Maximum entropy copula with given diagonal section
Cristina Butucea, Jean-François Delmas, Anne Dutfoy, Richard Fischer
aa r X i v : . [ m a t h . S T ] D ec MAXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION
CRISTINA BUTUCEA, JEAN-FRANC¸ OIS DELMAS, ANNE DUTFOY, AND RICHARD FISCHER
Abstract.
We consider copulas with a given diagonal section and compute the explicitdensity of the unique optimal copula which maximizes the entropy. In this sense, thiscopula is the least informative among the copulas with a given diagonal section. We give anexplicit criterion on the diagonal section for the existence of the optimal copula and give aclosed formula for its entropy. We also provide examples for some diagonal sections of usualbivariate copulas and illustrate the differences between them and the maximum entropycopula with the same diagonal section. Introduction
Dependence of random variables can be described by copula distributions. A copula isthe cumulative distribution function of a random vector U = ( U , . . . , U d ) with U i uniformlydistributed on I = [0 , Nelsen [16].The diagonal section δ of a d -dimesional copula C , defined on I as δ ( t ) = C ( t, . . . , t ) isthe cumulative distribution function of max ≤ i ≤ d U i . The function δ is non-decreasing, d -Lipschitz, and verifies δ ( t ) ≤ t for all t ∈ I with δ (0) = 0 and δ (1) = 1. It was shown that if afunction δ satisfies these properties, then there exists a copula with δ as diagonal section (see Bertino [2] or
Fredricks and Nelsen [12] for d = 2 and Cuculescu and Theodorescu [6] for d ≥ δ is thecumulative distribution function of the maximum of the marginals, it also characterizes thetail dependence of the copula (see Joe [14] p.33. and references in
Nelsen et al. [18],
Du-rante and Jaworski [8],
Jaworski [13]) as well as the generator for Archimedean copulas(
Sungur and Yang [25]). For d = 2, Bertino in [2] introduces the so-called Bertino copula B δ given by B δ ( u, v ) = u ∧ v − min u ∧ v ≤ t ≤ u ∨ v ( t − δ ( t )) for u, v ∈ I . Fredricks and Nelsen in[12] give the example called diagonal copula defined by K δ ( u, v ) = min( u, v, ( δ ( u ) + δ ( v )) / u, v ∈ I . In Nelsen et al. [17, 18] lower and upper bounds related to the pointwisepartial ordering are given for copulas with a given diagonal section. They showed that if C is a symmetric copula with diagonal section δ , then for every u, v ∈ I , we have: B δ ( u, v ) ≤ C ( u, v ) ≤ K δ ( u, v ) . Durante et al. [10] provide another construction of copulas for a certain class of diag-onal sections, called MT-copulas named after Mayor and Torrens and defined as D δ ( u, v ) =max(0 , δ ( x ∨ y ) − | x − y | ). Bivariate copulas with given sub-diagonal sections δ x : [0 , − x ] → [0 , − x ] , δ x ( t ) = C ( x + t, t ) are constructed from copulas with given diagonal sections in Date : August 21, 2018.2010
Mathematics Subject Classification.
Key words and phrases. copula, entropy, diagonal section.This work is partially supported by the French “Agence Nationale de la Recherche”,CIFRE n ◦ Quesada-Molina et al. [22].
Durante et al. [9] or [18] introduce the technique ofdiagonal splicing to create new copulas with a given diagonal section based on other suchcopulas. According to [8] for d = 2 and Jaworski [13] for d ≥
2, there exists an absolutelycontinuous copula with diagonal section δ if and only if the set Σ δ = { t ∈ I ; δ ( t ) = t } haszero Lebesgue measure. de Amo et al. [7] is an extension of [8] for given sub-diagonalsections. Further construction of possibly asymmetric absolutely continuous bidimensionalcopulas with a given diagonal section is provided in Erdely and Gonz´alez [11].Our aim is to find the most uninformative copula with a given diagonal section δ . Wechoose here to maximize the relative entropy to the uniform distribution on I d , among thecopulas with given diagonal section. This is equivalent to minimizing the Kullback-Leiblerdivergence with respect to the independent copula. The Kullback-Leibler divergence is finiteonly for absolutely continuous copulas. The previously introduced bivariate copulas B δ , K δ and D δ are not absolutely continuous, therefore their Kullback-Leibler divergence is infinite.Possible other entropy criteria, such as R´enyi, Tsallis, etc. are considered for example in Pougaza and Mohammad-Djafari [21]. We recall that the entropy of a d -dimensionalabsolutely continuous random vector X = ( X , . . . , X d ) can be decomposed as the sum of theentropy of the marginals and the entropy of the corresponding copula (see Zhao and Lin [26]) : H ( X ) = d X i =1 H ( X i ) + H ( U ) , where H ( Z ) = − R f Z ( z ) log f Z ( z ) dz is the entropy of the random variable Z with density f Z , and U = ( U , . . . , U d ) is a random vector with U i uniformly distributed on I , such that U has the same copula as X ; namely U is distributed as (cid:0) F − ( X ) , . . . F − d ( X d ) (cid:1) with F i thecumulative distribution function of X i . Maximizing the entropy of X with given marginalstherefore corresponds to maximizing the entropy of its copula. The maximum relative entropyapproach for copulas has an extensive litterature. Existence results for an optimal solution onconvex closed subsets of copulas for the total variation distance can be derived from Csisz´ar [5]. A general discussion on abstract entropy maximization is given by
Borwein et al. [3]. This theory was applied for copulas and a finite number of expectation constraints in
Bedford and Wilson [1]. Some applications for various moment-based constraints includerank correlation (
Meeuwissen and Bedford [15],
Chu [4],
Piantadosi et al. [20]) andmarginal moments (
Pasha and Mansoury [19]).We shall apply the theory developed in [3] to compute the density of the maximum entropycopula with a given diagonal section. We show that there exists a copula with diagonalsection δ and finite entropy if and only if δ satisfies: R I | log( t − δ ( t )) | dt < + ∞ . Notice thatthis condition is stronger than the condition of Σ δ having zero Lebesgue measure which isrequired for the existence of an absolutely continuous copula with diagonal section δ . Underthis condition, and in the case of Σ δ = { , } , the optimal copula’s density c δ turns out to beof the form, for x = ( x , . . . , x d ) ∈ I d : c δ ( x ) = b (max( x )) Y x i =max( x ) a ( x i ) , with the notation max( x ) = max ≤ i ≤ d x i , see Theorem 2.3. The optimal copula’s density inthe general case is given in Theorem 2.4. Notice that c δ is symmetric, that is it is invariantunder the permutation of the variables. This provides a new family of absolutely continuoussymmetric copulas with given diagonal section enriching previous work on this subject that AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 3 we discussed, see [2],[8],[9],[10],[11],[12],[18]. We also calculate the maximum entropy copulafor diagonal sections that arise from well-known families of bivariate copulas.The rest of the paper is organised as follows. Section 2 introduces the definitions andnotations used later on, and gives the main theorems of the paper. In Section 3 we study theproperties of the feasible solution c δ of the problem for a special class of diagonal sectionswith Σ δ = { , } . In Section 4, we formulate our problem as a linear optimization problemin order to apply the theory established in [3]. Then in Section 5 we give the proof for ourmain theorem showing that c δ is indeed the optimal solution when Σ δ = { , } . In Section6 we extend our results for the general case when Σ δ has zero Lebesgue measure. We givein Section 7 several examples with diagonals of popular bivariate copula families such as theGaussian, Gumbel or Farlie-Gumbel-Morgenstern copulas among others.2. Main results
Let d ≥ C defined on I d , with I = [0 , d -dimensionalcopula if there exists a random vector U = ( U , . . . , U d ) such that U i are uniform on I and C ( u ) = P ( U ≤ u ) for u ∈ I d , with the convention that x ≤ y for x = ( x , . . . .x d ) and y = ( y , . . . , y d ) elements of R d if and only if x i ≤ y i for all 1 ≤ i ≤ d . We shall say that C isthe copula of U . We refer to [16] for a monograph on copulas. The copula C is said absolutelycontinuous if the random variable U has a density, which we shall denote by c C . In this case,we have that a.e. c C ( u ) = ∂ du ,...,u d C ( u ) for u ∈ I d . When there is no confusion, we shallwrite c for the density c C associated to the copula C . We denote by C the set of d -dimensionalcopulas and by C the subset of the d -dimensional absolutely continuous copulas.The diagonal section δ C of a copula C is defined by: δ C ( t ) = C ( t, . . . , t ). Let us note, for u ∈ R d , max( u ) = max ≤ i ≤ d u i . Notice that if C is the copula of U , then δ C is the cumulativedistribution function of max( U ) as δ C ( t ) = P (max( U ) ≤ t ) for t ∈ I . We denote by D = { δ C , C ∈ C} the set of diagonal sections of d -dimensional copulas and by D = { δ C ; C ∈ C } the set of diagonal sections of absolutely continuous copulas. According to [12], a function δ defined on I belongs to D if and only if:(i) δ is a cumulative function on [0 , δ (0) = 0, δ (1) = 1 and δ is non decreasing;(ii) δ ( t ) ≤ t for t ∈ I and δ is d -Lipschitz: | δ ( s ) − δ ( t ) | ≤ d | s − t | for s, t ∈ I .For δ ∈ D , we shall consider the set C δ = { C ∈ C ; δ C = δ } of copulas with diagonal section δ , and the subset C δ = C δ T C of absolutely continuous copulas with section δ . Accordingto [8] and [13], the set C δ is non empty if and only if the set Σ δ = { t ∈ I ; δ ( t ) = t } has zeroLebesgue measure.For a non-negative measurable function f defined on I k , k ∈ N ∗ , we set I k ( f ) = Z I k f ( x ) log( f ( x )) dx. Since copulas are cumulative functions of probability measures, we will consider the Kull-back-Leibler divergence relative to the uniform distribution as a measure of entropy, see [5]: I ( C ) = ( I d ( c ) if C ∈ C ,+ ∞ if C
6∈ C ,with c the density associated to C when C ∈ C . Notice the Shannon-entropy introduced in[24] of the probability measure P defined on I d with cumulative distribution function C isdefined as H ( P ) = −I ( C ). Thus minimizing the Kullback-Leibler divergence I (w.r.t. the CRISTINA BUTUCEA, JEAN-FRANC¸ OIS DELMAS, ANNE DUTFOY, AND RICHARD FISCHER uniform distribution) is equivalent to maximizing the Shannon-entropy. It is well known thatthe copula Π with density c Π = 1, which corresponds to ( U i , ≤ i ≤ d ) being independent,minimizes I ( C ) over C .We shall minimize the entropy I over the set C δ or equivalently over C δ of copulas with agiven diagonal section δ ∈ D (in fact for δ ∈ D as otherwise C δ is empty). If C minimizes I on C δ , it means that C is the least informative (or the “most random”) copula with givendiagonal section δ .For δ ∈ D , let us denote:(1) J ( δ ) = Z I | log( t − δ ( t )) | dt. Notice that J ( δ ) ∈ [0 , + ∞ ] and it is infinite if δ
6∈ D . Since δ is d -Lipschitz, the derivative δ ′ of δ exists a.e. and since δ is non-decreasing we have a.e. 0 ≤ δ ′ ≤ d . This implies that I ( δ ′ ) and I ( d − δ ′ ) are well defined. Let us denote:(2) G ( δ ) = I ( δ ′ ) + I ( d − δ ′ ) − d log( d ) − ( d − . We have the rough upper bound:(3) sup δ ∈D |G ( δ ) | ≤ d + d log( d ) . The following Proposition gives an absolutely continuous copula whose diagonal section is δ . The proof of this Proposition can be found in Section 3 and Section 8 is dedicated to theproof of (6). Proposition 2.1.
Let δ ∈ D with Σ δ = { , } . We define, for r ∈ I : h ( r ) = r − δ ( r ) , F ( r ) = d − d Z r h ( s ) ds, (4) a ( r ) = d − δ ′ ( r ) d h ( r ) − /d e F ( r ) and b ( r ) = δ ′ ( r ) d h ( r ) − /d e − ( d − F ( r ) . Then c δ defined a.e. by (5) c δ ( x ) = b (max( x )) Y x i =max( x ) a ( x i ) for x = ( x , . . . , x d ) ∈ I d , is the density of a symmetric copula C δ with diagonal section δ . Furthermore, we have: (6) I ( C δ ) = ( d − J ( δ ) + G ( δ ) . This and (3) readily implies the following Remark.
Remark . Let δ ∈ D such that Σ δ = { , } . We have I ( C δ ) < + ∞ if and only if J ( δ ) < + ∞ .We can now state our main result in the simpler case Σ δ = { , } . It gives the necessaryand sufficient condition for C δ to be the unique optimal solution of the minimization problem.The proof is given in Section 5. Theorem 2.3.
Let δ ∈ D such that Σ δ = { , } . a) If J ( δ ) = + ∞ then min C ∈C δ I ( C ) = + ∞ . b) If J ( δ ) < + ∞ then min C ∈C δ I ( C ) < + ∞ and C δ is the unique copula such that I ( C δ ) = min C ∈C δ I ( C ) . AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 5
To give the answer in the general case where Σ δ has zero Lebesgue measure, we need someextra notations. Since δ is continuous, we get that I \ Σ δ can be written as the union ofnon-empty open intervals (( α j , β j ) , j ∈ J ), with α j < β j and J at most countable. Noticethat δ ( α j ) = α j and δ ( β j ) = β j . For J = ∅ and j ∈ J , we set ∆ j = β j − α j and for t ∈ I :(7) δ j ( t ) = δ ( α j + t ∆ j ) − α j ∆ j · It is clear that δ j satisfies (i) and (ii) and it belongs to D as Σ δ j = { , } . Let c δ j be definedby (5) with δ replaced by δ j . For δ ∈ D such that Σ δ = { , } , we define the function c δ by,for u ∈ I d :(8) c δ ( u ) = X j ∈ J j c δ j (cid:18) u − α j ∆ j (cid:19) ( α j ,β j ) d ( u ) , with = (1 , . . . , ∈ R d . It is easy to check that c δ is a copula density and that is zerooutside [ α j , β j ] d for j ∈ J . We state our main result in the general case whose proof is givenin Section 6. Theorem 2.4.
Let δ ∈ D . a) If J ( δ ) = + ∞ then min C ∈C δ I ( C ) = + ∞ . b) If J ( δ ) < + ∞ then min C ∈C δ I ( C ) < + ∞ and there exists a unique copula C δ ∈ C δ such that I ( C δ ) = min C ∈C δ I ( C ) . Furthermore, we have: I ( C δ ) = ( d − J ( δ ) + G ( δ ); the copula C δ is absolutely continuous, symmetric; its density c δ is given by (5) if Σ δ = { , } or by (8) if Σ δ = { , } .Remark . For δ ∈ D , notice the condition J ( δ ) < + ∞ implies that Σ δ has zero Lebesguemeasure, and therefore, according to [8] and [13], δ ∈ D . And if δ
6∈ D , then I ( C ) = + ∞ for all C ∈ C δ . Therefore, we could replace the condition δ ∈ D by δ ∈ D in Theorem 2.4.3. Proof of Proposition 2.1
We assume that δ ∈ D and Σ δ = { , } . We give the proof of Proposition 2.1, which statesthat C δ , with density c δ given by (5), is indeed a symmetric copula with diagonal section δ whose entropy is given by (6).Recall the definition of h, F, a, b and c δ from Theorem 2.3. Notice that by construction c δ is non-negative and well defined on I d . In order to prove that c δ is the density of a copula,we only have to prove that for all 1 ≤ i ≤ d , r ∈ I : Z I d c δ ( u ) { u i ≤ r } du = r, or equivalently Z I d c δ ( u ) { u i ≥ r } du = 1 − r. We define for r ∈ I :(9) A ( r ) = Z r a ( t ) dt. Elementary computations yield for r ∈ (0 , A ( r ) = h /d ( r ) e F ( r ) . CRISTINA BUTUCEA, JEAN-FRANC¸ OIS DELMAS, ANNE DUTFOY, AND RICHARD FISCHER
Notice that F (0) ∈ [ −∞ ,
0] which implies that A (0) = 0. A direct integration gives:(11) d Z I A d − ( s ) b ( s ) { s ≥ r } = 1 − δ ( r ) . We also have:( d − Z I A d − ( s ) b ( s ) ds { s ≥ r } = ( d − d Z I h − /d ( s ) e − F ( s ) { s ≥ r } ds = h − h − /d ( s ) e − F ( s ) i s = r = h − /d ( r ) e − F ( r ) , (12)where we used for the last step that h (1) = 0 and F (1) ∈ [0 , ∞ ]. We have: Z I d c δ ( u ) { u i ≥ r } du = Z I d b (max( u )) Y u j =max( u ) a ( u j ) { u i ≥ r } du = Z I A d − ( s ) b ( s ) { s ≥ r } ds + ( d − Z I A d − ( s ) b ( s )( A ( s ) − A ( r )) { s ≥ r } ds = d Z I A d − ( s ) b ( s ) { s ≥ r } ds − ( d − A ( r ) Z I A d − ( s ) b ( s ) { s ≥ r } ds = 1 − δ ( r ) − ( r − δ ( r ))= 1 − r, where we first divided the integral according to which u i was the maximum; then we used(9) for the second equality, finally (11) and (12) for the forth. This implies that c δ is indeedthe density of a copula. We denote by C δ the copula with density c δ . We check that δ is thediagonal section of C δ . Using (11), we get, for r ∈ I : Z I d c δ ( u ) { max( u ) ≤ r } du = Z I d b (max( u )) Y u i =max( u i ) a ( u i ) { max( u ) ≥ r } du = d Z I A d − ( s ) b ( s ) { s ≤ r } ds = δ ( r ) . The calculations which show that the entropy of C δ is given by (6) can be found in Section 8.4. The minimization problem
Let δ ∈ D . As a first step we will show, using [3], that the problem of a maximumentropy copula with a given diagonal section δ has at most a unique optimal solution. Toformulate this problem in the framework of [3], we introduce the continuous linear functional A = ( A i , ≤ i ≤ d + 1) : L ( I d ) → L ( I ) d +1 defined by, for 1 ≤ i ≤ d , f ∈ L ( I d ) and r ∈ I , A i ( f )( r ) = Z I d f ( u ) { u i ≤ r } du, and A d +1 ( f )( r ) = Z I d f ( u ) { max( u ) ≤ r } du. AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 7
We also define b δ = ( b i , ≤ i ≤ d + 1) ∈ L ( I ) d +1 with b d +1 = δ and b i = id I for 1 ≤ i ≤ d ,with id I the identity map on I . Notice that the conditions A i ( c ) = b i , 1 ≤ i ≤ d , and c ≥ c is the density of a copula C ∈ C . If we assume further that the condition A d +1 ( c ) = b d +1 holds then the diagonal section of C is δ (thus C ∈ C δ ).Since I is infinite outside C δ and the density of any copula in C belongs to L ( I d ), we getthat minimizing I over C δ is equivalent to the linear optimization problem ( P δ ) given by:( P δ ) minimize I d ( c ) subject to ( A ( c ) = b δ ,c ≥ c ∈ L ( I d ) . We say that a function f is feasible for ( P δ ) if f ∈ L ( I d ), f ≥ A ( f ) = b δ and I d ( f ) < + ∞ . Notice that any feasible f is the density of a copula. We say that f is anoptimal solution to ( P δ ) if f is feasible and I d ( f ) ≤ I d ( g ) for all g feasible. Proposition 4.1.
Let δ ∈ D . If there exists a feasible c , then there exists a unique optimalsolution to ( P δ ) and it is symmetric.Proof. Since A ( f ) = b δ implies A ( f )(1) = b (1) that is R I d f ( x ) dx = 1, we can directlyapply Corollary 2.3 of [3] which states that if there exists a feasible c , then there exists aunique optimal solution to ( P δ ). Since the constraints are symmetric and the functional I d is also symmetric, we deduce that the unique optimal solution is also symmetric. (cid:3) The next Proposition gives that the set of zeros of any non-negative solution c of A ( c ) = b δ contains:(13) Z δ = { u ∈ I d ; δ ′ (max( u )) = 0 or ∃ i such that u i < max( u ) and δ ′ ( u i ) = d } . Proposition 4.2.
Let δ ∈ D . If c is feasible then c = 0 a.e. on Z δ (that is c Z δ = 0 a.e.).Proof. Recall that 0 ≤ δ ′ ≤ d . Since c ∈ L ( I d ), the condition A d +1 ( c ) = b d +1 , that is for all r ∈ I Z I d c ( u ) { max( u ) ≤ r } du = Z r δ ′ ( s ) ds, implies, by the monotone class theorem, that for all measurable subset H of I , we have: Z I d c ( u ) H (max( u )) du = Z H δ ′ ( s ) ds. Since c ≥ c ( u ) { δ ′ (max( u ))=0 } = 0.Next, notice that for all r ∈ I , 1 ≤ i ≤ d , the symmetrical property of c gives: Z I d c ( u ) { u i < max( u ) ,u i ≤ r } du = Z I d c ( u ) { u i ≤ r } du − Z I d c ( u ) { u i =max( u ) ,u i ≤ r } du = r − δ ( r ) d = Z r (cid:18) − δ ′ ( s ) d (cid:19) ds. This implies that a.e. c ( u ) {∃ i such that u i < max( u ) ,δ ′ ( u i )= d } = 0. This gives the result. (cid:3) We define µ to be the Lebesgue measure restricted to Z cδ = I d \ Z δ : µ ( du ) = Z cδ ( u ) du .We define, for f ∈ L ( I d , µ ): I µ ( f ) = Z I d f ( u ) log( f ( u )) µ ( du ) . CRISTINA BUTUCEA, JEAN-FRANC¸ OIS DELMAS, ANNE DUTFOY, AND RICHARD FISCHER
From Proposition 4.2 we can deduce that if c is feasible then I µ ( c ) = I d ( c ). Let us alsodefine, for 1 ≤ i ≤ d , r ∈ I : A µi ( c )( r ) = Z I d c ( u ) { u i ≤ r } µ ( du ) , and A µd +1 ( c )( r ) = Z I d c ( u ) { max( u ) ≤ r } µ ( du ) . The corresponding optimization problem ( P δµ ) is given by :( P δµ ) minimize I µ ( c ) subject to ( A µ ( c ) = b δ ,c ≥ µ -a.e. and c ∈ L ( I d , µ ) , with A µ = ( A µi , ≤ i ≤ d + 1). For f ∈ L ( I d , µ ), we define: f µ = ( f on Z cδ , Z δ . Using Proposition 4.2, we easily get the following Corollary.
Corollary 4.3. If c is a solution of ( P δµ ) , then c µ is a solution of ( P δ ) . If c is a solution of ( P δ ) , then it is also a solution of ( P δµ ) . Proof of Theorem 2.3
Form of the optimal solution.
Let ( A µ ) ∗ : L ∞ ( I ) d +1 → L ∞ ( I d , µ ) be the adjoint of A µ . We will use Theorem 2.9. from [3] on abstract entropy minimization, which we recallhere, adapted to the context of ( P δµ ). Theorem 5.1 (Borwein, Lewis and Nussbaum) . Suppose there exists c > µ -a.e. which isfeasible for ( P δµ ) . Then there exists a unique optimal solution, c ∗ , to ( P δµ ) . Furthermore, wehave c ∗ > µ -a.e. and there exists a sequence ( λ n , n ∈ N ∗ ) of elements of L ∞ ( I ) d +1 suchthat: (14) Z I d c ∗ ( x ) | ( A µ ) ∗ ( λ n )( x ) − log( c ∗ ( x )) | µ ( dx ) −−−→ n →∞ . We first compute ( A µ ) ∗ . For λ = ( λ i , ≤ i ≤ d + 1) ∈ L ∞ ( I ) d +1 and f ∈ L ∞ ( I d , µ ), wehave: h ( A µ ) ∗ ( λ ) , f i = h λ, A µ ( f ) i = d X i =1 Z I dr λ i ( r ) Z I d f ( x ) { x i ≤ r } dµ ( x ) + Z I dr λ d +1 ( r ) Z I d f ( x ) { max( x ) ≤ r } dµ ( x )= Z I d dµ ( x ) f ( x ) d X i =1 Λ i ( x i ) + Λ d +1 (max( x )) ! , where we used the definition of the adjoint operator for the first equality, Fubini’s theoremfor the second, and the following notation for the third equality:Λ i ( x i ) = Z I λ i ( r ) { r ≥ x i } dr, and Λ d +1 ( t ) = Z I λ d +1 ( r ) { r ≥ t } dr. Thus, we can set for λ ∈ L ∞ ( I ) d +1 and x ∈ I d :(15) ( A µ ) ∗ ( λ )( x ) = d X i =1 Λ i ( x i ) + Λ d +1 (max( x )) . AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 9
Now we are ready to prove that the optimal solution c ∗ of ( P δµ ) is the product of measurableunivariate functions. Lemma 5.2.
Let δ ∈ D such that Σ δ = { , } . Suppose that there exists c > µ -a.e.whichis feasible for ( P δµ ) . Then there exist a ∗ , b ∗ non-negative, measurable functions defined on I such that c ∗ ( u ) = b ∗ (max( u )) Y u i =max( u ) a ∗ ( u i ) µ -a.e.with a ∗ ( s ) = 0 if δ ′ ( s ) = d and b ∗ ( s ) = 0 if δ ′ ( s ) = 0 .Proof. According to Theorem 5.1, there exists a sequence ( λ n , n ∈ N ∗ ) of elements of L ∞ ( I ) d +1 such that the optimal solution, say c ∗ , satisfies (14). This implies, thanks to (15), that thereexist d + 1 sequences (Λ ni , n ∈ N ∗ , ≤ i ≤ d + 1) of elements of L ∞ ( I ) such that the followingconvergence holds in L ( I d , c ∗ µ ):(16) d X i =1 Λ ni ( u i ) + Λ nd +1 (max( u )) −−−→ n →∞ log( c ∗ ( u )) . Arguing as in Proposition 4.1 and since Z cδ is symmetric, we deduce that c ∗ is symmetric.Therefore we shall only consider functions supported on the set △ = { u ∈ I d ; u d = max( u ) } .The convergence (16) holds in L ( △ , c ∗ µ ). For simplicity, we introduce the functions Γ ni ∈ L ∞ ( I ) defined by Γ ni = Λ ni for 1 ≤ i ≤ d −
1, and Γ nd = Λ nd + Λ nd +1 . Then we have in L ( △ , c ∗ µ ):(17) d X i =1 Γ ni ( u i ) −−−→ n →∞ log( c ∗ ( u )) . We first assume that there exist Γ and Γ d measurable functions defined on I such that µ -a.e.on △ :(18) d − X i =1 Γ( u i ) + Γ d ( u d ) = log( c ∗ ( u )) . The symmetric property of c ∗ ( u ) seen in Proposition 4.1 implies we can choose Γ i = Γ for1 ≤ i ≤ d − d . Set a ∗ = exp(Γ) and b ∗ = exp(Γ d ) so that µ -a.e.on △ :(19) c ∗ ( u ) = b ∗ ( u d ) d − Y i =1 a ∗ ( u i ) . Recall µ ( du ) = Z cδ ( u ) du . From the definition (13) of Z δ , we deduce that without loss ofgenerality, we can assume that a ∗ ( u i ) = 0 if δ ′ ( u i ) = d and b ∗ ( u d ) = 0 if δ ′ ( u d ) = 0. Use thesymmetry of c ∗ to conclude.To complete the proof, we now show that (18) holds for Γ and Γ d measurable functions. Weintroduce the notation u ( − i ) = ( u , . . . , u i − , u i +1 , . . . , u d ) ∈ I d − . Let us define the probabil-ity measure P ( dx ) = c ∗ ( x ) △ ( x ) µ ( dx ) / R △ c ∗ ( y ) µ ( dy ) on I d . We fix j , 1 ≤ j ≤ d −
1. In orderto apply Proposition 2 of [23], we first check that P is absolutely continuous with respect to P j ⊗ P j , where P j ( du ( − j ) ) = R u j ∈ I P ( du ( − j ) du j ) and P j ( du j ) = R u ( − j ) ∈ I d − P ( du ( − j ) du j ) are the marginals of P . Notice the following equivalence of measures:(20) P ( du ) ∼ △ ( u ) d − Y i =1 { δ ′ ( u i ) = d } { δ ′ ( u d ) =0 } du. Let B ⊂ I d − be measurable. We have: P ( B ) = 0 ⇐⇒ Z I d △ ( u ) d − Y i =1 { δ ′ ( u i ) = d } { δ ′ ( u d ) =0 } B ( u ( − j ) ) du = 0 . By Fubini’s theorem this last equiality is equivalent to:(21) Z I d − d − Y i =1 ,i = j (cid:0) { δ ′ ( u i ) = d } { u i ≤ u d } (cid:1) { δ ′ ( u d ) =0 } B ( u ( − j ) ) (cid:18)Z I { ≤ u j ≤ u d } { δ ′ ( u j ) = d } du j (cid:19) du ( − j ) = 0 . Since, for ε > δ ( ε ) < ε < dε , we have R I { ≤ u j ≤ s } { δ ′ ( u j ) = d } du j > s ∈ I . Therefore(21) is equivalent to Z I d − d − Y i =1 ,i = j (cid:0) { δ ′ ( u i ) = d } { u i ≤ u d } (cid:1) { δ ′ ( u d ) =0 } B ( u ( − j ) ) du ( − j ) = 0 . This implies that there exists h > I d − such that P j ( du ( − j ) ) = h ( u ( − j ) ) d − Y i =1 ,i = j (cid:0) { δ ′ ( u i ) = d } { u i ≤ u d } (cid:1) { δ ′ ( u d ) =0 } du ( − j ) . Similarly we have for B ′ ⊂ I that P j ( B ′ ) = 0 if and only if(22) Z I { δ ′ ( u j ) = d } B ′ ( u j ) Z I d − d − Y i =1 ,i = j (cid:0) { δ ′ ( u i ) = d } { u i ≤ u d } (cid:1) { δ ′ ( u d ) =0 } { u d ≥ u j } du ( − j ) du j = 0 . Since, for ε > δ (1) − δ (1 − ε ) > − (1 − ε ) = ε > g > I such that P j ( du j ) = g ( u j ) { δ ′ ( u j ) = d } du j . Therefore by (20) we deduce that P is absolutely continuouswith respect to P j ⊗ P j . Then according to Proposition 2 of [23], (17) implies that thereexist measurable functions Φ j and Γ j defined respectively on I d − and I , such that c ∗ µ -a.e.on △ : log( c ∗ ( u )) = Φ j ( u ( − j ) ) + Γ j ( u j ) . As µ -a.e. c ∗ >
0, this equality holds µ -a.e. on △ . Since we have such a representation forevery 1 ≤ j ≤ d −
1, we can easily verify that there exists a measurable function Γ d definedon I such that log( c ∗ ( u )) = P di =1 Γ i ( u i ) µ -a.e. on △ . (cid:3) Calculation of the optimal solution.
Now we prove that the optimal solution to( P δ ), if it exists, is indeed c δ . Proposition 5.3.
Let δ ∈ D such that Σ δ = { , } . If there exists an optimal solution to ( P δ ) , then it is c δ given by (5). AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 11
Proof.
In Lemma 5.2 we have already shown that if an optimal solution exists for ( P δ ), thenit is of the form c ∗ ( u ) = b ∗ (max( u )) Q u i =max( u ) a ∗ ( u i ). Here we will prove that the constraintsof ( P δ ) uniquely determine the functions a ∗ and b ∗ up to a multiplicative constant, giving c ∗ = c δ . We set for r ∈ I : A ∗ ( r ) = Z r a ∗ ( s ) ds which take values in [0 , + ∞ ]. From A d +1 ( c ∗ ) = b δd +1 , we have for r ∈ I : δ ( r ) = Z I d c ∗ ( u ) { max( u ) ≤ r } du = Z I d b ∗ (max( u )) Y u i =max( u ) a ∗ ( u i ) { max( u ) ≤ r } du = d Z I ( A ∗ ( s )) d − b ∗ ( s ) { s ≤ r } ds. (23)Taking the derivative with respect to r gives a.e. on I :(24) δ ′ ( r ) = d ( A ∗ ( r )) d − b ∗ ( r ) . This implies that A ∗ ( r ) is finite for all r ∈ [0 ,
1) and thus A ∗ (0) = 0. Similarly, using that A ( c ∗ ) = b δ , we get that for r ∈ I :1 − r = Z I d c ∗ ( u ) { u ≥ r } du = Z I d b ∗ (max( u )) Y u i =max( u ) a ∗ ( u i ) { u ≥ r } du = Z I d d Y i =2 (cid:0) a ∗ ( u i ) { u i ≤ u } (cid:1) b ∗ ( u ) { u ≥ r } du + ( d − Z I d a ∗ ( u ) d Y i =3 (cid:0) a ∗ ( u i ) { u i ≤ u } (cid:1) b ∗ ( u ) { u ≥ u ≥ r } du = Z I ( A ∗ ( s )) d − b ∗ ( s ) { s ≥ r } ds + ( d − Z I ( A ∗ ( s )) d − b ∗ ( s )( A ∗ ( s ) − A ∗ ( r )) { s ≤ r } ds = d Z I ( A ∗ ( s )) d − b ∗ ( s ) { s ≥ r } ds − ( d − A ∗ ( r ) Z I ( A ∗ ( s )) d − b ∗ ( s ) { s ≥ r } ds. Using this and (23) we deduce that for r ∈ I :(25) h ( r ) = ( d − A ∗ ( r ) Z I ( A ∗ ( s )) d − b ∗ ( s ) { s ≥ r } ds. Since r > δ ( r ) on (0 , A ∗ and R I ( A ∗ ( s )) d − b ∗ ( s ) { s ≥ r } ds are positive on (0 , r ∈ I : d − d δ ′ ( r ) h ( r ) = ( A ∗ ( r )) d − b ( r ) R I ( A ∗ ( r )) d − b ∗ ( s ) { r ≤ s ≤ } ds · We integrate both sides to get for r ∈ I : d − d log (cid:18) h ( r ) h (1 / (cid:19) − Z r / h ( s ) ds ! = log R I ( A ∗ ( s )) d − b ∗ ( s ) { r ≤ s ≤ } ds R I ( A ∗ ( s )) d − b ∗ ( s ) { / ≤ s ≤ } ds ! . Taking the exponential yields:(26) αh ( d − /d ( r ) e − F ( r ) = Z I ( A ∗ ( s )) d − b ∗ ( s ) { r ≤ s ≤ } ds, for some positive constant α . From (25) and (26), we derive:(27) A ∗ ( r ) = 1 α ( d − h /d ( r ) e F ( r ) . This proves that the function A ∗ is uniquely determined up to a multiplicative constant andso is a ∗ . With the help of (24) and (27), we can express b ∗ as, for r ∈ I :(28) b ∗ ( r ) = δ ′ ( r )( α ( d − d − d e − ( d − F ( r ) . The function b ∗ is also uniquely determined up to a multiplicative constant. Therefore (24)implies that there is a unique c ∗ of the form (19) which solves A ( c ) = b δ . (Notice howeverthat the functions a ∗ and b ∗ are defined up to a multiplicative constant.) Then according toProposition 2.1 we get that c δ defined by (19) with a and b defined by (4) solves A ( c ) = b δ ,implying that c ∗ is equal to c δ . (cid:3) Proof of Theorem 2.3.
Let δ ∈ D such that Σ δ = { , } . Thanks to Proposition 5.3,we deduce that if there exists an optimal solution to ( P δ ) then it is c δ given by (19). Byconstruction, we have µ -a.e. c δ >
0. According to Corollary 2.2, c δ is feasible for ( P δ ) if andonly if J ( δ ) < + ∞ . Therefore if J ( δ ) < + ∞ , then c δ is the optimal solution. If J ( δ ) = + ∞ then there is no optimal solution.6. Proof of Theorem 2.4
We first state an elementary Lemma, whose proof if left to the reader. For f a functiondefined on I d and 0 ≤ s < t ≤
1, we define f s,t by, for u ∈ I d : f s,t ( u ) = ( t − s ) f ( s + u ( t − s )) . Lemma 6.1. If c is the density of a copula C such that δ C ( s ) = s and δ C ( t ) = t for somefixed ≤ s < t ≤ , then c s,t is also the density of a copula, and its diagonal section, δ s,t , isgiven by, for r ∈ I : δ s,t ( r ) = δ C ( s + r ( t − s )) − st − s · According to Remark 2.5, it is enough to consider the case δ ∈ D , that is Σ δ with zeroLebesgue measure. We shall assume that Σ δ = { , } . Since δ is continuous, we get that I \ Σ δ can be written as the union of non-empty open intervals (( α j , β j ) , j ∈ J ), with α j < β j and J non-empty and at most countable. Set ∆ j = β j − α j . Since Σ δ is of zero Lebesguemeasure, we have P j ∈ J ∆ j = 1. We define also S = S j ∈ J [ α j , β j ] d For s ∈ Σ δ , notice that any feasible function c of ( P δ ) satisfies for all 1 ≤ i ≤ d : Z I d c ( u ) { u i
0. We have: I d ( c δ ) = lim ε ↓ X j ∈ J ∆ j (cid:0) I d ( c j ) − log(∆ j ) (cid:1) { ∆ j >ε } = lim ε ↓ X j ∈ J ∆ j (cid:0) ( d − J ( δ j ) − log(∆ j ) (cid:1) { ∆ j >ε } + X j ∈ J ∆ j G ( δ j )= X j ∈ J ∆ j (cid:0) ( d − J ( δ j ) − log(∆ j ) (cid:1) + X j ∈ J ∆ j G ( δ j ) , where we used the monotone convergence theorem for the first equality, (6) for the secondand the fact that G ( δ ) is uniformly bounded over D and the monotone convergence theoremfor the last. Elementary computations yields:( d − J ( δ ) = X j ∈ J ∆ j (cid:0) ( d − J ( δ j ) − log(∆ j ) (cid:1) and G ( δ ) = X j ∈ J ∆ j G ( δ j ) . So, we get: I d ( c δ ) = ( d − J ( δ ) + G ( δ ) . Since G ( δ ) is uniformly bounded over D , we get that I d ( c δ ) is finite if and only if J ( δ ) is finite.To end the proof, recall the definition of I ( C δ ) to conclude that I ( C δ ) = ( d − J ( δ ) + G ( δ ).7. Examples for d = 2In this section we compute the density of the maximum entropy copula for various diagonalsections of popular bivariate copula families. In this Section, u and v will denote elements of I . The density for d = 2 is of the form c δ ( u, v ) = a (min( u, v )) b (max( u, v )). We illustrate thesedensities by displaying their isodensity lines or contour plots, and their diagonal cross-section ϕ defined as ϕ ( t ) = c ( t, t ), t ∈ I .7.1. Maximum entropy copula for a piecewise linear diagonal section.
Let α ∈ (0 , / δ ( r ) = ( r − α ) ( α, − α ) ( r ) + (2 r − [1 − α, ( r ) . tδ ( t ) α − α Figure 1.
Graph of δ with α = 0 . α = 0 and α = 1 / α = 0, Σ δ = I , therefore every copula C with this diagonal section gives I ( C ) = + ∞ .(In fact the only copula that has this diagonal section is the Fr´echet-Hoeffding upper bound M defined by M ( u, v ) = min( u, v ), u, v ∈ I .) When α ∈ (0 , / J ( δ ) < + ∞ is satisfied,therefore we can apply Theorem 2.3 to compute the density of the maximum entropy copula.The graph of δ can be seen in Figure 1 for α = 0 .
2. We compute the functions F , a and b : F ( r ) = log( rα ) − α + if r ∈ [0 , α ) r α − α if t ∈ [ α, − α ) log (cid:16) α − r (cid:17) + α − if t ∈ [1 − α, a ( r ) = 1 √ α e − α + [0 ,a ] ( r ) + 12 √ α e r α − α ( a, − a ) ( r )and: b ( r ) = 12 √ α e − r α + α ( a, − a ) ( r ) + 1 √ α e − α + [1 − a, ( r )The density c δ ( u, v ) consists of six distinct regions on △ = { ( u, v ) ∈ I , u ≤ v } as shown inFigure 2a and takes the values:(31) c δ ( u, v ) = α e α − v α in II, α e u − v α in III, α e α − α in IV, α e u + α − α in V,0 in VI.Figure 2b shows the isodensity lines of c δ . In the limiting case of α = , the diagonalsection is given by δ ( t ) = max(0 , t − D . Accordingly, it is the diagonal section of the Fr´echet-Hoeffding lower bound copula W AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 15 α − α α − α I.II. III.IV. V. VI. (a)
Partition for c δ u v . . . . . . . . . . . . . . . . . . . . . . . . (b) Isodensity lines of c δ Figure 2.
The partition and the isodensity lines of c δ .given by W ( u, v ) = max(0 , u + v −
1) for u, v ∈ I . All copulas having this diagonal sectionare of the following form: D C ,C ( u, v ) = W ( u, v ) if ( u, v ) ∈ [0 , / ∪ [1 / , , C (2 u, v −
1) if ( u, v ) ∈ [0 , / × [1 / , , C (2 u − , v ) if ( u, v ) ∈ [1 / , × [0 , / , where C and C are copula functions. Recall that the independent copula Π with uniformdensity c Π = 1 on I minimizes I ( C ) over C . According to (31), the maximum entropy copulawith diagonal section δ is D Π , Π . This corresponds to choosing the maximum entropy copulason [0 , / × [1 / ,
1] and [1 / , × [0 , / Maximum entropy copula for δ ( t ) = t α . Let α ∈ (1 , δ ( t ) = t α . This corresponds to the Gumbel family of copulas andalso to the family of Cuadras-Aug´e copulas. The Gumbel copula with parameter θ ∈ [1 , ∞ )is an Archimedean copula defined as, for u, v ∈ I : C G ( u, v ) = ϕ − θ ( ϕ θ ( u ) + ϕ θ ( v ))with generator function ϕ θ ( t ) = ( − log( t )) θ . Its diagonal section is given by δ G ( t ) = t θ = t α with α = 2 θ . The Cuadras-Aug´e copula with parameter γ ∈ (0 ,
1) is defined as, for u, v ∈ I : C CA ( u, v ) = min( uv − γ , u − γ v ) . It is a subclass of the two parameter Marshall-Olkin family of copulas given by C M ( u, v ) =min( u − γ v, uv − γ ). The diagonal section of C CA is given by δ ( t ) = t − γ = t α with α = 2 − γ .While the Gumbel copula is absolutely continuous, the Cuadras-Aug´e copula is not, althoughit has full support. Since J ( δ ) < + ∞ , we can apply Theorem 2.3. To give the density ofthe maximum entropy copula, we have to calculate F ( v ) − F ( u ). Elementary computations yield: F ( v ) − F ( u ) = 12 Z vu dss − s α = 12 log (cid:16) vu (cid:17) − α − (cid:18) − v α − − u α − (cid:19) . The density c δ is therefore given by, for ( u, v ) ∈ △ : c δ ( u, v ) = α − αu α − (1 − u α − ) α/ (2 α − v α − (1 − v α − ) (2 − α ) / (2 α − . Figure 3 represents the isodensity lines of the Gumbel and the maximum entropy copula c δ with common parameter α = 2 , which corresponds to θ = 3 for the Gumbel copula. Wehave also added a graph of the diagonal cross-section of the two densities. In the limitingcase of α = 2, the above formula gives c δ ( u, v ) = 1, which is the density of the independentcopula Π, which is also maximizes the entropy on the entire set of copulas. u v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Gumbel u v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) c δ t c ( t,t ) GumbelMI (c)
Diagonal cross-section
Figure 3.
Isodensity lines and the diagonal cross-section of copulas withdiagonal section δ ( t ) = t α , α = 2 .7.3. Maximum entropy copula for the Farlie-Gumbel-Morgenstern diagonal sec-tion.
Let θ ∈ [ − , C ( u, v ) = uv + θuv (1 − u )(1 − v ) . These copulas are absolutely continuous with densities c ( u, v ) = 1 + θ (1 − u )(1 − v ). Itsdiagonal section δ θ is given by: δ ( t ) = t + θt (1 − t ) = θt − θt + (1 + θ ) t . Since δ θ ( t ) < t on (0 ,
1) and it verifies J ( δ ) < + ∞ , we can apply Theorem 2.3 to calculatethe density of the maximum entropy copula. For F ( r ), we have: F ( r ) = log (cid:16) r − r (cid:17) + θ √ θ − θ arctan (cid:16) θr − θ √ θ − θ (cid:17) if θ ∈ (0 , , log (cid:16) r − r (cid:17) if θ = 0 , log (cid:16) r − r (cid:17) − θ √ θ − θ arctanh (cid:16) θr − θ √ θ − θ (cid:17) if θ ∈ [ − , . The density c δ is given by, for θ ∈ (0 ,
1] and ( u, v ) ∈ △ : AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 17 c δ ( u, v ) = (cid:0) − θu + 3 θu + (1 + θ ) u (cid:1) (1 − u ) √ θu − θu + 1 (cid:0) θv + 3 θv + (1 + θ ) (cid:1) √ θv − θv + 1exp (cid:18) − θ √ θ − θ (cid:18) arctan (cid:18) θv − θ √ θ − θ (cid:19) − arctan (cid:18) θu − θ √ θ − θ (cid:19)(cid:19)(cid:19) Figure 4 illustrates the isodensities of the FGM copula and the maximum entropy copulawith the same diagonal section for θ = 0 . u v . . . . . . . . . . . . . . . . . . . . . . (a) FGM u v . . . . . . . . . . . . . . . . . . (b) c δ . . . . . . t c ( t,t ) FGMMI (c)
Diagonal cross-section
Figure 4.
Isodensity lines and the diagonal cross-section of copulas withdiagonal section δ ( t ) = θt − θt + (1 + θ ) t , θ = 0 . θ = 0 corresponds once again to the diagonal section δ ( t ) = t , and the formulagives the density of the independent copula Π, accordingly.7.4. Maximum entropy copula for the Gaussian diagonal section.
The Gaussian(normal) copula takes the form: C ρ ( u, v ) = Φ ρ (cid:0) Φ − ( u ) , Φ − ( v ) (cid:1) , with Φ ρ the joint cumulative distribution function of a two-dimensional normal random vari-able with standard normal marginals and correlation parameter ρ ∈ [ − , − thequantile function of the standard normal distribution. The density c ρ of C ρ can be writtenas: c ρ ( u, v ) = ϕ ρ (cid:0) Φ − ( u ) , Φ − ( v ) (cid:1) ϕ (Φ − ( u )) ϕ (Φ − ( v )) , where ϕ and ϕ ρ stand for respectively the densities of a standard normal distribution and atwo-dimensional normal distribution with correlation parameter ρ , respectively. The diagonalsection and its derivative are given by:(32) δ ρ ( t ) = Φ ρ (cid:0) Φ − ( t ) , Φ − ( t ) (cid:1) , δ ′ ρ ( t ) = 2Φ (cid:18)r − ρ ρ Φ − ( t ) (cid:19) . Since δ ρ verifies δ ρ ( t ) < t on (0 ,
1) and J ( δ ρ ) < + ∞ , we can apply Theorem 2.3 to calculatethe density of the maximum entropy copula. We have calculated numerically the density ofthe maximum entropy copula with diagonal section δ ρ for ρ = 0 . , . , − . − .
95. Thecomparison between these densities and the densities of the corresponding normal copula can be seen in Figures 5,6 and 7. We observe a very different behaviour of c ρ and c δ ρ inthe case of ρ <
0. In the limiting case when ρ goes down to −
1, we retrieve the diagonal δ ( t ) = max(0 , t − u v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Normal, ρ = 0 . u v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) c δ , ρ = 0 . t c ( t,t ) NormalMI (c)
Diagonal cross-section u v . . . . . . . . . . . . . . . . . . . . . . . . (d) Normal, ρ = 0 . u v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (e) c δ , ρ = 0 . t c ( t,t ) NormalMI (f)
Diagonal cross-section
Figure 5.
Isodensity lines and the diagonal cross-section of copulas withdiagonal section given by (32), with ρ = 0 . ρ = 0 . Appendix - Calculation of the entropy of C δ Let us first introduce some notations. Let ε ∈ (0 , / x log( x ) ≥ − / e for x > I ( C δ ) = lim ε ↓ I ε ( C δ ) , with: I ε ( C δ ) = Z [ ε, − ε ] d c δ ( x ) log( c δ ( x )) dx. Using δ ≤ t and that δ is a non-decresing, d -Lipschitz function, we get that for t ∈ I :(34) 0 ≤ h ( t ) ≤ min( t, ( d − − t )) ≤ ( d −
1) min( t, − t ) . We set:(35) w ( t ) = a ( t ) e − F ( t ) = d − δ ′ ( r ) d h − /d ( r ) . AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 19 u v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Normal, ρ = − . u v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) c δ , ρ = − . . . . . . . . t c ( t,t ) NormalMI (c)
Diagonal cross-section u v . . . . . . . . . . . . . . . . . . . . . . . . (d) Normal, ρ = − . u v . . . . . . (e) c δ , ρ = − . . . . . . . . t c ( t,t ) NormalMI (f)
Diagonal cross-section
Figure 6.
Isodensity lines and the diagonal cross-section of copulas withdiagonal section given by (32), with ρ = − . ρ = − . c δ , we have that(36) I ε ( C δ ) = J ( ε ) + J ( ε ) − J ( ε ) , with: J ( ε ) = d Z [ ε, − ε ] d c δ ( x ) { max( x )= x d } d − X i =1 log ( w ( x i )) ! dx,J ( ε ) = d Z [ ε, − ε ] d c δ ( x ) { max( x )= x d } log (cid:18) δ ′ ( x d ) d h − /d ( x d ) (cid:19) dx,J ( ε ) = d Z [ ε, − ε ] d c δ ( x ) { max( x )= x d } ( d − F ( x d ) − d − X i =1 F ( x i ) ! dx. . . . . . . Gaussian (−0.95) u v (a) Normal, ρ = − . . . . . . . Gaussian c_delta (−0.95) u v (b) C δ , ρ = − . Figure 7.
Sample of 500 drawn from the Gaussian copula with ρ = − . C δ We introduce A ε ( r ) = R rε a ( x ) dx . For J ( ε ), we have: J ( ε ) = d ( d − Z [ ε, − ε ] { max( x ) = x d } b ( x d ) d − Y j =1 a ( x j ) log ( w ( x )) dx = d ( d − Z [ ε, − ε ] Z [ t, − ε ] A d − ε ( s ) b ( s ) ds ! a ( t ) log ( w ( t )) dt. Notice that using (10) and (12), we have: Z [ t, − ε ] A d − ε ( s ) b ( s ) ds = Z [ t, A d − ( s ) b ( s ) ds − Z [ t, (cid:16) A d − ( s ) − A d − ε ( s ) (cid:17) b ( s ) ds − Z [1 − ε, A d − ε ( s ) b ( s ) ds. = h ( t )( d − A ( t ) − Z t (cid:16) A d − ( s ) − A d − ε ( s ) (cid:17) b ( s ) ds − Z [1 − ε, A d − ε ( s ) b ( s ) ds. By Fubini’s theorem, we get: J ( ε ) = J , ( ε ) − J , ( ε ) − J , ( ε ) , AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 21 with: J , ( ε ) = Z [ ε, − ε ] ( d − δ ′ ( t )) log ( w ( t )) dtJ , ( ε ) = d ( d − Z [1 − ε, A d − ε ( s ) b ( s ) ds ! Z [ ε, − ε ] a ( t ) log ( w ( t )) dtJ , ( ε ) = d ( d − Z [ ε, − ε ] (cid:18)Z t (cid:16) A d − ( s ) − A d − ε ( s ) (cid:17) b ( s ) ds (cid:19) a ( t ) log ( w ( t )) dt. To study J , , we first give an upper bound for the term R [1 − ε, A d − ε ( s ) a ( s ) b ( s ) ds : Z [1 − ε, A d − ε ( s ) b ( s ) ds ≤ Z [1 − ε, A d − ( s ) b ( s ) ds = 1( d − h − /d (1 − ε ) e − F (1 − ε ) ≤ ( d − − /d ε − /d , (37)where we used that A ε ( s ) ≤ A ( s ) for s > ε for the first inequality, (12) for the first equality,and (34) for the last inequality. Since t log( t ) ≥ − / e, we have, using (35): J , ( ε ) ≥ − d ( d − Z [1 − ε, A d − ε ( s ) b ( s ) ds ! Z [ ε, − ε ] e F ( t ) dt ≥ − d e h − /d (1 − ε ) Z [ ε, − ε ] e F ( t ) − F (1 − ε ) dt ≥ − d e (( d − ε ) − /d , where we used (12) for the second inequality, and that F is non-decreasing and (37) for thethird inequality. On the other hand, we have t log( t ) ≤ t − /d , if t ≥
0, which gives: J , ( ε ) ≤ d ( d − Z [1 − ε, A d − ε ( s ) b ( s ) ds ! Z [ ε, − ε ] e F ( t ) (cid:16) d − δ ′ ( t ) d (cid:17) − /d h ( t ) dt = dh (1 − ε ) − /d Z [ ε, − ε ] e F ( t ) − F (1 − ε ) h ( t ) dt = dh (1 − ε ) − /d (cid:16) − e F ( ε ) − F (1 − ε ) (cid:17) ≤ d (( d − ε ) − /d , where we used (37) and t − /d ≤ t ∈ I for the first inequality, and that F is non-decreasing for the last. This proves that lim ε → J , ( ε ) = 0. For J , ( ε ), we first observe thatfor s ∈ [ ε, − ε ] we have A ε ( s ) ≤ A ( s ) and thus:(38) (cid:16) A d − ( s ) − A d − ε ( s ) (cid:17) = A ( ε ) d − X i =0 A i ( s ) A d − − iε ( s ) ≤ ( d − A ( ε ) A d − ( s ) . Using the previous inequality we obtain: J , ( ε ) = d ( d − Z [ ε, − ε ] (cid:18)Z t (cid:16) A d − ( s ) − A d − ε ( s ) (cid:17) b ( s ) ds (cid:19) a ( t ) log ( w ( t )) dt ≥ − d ( d − Z [ ε, − ε ] (cid:18)Z t (cid:16) A d − ( s ) − A d − ε ( s ) (cid:17) b ( s ) ds (cid:19) e F ( t ) dt ≥ − d ( d − d − A ( ε )e Z [ ε, − ε ] (cid:18)Z t A d − ( s ) b ( s ) ds (cid:19) e F ( t ) dt ≥ − d ( d − d − A ( ε )e Z [ ε, − ε ] (cid:16)R t A d − ( s ) b ( s ) ds (cid:17) A ( t ) e F ( t ) dt = − d ( d − A ( ε )e Z [ ε, − ε ] h ( t ) A ( t ) e F ( t ) dt = − d ( d − h /d ( ε )e Z [ ε, − ε ] h ( t ) − /d e F ( ε ) − F ( t ) dt ≥ − d ( d − d − − /d ε /d e , where we used t log( t ) ≥ − / e for the first inequality, (38) for the second, (10) and (12) inthe following equality, and (34) to conclude. For an upper bound, we have after noticing that t log( t ) ≤ t : J , ( ε ) = d ( d − Z [ ε, − ε ] (cid:18)Z t (cid:16) A d − ( s ) − A d − ε ( s ) (cid:17) b ( s ) ds (cid:19) a ( t ) log ( w ( t )) dt ≤ d ( d − Z [ ε, − ε ] (cid:18)Z t (cid:16) A d − ( s ) − A d − ε ( s ) (cid:17) b ( s ) ds (cid:19) e F ( t ) w ( t ) dt ≤ d ( d − d − A ( ε ) Z [ ε, − ε ] (cid:16)R t A d − ( s ) b ( s ) ds (cid:17) A ( t ) e F ( t ) h − /d ( t ) dt = d ( d − A ( ε ) Z [ ε, − ε ] e − F ( t ) h ( t ) dt = d ( d − h /d ( ε )(1 − e F ( ε ) − F (1 − ε ) ) ≤ d ( d − d − /d ε /d , where we used (38) and 0 ≤ ( d − δ ′ ( t )) /d ≤ AXIMUM ENTROPY COPULA WITH GIVEN DIAGONAL SECTION 23 lim ε → J , ( ε ) = 0. Similarly, for J ( ε ), we get: J ( ε ) = Z [ ε, − ε ] d { max( x ) = x d } b ( x d ) d − Y j =1 a ( x j ) log (cid:18) δ ′ ( x d ) d h − /d ( x d ) (cid:19) dx = d Z [ ε, − ε ] A d − ε ( t ) b ( t ) log (cid:18) δ ′ ( t ) d h − /d ( t ) (cid:19) dt = d Z [ ε, − ε ] A d − ( t ) b ( t ) log (cid:18) δ ′ ( t ) d h − /d ( t ) (cid:19) dt − d Z [ ε, − ε ] (cid:16) A d − ( t ) − A d − ε ( t ) (cid:17) b ( t ) log (cid:18) δ ′ ( t ) d h − /d ( t ) (cid:19) dt = J , ( ε ) − J , ( ε )with J , ( ε ) and J , ( ε ) given by, using (11): J , ( ε ) = d Z [ ε, − ε ] A d − ( t ) b ( t ) log (cid:18) δ ′ ( t ) d h − /d ( t ) (cid:19) dtJ , ( ε ) = d Z [ ε, − ε ] (cid:16) A d − ( t ) − A d − ε ( t ) (cid:17) b ( t ) log (cid:18) δ ′ ( t ) d h − /d ( t ) (cid:19) dt. By (11), we have:(39) J , ( ε ) = Z [ ε, − ε ] δ ′ ( t ) log (cid:18) δ ′ ( t ) d h − /d ( t ) (cid:19) dt. Similarly to J , ( ε ) we can show that lim ε → J , ( ε ) = 0.Adding up J ( ε ) and J ( ε ) gives J ( ε ) + J ( ε ) = J ε ( δ ) + J ( ε ) − d log( d )(1 − ε ) − J , ( ε ) − J , ( ε ) − J , ( ε )with J ε ( δ ) = ( d − Z − εε | log ( h ( t )) | dt,J ( ε ) = Z − εε (cid:0) d − δ ′ ( t ) (cid:1) log (cid:0) d − δ ′ ( t ) (cid:1) dt + Z − εε δ ′ ( t ) log (cid:0) δ ′ ( t ) (cid:1) dt. Notice that J ε ( δ ) is non-decreasing in ε > J ( δ ) = lim ε → J ε ( δ ) . Since δ ′ ( t ) ∈ [0 , d ], we deduce that ( d − δ ′ ) log( d − δ ′ ) and δ ′ log( δ ′ ) are bounded on I fromabove by d log( d ) and from below by − / e and therefore integrable on I . This implies :lim ε → J ( ε ) = I ( δ ′ ) + I ( d − δ ′ ) . As for J ( ε ), we have by integration by parts: J ( ε ) = d Z [ ε, − ε ] d { max( x )= x d } b ( x d ) d − Y i = i a ( x i ) ( d − F ( x d ) − d − X i =1 F ( x i ) ! dx = d ( d − Z [ ε, − ε ] A d − ε ( t ) b ( t ) F ( t ) dt − d ( d − Z [ ε, − ε ] A d − ε ( t ) b ( t ) (cid:18)Z tε a ( s ) F ( s ) (cid:19) dt = d ( d − Z [ ε, − ε ] A d − ε ( t ) b ( t ) F ( t ) dt − d ( d − Z [ ε, − ε ] A d − ε ( t ) b ( t ) (cid:18) A ε ( t ) F ( t ) − d − d Z tε A ε ( s ) h ( s ) ds (cid:19) dt = ( d − Z [ ε, − ε ] (cid:18)Z − εt A d − ε ( s ) b ( s ) (cid:19) A ε ( t ) h ( t ) dt. By the monotone convergence theorem, (10) and (12) we have:lim ε → J ( ε ) = ( d − Z I (cid:18)Z t A d − ( s ) b ( s ) (cid:19) A ( t ) t − δ ( t ) dt = d − . Summing up all the terms and taking the limit ε = 0 give : I ( C δ ) = ( d − Z I | log( t − δ ( t )) | dt + I ( δ ′ ) + I ( d − δ ′ ) − d log( d ) − ( d − d − J ( δ ) + G ( δ ) . References [1] T. Bedford and K. Wilson. On the construction of minimum information bivariate copula families.
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Cristina Butucea, Universit´e Paris-Est, LAMA (UPE-MLV), F-77455 Marne La Vall´ee, France.
E-mail address : [email protected] Jean-Franc¸ois Delmas, Universit´e Paris-Est, CERMICS (ENPC), F-77455 Marne La Vall´ee,France.
E-mail address : [email protected] Anne Dutfoy, EDF Research & Development, Industrial Risk Management Department, 92141Clamart Cedex, France.
E-mail address : [email protected] Richard Fischer, Universit´e Paris-Est, CERMICS (ENPC), F-77455 Marne La Vall´ee, France,EDF Research & Development, Industrial Risk Management Department, 92141 Clamart Cedex,France.
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