , · · · , h (ud))), 0 ≤ ui ≤ 1

is a copula. Those functions will be called distortion functions.Example4A classical example is obtained when h is a power function, and when the power is theinverse of an integer, hn(x) = x1/n, i.e.

Dhn(C) (u, v) = Cn(u1/n, v1/n), 0 ≤ u, v ≤ 1 and n ∈ N.

Then this copula is the survival copula of the componentwise maxima : the copula of

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(max{X1, · · · , Xn},max{Y1, · · · , Yn}) is Dhn(C), where {(X1, Y1), · · · , (Xn, Yn)}

is an i.i.d. sample, and the (Xi, Yi)’s have copula C.

A max-stable copula is a copula C such that ∀n ∈ N,

Cn(u1/n1 , · · · , u1/n

d ) = C(u1, · · · , ud).

Example5Let φ denote a convex decreasing function on (0, 1] such that φ(1) = 0, and defineC(u, v) = φ−1(φ(u) +φ(v)) = Dexp[−φ](C⊥). This function is an Archimedean copula.

Example6A distorted version of the comonontonic copula is the comonotonic copula,

h−1[min{h(u1), · · · , h(ud)}] = min{u1, · · · , ud}

Example7Following the idea of Capéraà, Fougères & Genest (2000), it is possible to constructArchimax copulas as distortions of max-stable copulas. In dimension d = 2, max-stable

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copulas are characterized through a generator A such that

C(u, v) = exp[log(uv)A

(log(u)log(uv)

)]Here consider φ an Archimedean generator, then Archimax copulas are defined as

C(u, v) = φ−1[[φ(u) + φ(v)]A

(φ(u)

φ(u) + φ(v)

)]In the bivariate case, h need not be differentiable, and concavity is a sufficientcondition.

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With nonconcave distortion function, distorted copulas are semi-copulas, fromBassan & Spizzichino (2001).Definition4Function S : [0, 1]d → [0, 1] is a semi-copula if 0 ≤ ui ≤ 1, i = 1, · · · , d,

S(1, ..., 1, ui, 1, ..., 1) = ui, (3)

S(u1, ..., ui−1, 0, ui+1, ..., ud) = 0, (4)and s 7→ S(u1, ..., ui−1, s, ui+1, ..., ud) is increasing on [0, 1].

Let Hd denote the set of continuous strictly increasing functions [0, 1]→ [0, 1]such that h (0) = 0 and h (1) = 1, C ∈ C,

Dh (C) (u1, · · · , ud) = h−1 (C (h (u1) , · · · , h (ud))) , 0 ≤ ui ≤ 1

is a copula, called distorted copula.Hd-copulas will be functions Dh (C) for some distortion function h and somecopula C.d-increasingness of function Dh (C) is obtained when h ∈ Hd, i.e. h is continuous,

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with h (0) = 0 and h (1) = 1, and such that h(k)(x) ≤ 0 for all x ∈ (0, 1) andk = 2, 3, · · · , d (see Theorem 2.6 and 4.4 in Morillas (2005)).

As a corollary, note that if φ ∈ Φd, then h(x) = exp(−φ(x)) belongs to Hd.Further, observe that for h, h′ ∈ Hd,

Dh◦h′ (C) (u1, · · · , ud) = (Dh ◦ Dh′) (C) (u1, · · · , ud) , 0 ≤ ui ≤ 1.

Again, it is possible to get an intuitive interpretation of that distortion.

Consider a max-stable copula C. Let X be a random vector such that X given Θhas copula C and P (Xi ≤ xi|Θ) = Gi (xi)Θ, i = 1, · · · , d.

Then, the (unconditional) joint distribution function of X is given by

F (x) = E (P (X1 ≤ x1, · · · , Xd ≤ xd|Θ))= E (C (P (X1 ≤ xi|Θ) , · · · ,P (Xd ≤ xd|Θ)))

= E(C(G1 (x1)Θ

, · · · , Gd (xd)Θ))

= E(CΘ (G1 (x1) , · · · , Gd (xd))

)= ψ (− logC (G1 (x1) , · · · , Gd (xd))) ,

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where ψ is the Laplace transform of the distribution of Θ, i.e.ψ (t) = E (exp (−tΘ)), since C is a max-stable copula, i.e.

C(G1 (x1)Θ

, · · · , Gd (xd)Θ)

= CΘ (G1 (x1) , · · · , Gd (xd)) .

The unconditional marginal distribution functions are Fi (xi) = ψ (− logGi (xi)),and therefore

CX (x1, · · · , xd) = ψ(− log

(C(exp

[−ψ−1 (x)

], exp

[−ψ−1 (y)

]))).

Note that since ψ−1 is completly montone, then h belongs to Hd.

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Remark2It is possible to use distortion to obtain stronger tail dependence (with results that can berelated to C & Segers (2007)). Recall that

λL = limu→0

C(u, u)u

and λU = limu→1

1− C(u, u)1− u .

If h−1 is regularly varying in 0 with exponent α > 0, i.e. h−1(t) ∼ L0tα in 0, then

λL(Dh(C)) = [λL(C)]α.

If h−1 is regularly varying in 1 with exponent β > 0, i.e. 1− h−1(t) ∼ L0[1− t]β in 1,then λU (Dh(C)) = 2− [2− λU (C)]β .

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Arthur CHARPENTIER, Distortion in actuarial sciences 5 APPLICATION TO MULTIVARIATE RISK MEASURE

5 Application to multivariate risk measure

Wang (1996) proposed the risk measure based on distortion functiong(t) = Φ(Φ−1(t)− λ), with λ ≥ 0 (to be convex).

Valdez (2009) suggested a multivariate distortion.

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Arthur CHARPENTIER, Distortion in actuarial sciences 6 APPLICATION TO AGING PROBLEMS

6 Application to aging problems

Let T = (T1, · · · , Td) denote remaining lifetime, at time t = 0. Consider theconditional distribution

(T1, · · · , Td) given T1 > t, · · · , Td > t

for some t > 0.

Let C denote the survival copula of T ,

P(T1 > t1, · · · , Td > td) = C(P(T1 > t1), · · · ,P(T1 > tc)).

The survival copula of the conditional distribution is the copula of

(U1, · · · , Ud) given U1 <F 1(t)︸︷︷︸u1

, · · · ,underbraceF d(t)ud

where (U1, · · · , Ud) has distribution C , and where Fi is the distribution of Ti

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Let C be a copula and let U be a random vector with joint distribution functionC. Let u ∈ (0, 1]d be such that C(u) > 0. The lower tail dependence copula of Cat level u is defined as the copula, denoted Cu, of the joint distribution of U

conditionally on the event {U ≤ u} = {U1 ≤ u1, · · · , Ud ≤ ud}.

6.1 Aging with Archimedean copulas

If C is a strict Archimedean copula with generator φ (i.e. φ(0) =∞), then thelower tail dependence copula relative to C at level u is given by the strictArchimedean copula with generator φu defined by

φu(t) = φ(t · C(u))− φ(C(u)), 0 ≤ t ≤ 1,

where C(u) = φ−1[φ(u1) + · · ·+ φ(ud)] (see Juri & Wüthrich (2002) or C & Juri(2007)).

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Example8Gumbel copulas have generator φ (t) = [− ln t]θ where θ ≥ 1. For any u ∈ (0, 1]d, thecorresponding conditional copula has generator

φu (t) =[M1/θ − ln t

]θ−M where M = [− ln u1]θ + · · ·+ [− ln ud]θ .

Example9Clayton copulas C have generator φ (t) = t−θ − 1 where θ > 0. Hence,

φu (t) = [t·C(u)]−θ−1−φ(C(u)) = t−θ·C(u)−θ−1−[C(u)−θ−1] = C(u)−θ·[t−θ−1],

hence φu (t) = C(u)−θ · φ(t). Since the generator of an Archimedean copula is uniqueup to a multiplicative constant, φu is also the generator of Clayton copula, withparameter θ.

Theorem3Consider X with Archimedean copula, having a factor representation, and let ψ denotethe Laplace transform of the heterogeneity factor Θ. Let u ∈ (0, 1]d, then X givenX ≤ F−1

X (u) (in the pointwise sense, i.e. X1 ≤ F−11 (u1), · · · ., Xd ≤ F−1

d (ud)) is an

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Archimedean copula with a factor representation, where the factor has Laplace transform

ψu (t) =ψ(t+ ψ−1 (C(u))

)C(u) .

6.2 Aging with distorted copulas copulas

Recall that Hd-copulas are defined as

Dh(C)(u1, · · · , ud) = h−1(C(h(u1), · · · , h(ud))), 0 ≤ ui ≤ 1,

where C is a copula, and h ∈ Hd is a d-distortion function.

Assume that there exists a positive random variable Θ, such that, conditionallyon Θ, random vector X = (X1, · · · , Xd) has copula C, which does not depend onΘ. Assume moreover that C is in extreme value copula, or max-stable copula (seee.g. Joe (1997)) : C

(xh1 , · · · , xhd

)= Ch (x1, · · · , xd) for all h ≥ 0. The following

result holds,Lemma1

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Let Θ be a random variable with Laplace transform ψ, and consider a random vectorX = (X1, · · · , Xd) such that X given Θ has copula C, an extreme value copula.Assume that, for all i = 1, · · · , d, P (Xi ≤ xi|Θ) = Gi (xi)Θ where the Gi’s aredistribution functions. Then X has copula

CX (x1, · · · , xd) = ψ(− log

(C(exp

[−ψ−1 (x1)

], · · · , exp

[−ψ−1 (xd)

]))),

whose copula is of the form Dh(C) with h(·) = exp[−ψ−1 (·)

].

Theorem4Let X be a random vector with anHd-copula with a factor representation, let ψ denotethe Laplace transform of the heterogeneity factor Θ, C denote the underlying copula, andGi’s the marginal distributions.

Let u ∈ (0, 1]d, then, the copula of X given X ≤ F−1X (u) is

CX,u (x) = ψu

(− log

(Cu

(exp

[−ψ−1

u (x1)], · · · , exp

[−ψ−1

u (xd)])))

= Dhu(Cu)(x),

where hu(·) = exp[−ψ−1

u (·)], and where

– ψu is the Laplace transform defined as ψu (t) = ψ (t+ α) /ψ (α) whereα = − log (C (u∗)), u∗i = exp

[−ψ−1 (ui)

]for all i = 1, · · · , d. Hence, ψu is the

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Laplace transform of Θ given X ≤ F−1X (u),

– P(Xi ≤ xi|X ≤ F−1

X (u) ,Θ)

= G′i (xi)Θ for all i = 1, · · · , d, where

G′i (xi) = C (u∗1, u∗2, · · · , Gi (xi) , · · · , u∗d)C (u∗1, u∗2, · · · , u∗i , · · · , u∗d)

,

– and Cu is the following copula

Cu (x) =C(G1(G′1−1 (x1)

), · · · , Gd

(G′d−1 (xd)

))C(G1(F−1

1 (u1)), · · · , Gd

(F−1d (ud)

)) .

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