Maximum entropy distribution of order statistics with given marginals

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DOI:10.3150/16-BEJ868

Maximum entropy distribution of order statistics with given marginals

C R I S T I NA BU T U C E A¹, J E A N - F R A N Ç O I S D E L M A S^2,*, A N N E D U T F OY³and R I C H A R D F I S C H E R^1,2,3,**

1LAMA (UPE-MLV), UPE, Marne La Vallée, France. E-mail:[email protected] 2CERMICS, École des Ponts, UPE, Champs-sur-Marne, France.

E-mail:^*[email protected];^**[email protected]

3EDF Research & Development, Industrial Risk Management Department, Palaiseau, France.

E-mail:[email protected]

We consider distributions of ordered random vectors with given one-dimensional marginal distributions.

We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.

Keywords:copula; entropy; maximum entropy; order statistics

1. Introduction

Order statistics, an almost surely non-decreasing sequence of random variables, have received a lot of attention due to the diversity of possible applications. If X =(X1, . . . , Xd) is a d-dimensional random vector, then its order statisticsX^OS=(X₍₁₎, . . . , X_(d))corresponds to the permutation of the components of X in the non-decreasing order, so that X₍₁₎≤X₍₂₎≤

· · · ≤X_(d). The components of the underlying random vectorX are usually, but not necessar- ily, independent and identically distributed (i.i.d.). Special attention has been given to extreme valuesX₍₁₎andX_(d), the rangeX_(d)−X₍₁₎, or the median value. Direct application of the distribution of thekth largest order statistic occurs in various fields, such as climatology, extreme events, reliability, insurance, financial mathematics. We refer to the monographs of David and Nagaraja [8] and Arnold, Balakrishnan and Nagaraja [1] for a general overview on the subject of order statistics. We are interested in the dependence structure of order statistics, which has received great attention. In the i.i.d. case, Bickel [3] showed that any two order statistics are pos- itively correlated. The copula of the joint distribution ofX₍₁₎andX_(d)is derived in Schmitz [19]

with exact formulas for Kendall’sτ and Spearman’sρ. In Avérous, Genest and Kochar [2], it is shown that the dependence of thejth order statistic on theith order statistic decreases as the distance betweeniandj increases according to the bivariate monotone regression dependence ordering. The copula connecting the limit distribution of the two largest order statistics, called bi-extremal copula, is given by de Melo Mendes and Sanfins [9] with some additional properties.

Exact expressions for Pearson’s correlation coefficient, Kendall’sτ and Spearman’s ρ for any

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two order statistics are obtained in Navarro and Balakrishnan [16]. For the non i.i.d. case, Kim and David [14] show that some pairs of order statistics can be negatively correlated, if the underlying random vector is sufficiently negatively dependent. Positive dependence measures for two order statistics are considered in Bolandet al.[4] when the underlying random variables are independent but arbitrarily distributed or when they are identically distributed but not independent. A generalization of these results for multivariate dependence properties is given by Hu and Chen [11]. See also Dubhashi and Häggström [10] for conditional distribution of order statistics.

Here, we focus on the cumulative distribution function (c.d.f.) of order statistics without refer- ring to an underlying distribution. That is, we consider random vectorsX=(X₁, . . . , X_d)∈R^d such that a.s. X₁≤ · · · ≤X_d and we suppose that the one-dimensional marginal distributions F=(Fi,1≤i≤d)are given, whereFi is the c.d.f. ofX_i. A necessary and sufficient condition for the existence of a joint distribution of order statistics with one-dimensional marginalsFis that they are stochastically ordered, that isFi−1(x)≥Fi(x)for all 2≤i≤d, x∈R. With the marginals fixed, the joint distribution of the order statistics can be characterized by the connecting copula of the random vector, which contains all information on the dependence structure of the order statistics. Copulas of order statistics derived from an underlying i.i.d. sample were considered in [2] in order to calculate measures of concordance between any two pairs of order statistics. For order statistics derived from a general parent distribution, Navarro and Spizzichino [17]

shows that the copula of the order statistics depends on the marginals and the copula of the parent distribution through an exchangeable copula and the average of the marginals. Construction of some copula of order statistics with given marginals were given in Lebrun and Dutfoy [15].

Our aim is to find the c.d.f. of order statistics of dimensiond with fixed marginals which maximizes the relative entropyH_h defined by (13). In an information-theoretic interpretation, the maximum entropy distribution is the least informative among order statistics with given marginals. This problem appears in models where the one-dimensional marginals are well known (either from different experimentation or from physical models) but the dependence structure is unknown, see Butuceaet al. [7]. In [6], the same authors gave, when it exists, the maximum entropy distribution of(X₁, . . . , X_d)such thatX_iis uniformly distributed on[0,1]for 1≤i≤d and the distribution ofX_(d)=max1≤i≤dX_i is given, see Remark4.8.

For ad-dimensional random variable X=(X₁, . . . , X_d)with c.d.f. F and copulaC_F, the relative entropy ofF can be decomposed into the sum of the relative entropy (with respect to a one-dimensional probability densityh) of its one-dimensional marginals plus the entropy ofC_F, see Lemma2.1. In our case, since the marginalsF=(Fi,1≤i≤d)are fixed, maximizing the entropy of the joint distributionF of an order statistics is equivalent to maximizing the entropy of its copulaC_F under constraints, (see Section2.4). Therefore, we shall find the maximum entropy copula for order statistics with fixed marginal distributions.

The main result of this paper is given by Theorem5.4. It states that there exists a unique maximum entropy c.d.f.F_Fgiven by (52) if and only if:

d i=1

H_h(Fi)− d i=2

RFi(dt)log

Fi−1(t)−Fi(t)>−∞.

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In this case,F_Fis absolutely continuous with densityf_Fdefined as, forx=(x₁, . . . , x_d)∈R^d: f_F(x)=f1(x₁)

d i=2

fi(xi)

Fi−1(xi)−Fi(xi)exp

− _x_i

x_i−1

fi(s)

Fi−1(s)−Fi(s)ds 1_LF(x), wherefiis the density function ofFi andL^F⊂R^dis the set of ordered vectors(x₁, . . . , x_d), that isx1≤ · · · ≤xd, such thatFi−1(t) >Fi(t)for allt∈(xi−1, xi)and 2≤i≤d. See Example5.8 for an illustrative example.

The rest of the paper is organized as follows. In Section2, we introduce the basic notations and give the definition of the objects used in later parts. Section3describes the connection between copulas of order statistics with fixed marginals, and symmetric copulas with fixed multidiagonals. The multidiagonal, given by Definition3.6, is the generalization of the diagonal section for copulas, which received great attention in copula literature. We show that there exists a one-to- one map between these two sets of copulas, see Corollary3.14. This bijection has good properties with respect to the entropy as explained in Proposition3.22. In Section4, we determine the maximum entropy copula with fixed multidiagonal, see Theorem4.7. Since we obtain a symmetric copula as a result, this is also the maximum entropy symmetric copula with fixed multidiagonal.

In Section5, we use the one-to-one map between the two sets of copulas established in Section3 to give the maximum entropy copula of order statistics with fixed marginals. We finally obtain the density of the maximum entropy distribution for order statistics with fixed marginals by com- posing the maximum entropy copula with the marginals, see Theorem5.4. Section6contains the detailed proofs of Theorem4.7and other results from Section4. Section7 collects the main notations of the paper to facilitate reading.

2. Notations and definitions

2.1. Notations in R

^d

and generalized inverse

For a Borel setA⊂R^d, we write|A|for its Lebesgue measure. Forx=(x1, . . . , xd)∈R^d and y=(y₁, . . . , y_d)∈R^d, we writex≤yifx_i≤y_ifor all 1≤i≤d. We define minx=min{x_i,1≤ i≤d}and maxx=max{x_i,1≤i≤d}forx=(x₁, . . . , x_d)∈R^d. IfJ is a real-valued function defined onR, we setJ (x)=(J (x₁), . . . , J (x_d)). We shall consider the following subsets ofR^d:

S=

(x₁, . . . , x_d)∈R^d, x₁≤ · · · ≤x_d

and =S∩I^d,

withI = [0,1]. In what follows, usuallyx, y will belong to R^d, ands, t toRorI. For a set A⊂R, we note byA^c=R\Aits complementary set.

IfJ is a bounded non-decreasing càd-làg function defined onR. Its generalized inverseJ⁻¹ is given byJ⁻¹(t)=inf{s∈R;J (s)≥t}, fort∈R, with the convention that inf∅= +∞and infR= −∞. We have fors, t∈R:

J (t )≥s⇔t≥J⁻¹(s), J⁻¹◦J (t )≤t and J◦J⁻¹◦J (t )=J (t ). (1)

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We define the set of points whereJ is increasing on their left:

I_g(J )=

t∈R;u < t⇔J (u) < J (t )

. (2)

We have:

1(Ig(J ))^cdJ =0 a.e., (3)

J⁻¹(R)⊂I_g(J )∪ {±∞} (4) and fors∈R,t∈I_g(J ):

J (t )≤s⇔t≤J⁻¹(s) and J⁻¹◦J (t )=t. (5) Notice that ifJ is continuous in addition, then we have fort∈J (R):

J◦J⁻¹(t)=t. (6)

2.2. C.d.f. and copula

LetX=(X₁, . . . , X_d)be a random vector onR^d. Its cumulative distribution function (c.d.f.), denoted by F is defined by: F (x)=P(X≤x),x∈R^d. The corresponding one-dimensional marginal c.d.f.s are(F_i,1≤i≤d)withF_i(t)=P(X_i≤t ),t∈R. The c.d.f.F is called a copula ifX_i is uniform onI= [0,1]for all 1≤i≤d. (Notice a copula is characterized by its values on I^donly.)

We defineLdas the set of c.d.f.s onR^d whose one-dimensional marginal c.d.f.s are continuous, andC⊂Ldas the subset of copulas. We setL⁰_d(resp.C⁰) the subset of absolutely continuous c.d.f. (resp. copulas) onR^d.

Let us define for a c.d.f.F with one-dimensional marginals(Fi,1≤i≤d)the functionCF

defined onI^d:

C_F(y)=F

F₁⁻¹(y₁), . . . , F_d⁻¹(y_d)

, y=(y₁, . . . , y_d)∈I^d. (7) IfF ∈Ld, then C_F defined by (7) is a copula thanks to (6). According to Sklar’s theorem,F is then completely characterized by its one-dimensional marginal c.d.f.s(F_i,1≤i≤d)and the associated copulaCF which contains all information on the dependence:

F (x)=C_F

F₁(x₁), . . . , F_d(x_d)

, x=(x₁, . . . , x_d)∈R^d. (8) Equivalently, ifX=(X₁, . . . , X_d)has c.d.f.F, thenC_F is the c.d.f. of the random vector:

F₁(X₁), . . . , F_d(X_d)

. (9)

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2.3. Order statistics

For a d-dimensional c.d.f. F, we write PF for the distribution of a random vector X = (X1, . . . , Xd)with c.d.f.F. Ad-dimensional c.d.f.F is a c.d.f. of order statistics (and we shall say thatX is a vector of order statistics) ifPF(X₁≤X₂≤ · · · ≤X_d)=1. Let us denote by L^OS_d ⊂Ld the set of all c.d.f.s of order statistics with continuous one-dimensional marginal c.d.f.s. Thed-tuples(F_i,1≤i≤d)of marginal c.d.f.s then verifyF_i₋₁≥F_i for all 2≤i≤d. LetFdbe the set ofd-tuples of continuous one-dimensional c.d.f.s compatible with the marginal c.d.f.s of order statistics:

Fd=

F=(Fi,1≤i≤d)∈(L1)^d;Fi−1≥Fi,∀2≤i≤d

. (10)

For a givenF=(Fi,1≤i≤d) inFd, we define the set of c.d.f.s F of order statistics with marginal c.d.f.sF:

L^OSd (F)=

F ∈L^OSd ;F_i=Fi,1≤i≤d

. (11)

IfF∈Fd, then we haveLÔS_d (F)=∅, since the c.d.f. of(F⁻₁¹(U ), . . . ,F⁻_d¹(U )),U uniformly distributed onI, belongs toLÔS_d (F). We defineCÔS(F)the set of copulas of order statistics with marginalsF:

C^OS(F)=

C_F∈C;F ∈L^OSd (F)

. (12)

According to Sklar’s theorem, the mapF →C_F is a bijection betweenL^OS_d (F)andC^OS(F)if F∈Fd.

2.4. Entropy

Leth be a reference probability density function onR. We define h^⊗^d(x)=_d

i=1h(x_i)for x=(x1, . . . , xd)∈R^d. The relative Shannon-entropy for a c.d.f.F∈Ldis given by:

H_h(F )=

⎧⎨

⎩

−∞, ifF ∈Ld\L⁰_d,

−

R^dflog f/ h^⊗^d

, ifF ∈L⁰_d, (13)

withf the density ofF. Notice thatHh(F )∈ [−∞,0]is well defined. We will use the notation H_h(X)=H_h(F )ifXis a random vector with c.d.f.F andH_h(f )=H_h(F )ifF has densityf. We shall simply writeH (F )(resp.H (X)andH (f )) instead ofH_h(F )(resp.H_h(X)andH_h(f )) whenh=1_[0,1]. Note thatH (F )can be finite only ifF is the c.d.f. of a probability distribution on[0,1]^d.

According to the next lemma, the relative entropy of anyF ∈L^1c_d can be decomposed into the relative entropy of the one-dimensional marginals c.d.f.(F_i,1≤i≤d)and the entropy of the associated copulaC_F.

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Lemma 2.1. LetF∈L^1c_d .We have:

H_h(F )=H (C_F)+ d i=1

H_h(F_i). (14)

Proof. It is left to the reader to check thatF has a density, sayf, if and only ifFi has a density, sayf_i, for 1≤i≤dandC_F has a density, sayc_F. Furthermore, in this case, we have:

f (x)=c_F

F₁(x₁), . . . , F_d(x_d)^d

i=1

f_i(x_i) a.e. forx=(x₁, . . . , x_d)∈R^d, as well as, with the convention 0/0=0,

cF(u)=f (F₁⁻¹(u1), . . . , F_d⁻¹(ud)) _d

i=1f_i(F_i⁻¹(u_i)) a.e. foru=(u1, . . . , ud)∈I^d.

On the one hand, ifF does not have a density then we haveH_h(F )= −∞. Since F does not have a density, then one of theFi orCF does not have a density either, and thenH (CF)+ _d

i=1H_h(F_i)= −∞. Thus (14) holds.

On the other hand, let us assume thatF has a density, sayf. Elementary computations give withx=(x₁, . . . , x_d)and 1≤i≤d:

Hh(Fi)= −

Rfi(xi)log

(fi/ h)(xi)

dxi= −

R^df (x)log

(fi/ h)(xi) dx.

We also have withu=(u₁, . . . , u_d)andx=(x₁, . . . , x_d):

H (CF)= −

[0,1]^dcFlog(cF)= −

f (F₁⁻¹(u1), . . . , F_d⁻¹(ud)) _d

i=1f_i(F_i⁻¹(u_i)) log cF(u)

du

= −

f (x)log cF

F1(x1), . . . , Fd(xd) dx,

where, for the last equality, we used the change of variableFi(xi)=ui (forxi ∈Ig(Fi)) so that F_i⁻¹(ui)=F_i⁻¹◦Fi(xi)=xi holdsfi(xi) dxi-a.e and thatf (x) dx=0 on(⊗^d_i₌₁Ig(Fi))^c. Then use thatf (x)=cF(F1(x1), . . . , Fd(xd)) d

i=1fi(xi)a.e. forx =(x1, . . . , xd)∈R^d to deduce thatH (C_F)+_d

i=1H_h(F_i)= −

flog(f/ h^⊗^d)=H_h(F ).

Remark 2.2. Notice that ifFi has densityfi for 1≤i≤d, then one can choose the reference probability densityh(t)=¹_d_d

i=1f_i(t)so thatH_h(F_i)≥ −log(d). In this case,H_h(F )is finite if and only ifH (C_F)is finite, and we have:

H (C_F)=H_h(F )− d i=1

H_h(F_i)= −

f (x)log

f (x)

_d

i=1f_i(x_i) dx.

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Thus,H (C_F)is the relative entropy of the c.d.f.F with respect to the probability distribution with c.d.f.⊗^d_i₌₁Fi of independent real valued random variables with the same one-dimensional marginal as the one with c.d.f.F. This emphasizes the fact thatH_h(F )−d

i=1H_h(F_i), when it is well defined, does not depend onh.

ForF=(Fi,1≤i≤d)∈Fd, we defineJ(F)taking values in[0,+∞]by:

J(F)= d

i=2

RFi(dt)log

Fi−1(t)−Fi(t). (15) Our aim is to find the c.d.f.F^∗∈LÔS_d (F)which maximizes the entropyH_h. We shall see that this is possible if and only ifJ(F)is finite. From an information theory point of view, this is the distribution which is the least informative among distributions of order statistics with given one-dimensional marginal c.d.f.sF. Since the vector of marginal distribution functionsFis fixed, thanks to (14), we notice thatH_h(F )is maximal onLÔS_d (F)if and only ifH (C_F)is maximal on CÔS(F). Therefore, we focus on finding the copulaC^∗∈CÔS(F)which maximizes the entropyH. We will give the solution of this problem in Section5under some additional hypotheses onF.

3. Symmetric copulas with given order statistics

In this section, we introduce an operator on the setCÔS(F)of copulas of order statistics with fixed marginal c.d.f.sF. This operator assigns to a copulaC∈CÔS(F)the copula of the exchangeable random vector associated to the order statistics with marginal c.d.f.sFand copulaC. We show that this operator is a bijection betweenCÔS(F)and a set of symmetric copulas which can be characterized by their multidiagonal, which is a generalization of the well-known diagonal section of copulas. This bijection has good properties with respect to the entropyH, giving us a problem equivalent to maximizingHonCÔS(F). We shall solve this problem in Section4.

3.1. Symmetric copulas

Forx=(x₁, . . . , x_d)∈R^dwe definex^OS=(x₍₁₎, . . . , x_(d))the ordered vector (increasing order) ofx, wherex(1)≤ · · · ≤x(d)andd

i=1δˆx_i=d

i=1δˆx_(i), withδˆt the Dirac mass att∈R.

LetSdbe the set of permutations on{1, . . . , d}. Forx=(x₁, . . . , x_d)∈R^dandπ∈Sd, we set xπ=(xπ(1), . . . , xπ(d)). A functionhdefined onR^dis symmetric ifh(xπ)=h(x)for allπ∈Sd. A random vectorXtaking values inR^dis exchangeable ifX_π is distributed asXfor allπ∈Sd. In particular a random vectorXtaking values in R^d is exchangeable if and only if its c.d.f. is symmetric. LetL^sym_d ⊂Ld (resp.C^sym⊂Ld) denote the set of symmetric c.d.f. (resp. copulas) onR^d.

LetF ∈Ldand define its symmetrizationF^sym∈L^sym_d by:

F^sym(x)= 1 d!

π∈Sd

F (x_π), x∈R^d. (16)

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In particular, ifXis a random vector taking values inR^dwith c.d.f.F andis a random variable independent ofX, uniformly distributed onSd, thenXis exchangeable with c.d.f.F^sym.

We define the following operator on the set of copulas of order statistics.

Definition 3.1. LetF∈(L1)^d.ForC∈C we defineS_F(C)as the copula of the exchangeable random variableX,whereXis a random vector onR^dwith one-dimensional marginal c.d.f.s Fand copulaCandis an independent random variable uniform onSd.

The applicationSFis well-defined onCand takes values inC^sym. In the above definition, with F=(Fi,1≤i≤d), the one-dimensional marginal c.d.f.s ofXare equal to:

G=1 d

d i=1

Fi. (17)

Since the one-dimensional marginal c.d.f.sFi are continuous, we get thatGis continuous and thus the c.d.f. ofXbelongs toLd. In particular, thanks to Sklar’s theorem, the copula ofXis indeed uniquely defined.

Combining (8), (7) and (16), we can give an explicit formula forSF(C):

S_F(C)(u)= 1 d!

π∈Sd

C F1

G⁻¹(u_π(1)) , . . . ,Fd

G⁻¹(u_π(d))

, u∈I^d. (18)

Remark 3.2. The copula S_F(C) is not equal in general to the exchangeable copulaC^sym defined similarly to (16) by C^sym=(1/d!)

π∈SdC(x_π). However, this is the case if the one- dimensional marginal c.d.f.sFiare all equal, in which caseFi=Gfor all 1≤i≤d.

IfXis a random vector onR^d, letX^OS=(X(1), . . . , X(d))be the order statistics ofX. The proof of the next lemma is elementary.

Lemma 3.3. LetXbe a random vector onR^dwith c.d.f.F anda random variable indepen- dent ofX,uniformly distributed onSd.We have:

• IfF ∈L^OS_d ,then a.s.(X)^OS=X.

• IfF ∈L^sym_d ,then(X^OS)has the same distribution asX.

ForF∈Fd, we define the set of copulasC^sym(F)⊂C^sym as the image of C^OS(F) by the symmetrizing operatorS_F:

C^sym(F)=SF

C^OS(F)

. (19)

The following lemma is one of the main result of this section.

Lemma 3.4. LetF∈Fd.The symmetrizing operatorS_Fis a bijection fromC^OS(F)ontoC^sym(F).

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Proof. LetC₁, C₂∈CÔS(F)withS_F(C₁)=S_F(C₂). LetXandY be random vectors with one- dimensional marginal c.d.f.sFand copulaC1, C2, respectively. SinceC1, C2∈CÔS(F), we get thatXandY are order statistics. NoticeX andYhave the same one-dimensional marginals according to (17) and same copula given byS_F(C₁)=S_F(C₂). Therefore,X andYhave the same distribution. Thus, their corresponding order statistics(X)ÔSand(Y)ÔShave the same distribution. By Lemma3.3, we get thatX andY have the same distribution as well, which

impliesC1=C2.

Remark 3.5. We have in general C^sym(F)=C^OS(F)∩C^sym. One exception being when the marginal c.d.f.sFiare all equal. In this case, both sides reduce to one copula which is the Fréchet- Hoeffding upper bound copula:C⁺(u)=minu,u∈I^d.

3.2. Multidiagonals and characterization of C

^sym

(F)

LetC∈Cbe a copula andUa random vector with c.d.f.C. The mapt→C(t, . . . , t)fort∈I, which is called the diagonal section ofC, is the c.d.f. of maxU. We shall consider a generalization of the diagonal section ofC in the next definition.

Definition 3.6. LetC∈Cbe a copula onR^d andU a random vector with c.d.f.C.The multi- diagonal of the copulaC,δC=(δ(i),1≤i≤d),is thed-tuple of the one-dimensional marginal c.d.f.s ofU^OS=(U₍₁₎, . . . , U_(d))the order statistics ofU:for1≤i≤d

δ(i)(t)=P(U(i)≤t ), t∈I.

We denote byD= {δ_C;C∈C}the set of multidiagonals. Notice thatD⊂Fd, see Remark3.7.

Forδ∈Da multidiagonal, we defineCδ= {C;δC=δ}the set of copulas with multidiagonalδ.

A characterization of the setD is given by Theorem 1 of [13]: a vector of functions δ= (δ₍₁₎, . . . , δ_(d)) belongs toD if and only if δ_(i) is a one-dimensional c.d.f. and the following conditions hold:

δ_(i)≥δ_(i₊₁₎, 1≤i≤d−1, (20) d

i=1

δ_(i)(s)=ds, 0≤s≤1. (21)

Remark 3.7. The condition (21) implies thatδ_(i)∈L1, 1≤i≤d, moreover they ared-Lipschitz.

Also, it is enough to knowd−1 functions fromδ_(i), 1≤i≤d, the remaining one is implicitly defined by (21). Condition (20) along with the continuity ofδ_(i)implies that any multidiagonalδ_C is compatible with the continuous marginal distributions of an order statistics, thereforeD⊂Fd. Remark 3.8. Sinceδ_(i), 1≤i≤dare non-decreasing andd-Lipschitz, we have for almost every t∈I: 0≤(δ_(i))(t)≤d and thus|(δ_(i))(t)log((δ(i))(t))| ≤dlog(d)ford≥2. We deduce that ford≥2:

H (δ_(i))≤dlog(d). (22)

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Remark 3.9. LetC∈C^symbe a symmetric copula onR^dandU a random vector with c.d.f.C.

We check that the multidiagonalδ_C=(δ_(i),1≤i≤d)can be expressed in terms of the diagonal sections(C_{i},1≤i≤d)where for 1≤i≤d:

C_{i}(t)=P max

1≤k≤iUk≤t

=C(t, . . . , t

iterms

,1, . . . , 1

d−iterms

), t∈I.

According to 2.8 of [13], we have for 1≤i≤d: δ_(i)(t)=

d j=i

(−1)^j⁻ⁱ j−1

i−1 d

j C_{_j_}(t), t∈I.

Conversely, we can express the functions(C_{_i_},1≤i≤d)withδ_C. For 1≤i≤d andt∈I, we have:

C_{_i_}(t)=P

1max≤k≤iU_k≤t

= d j=i

P

U_{(j )}≤t| max

1≤k≤iU_k=U_{(j )} P

1max≤k≤iU_k=U_{(j )}

= d j=i

_j₋₁

i−1

_d

i

δ_{(j )}(t),

where we used the definition ofδ_(i)and the exchangeability ofU for the third equality.

The next technical lemma will be used in forthcoming proofs. Recall thatJ⁻¹ denotes the generalized inverse of a non-decreasing functionJ, see Section2.1for its definition and properties, in particular,J⁻¹◦J (t )≤tfort∈R. Recall also that forx=(x₁, . . . , x_d)∈R^d, we write G(x)=(G(x1), . . . , G(xd)).

Lemma 3.10. LetX=(X₁, . . . , X_d)be a random vector onR^dwith one-dimensional marginals c.d.f.(Fi,1≤i≤d).SetG=d

i=1Fi/d.We have for1≤i≤d: P

X_i≤G⁻¹◦G(t )

=P(X_i≤t ), t∈R, that isF_i◦G⁻¹◦G=F_i. (23) We also have forx∈R^d:

P

G(X)≤x

=P

X≤G⁻¹(x)

. (24)

Proof. Since G is the average of the non-decreasing functionsF_i, if G(s)=G(s) for some s, s∈R, then we haveF_i(s)=F_i(s)for every 1≤i≤d. Thanks to (1), we haveG◦G⁻¹◦ G(t )=G(t )and thusFi◦G⁻¹◦G(t )=Fi(t). This gives (23).

Recall definition (2) forI_g(J )the set of points where the function J is increasing on their left. Since G is the average of the non-decreasing functions F_i, we deduce that I_g(G)=

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1≤i≤dI_g(F_i). Notice that a.s.X_i belongs toI_g(F_i). Thanks to (5), we get that a.s.{G(X)≤ x} = {X≤G⁻¹(x)}. This gives (24).

We will also require the following lemma.

Lemma 3.11. LetX=(X₁, . . . , X_d)be a random vector onR^dwith one-dimensional marginals c.d.f.(F_i,1≤i≤d).SetG=_d

i=1F_i/d.We have for1≤i≤d: F_i◦G⁻¹₋1

=G◦F_i⁻¹. (25) Proof. Recall Definition (2) forI_g(J )the set of points where the functionJis increasing on their left. Let 1≤i≤d. Thanks to (4), we haveF_i⁻¹(R)⊂Ig(Fi)∪ {±∞}. SinceGis the average of the non-decreasing functionsF_i, we deduce thatI_g(G)=

1≤i≤dI_g(F_i). Thus we get:

F_i⁻¹(R)⊂I_g(G)∪ {±∞}, (26) for all 1≤i≤d. The functionFi◦G⁻¹is also bounded, non-decreasing and càd-làg therefore we have fort, s,∈R:

t≥

F_i◦G⁻¹₋1

(s) ⇐⇒ F_i◦G⁻¹(t)≥s ⇐⇒ G⁻¹(t)≥F_i⁻¹(s)

⇐⇒ t≥G◦F_i⁻¹(s),

where we used the equivalence of (1) for the first and second equivalence, (26) and the equivalence of (5) for the last. This gives that(F_i◦G⁻¹)⁻¹=G◦F_i⁻¹. In the following lemma, we show that forF∈Fd, all copulas in C^sym(F)share the same multidiagonal denoted byδ^F.

Lemma 3.12. LetF=(Fi,1≤i≤d)∈Fd.LetC∈C^OS(F)andU be a random vector with c.d.f.S_F(C).Letδ^F=(δ_(i),1≤i≤d)be the multidiagonal ofS_F(C),that is the one-dimensional marginal c.d.f.s ofU^OS,the order statistics ofU.We have thatδ^Fdoes not depend onCand for 1≤i≤d:

δ(i)=Fi◦G⁻¹ and δ⁻_(i)¹=G◦F⁻_i ¹, (27) withGgiven by(17).Furthermore,Cis the unique copula ofU^OS.

With obvious notation, we might simply writeδ^F=F◦G⁻¹, withGgiven by (17).

Proof of Lemma3.12. LetXbe a random vector of order statistics with marginalsF∈Fdand copulaC. ThenS_F(C)is the copula of the exchangeable random vectorX, whereis uniform onSd and independent ofX. We have already seen in (17) that the one-dimensional marginals ofX have the same distribution given byG∈L1. Thanks to (9), we deduce that the random vectorU, with c.d.f.S_F(C), has the same distribution asG(X). SinceGis non-decreasing, this

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implies that the order statistics ofU,U^OS, has the same distribution asG((X)^OS)that is as G(X), thanks to Lemma3.3. Then use (24) to get forx∈R^d:

P

U^OS≤x

=P

G(X)≤x

=P

X≤G⁻¹(x)

. (28)

This gives the first part of the lemma as the multidiagonal ofUis the one-dimensional marginal c.d.f.s of its order statistics. The second equation in (27) is due to Lemma3.11. The fact thatCis the copula ofU^OSand its uniqueness are due to (28) and the continuity ofδ_(i), see Remark3.7.

The next proposition shows that the setC^sym(F)is actually the set of symmetric copulas with diagonal sectionδ^F. This yields the main result of this section given by the subsequent corollary.

Proposition 3.13. LetF∈Fd.We haveC^sym(F)=Cδ^F∩C^sym. Proof. By Lemma3.12, we haveC^sym(F)⊂C_δ^F∩C^sym.

LetC∈C_δ^F ∩C^sym andU be a random vector with c.d.f. C. LetG be given by (17). No- tice that X =G⁻¹(U ) is an exchangeable random vector with marginals G and copula C.

Thanks to Lemma3.3, the proof will be complete as soon as we prove that the one-dimensional marginal c.d.f.s of XÔS=(X₍₁₎, . . . , X_(d)), the order statistics of X, is given by F. Notice XÔS=G⁻¹(UÔS), withUÔSthe order statistics ofU whose one-dimensional marginal c.d.f.s are given byδ^F. We have for 1≤i≤dandt∈R:

P(X(i)≤t )=P

G⁻¹(U(i))≤t

=P

U(i)≤G(t )

=Fi◦G⁻¹◦G(t )=Fi(t),

where we used (1) for the second equality, (27) for the third, and (23) for the last. This finishes

the proof.

Corollary 3.14. According to Proposition3.13, (19)and Lemma3.4,we get that for anyF∈Fd, the symmetrizing operatorSFis a bijection betweenC^OS(F)andCδ^F∩C^sym.

We end this section by an ancillary result we shall use later.

Lemma 3.15. LetF∈Fd.We haveJ(F)=J(δ^F).

Proof. LetF=(Fi,1≤i≤d). We get, using (27) and the change of variables=G⁻¹(t)that:

I

δ_(i)(t)log

δ_(i₋₁₎(t)−δ_(i)(t)dt=

G⁻¹((0,1))

Fi(ds)log

Fi−1(s)−Fi(s).

SincedFi=0 outsideG⁻¹((0,1))(asGis increasing as soon asFi is increasing), we get that the last integration above is also overR. We deduce that:

J δ^F

= d

i=2

RFi(ds)log

Fi−1(s)−Fi(s)=J(F).

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3.3. Density and entropy of copulas in C

^sym

(F)

We prove in this section thatS_Fpreserves the absolute continuity onC^OS(F)forF∈Fd and the entropy up to a constant. Let us introduce some notation. For marginalsF∈Fd, let

_i^F=

s∈R,Fi−1(s) >Fi(s)

for 2≤i≤d. (29)

The complementary set(_i^F)^cis the collection of the points whereFi−1=Fi. We define^F⊂I as:

^F= d i=2

Fi

_i^Fc

. (30)

By Remark3.7, we haveD⊂Fd, then the definitions (29) and (30) apply for allδ∈D. In partic- ular, forδ=(δ(1), . . . , δ(d))∈Dthe sets_i^δ, 2≤i≤dare open subsets ofI, therefore(_i^δ)^c∩I is a compact subset. This and the continuity ofδ_(i)imply thatδ_(i)((_i^δ)^c)=δ_(i)((_i^δ)^c∩I )is also compact, hence^δis compact. Notice that{0,1} ⊂^δalways holds. We defineC⁰_δ=Cδ∩C⁰the subset of absolutely continuous copulas with multidiagonalδand the subsetD⁰= {δ∈D,C_δ⁰=

∅}of multidiagonals of absolutely continuous copulas. According to Theorem 2 of [13], the multidiagonalδbelongs toD⁰if and only if it belongs toDand the Lebesgue measure of^δis zero:|^δ| =0.

Lemma 3.16. Letδ∈D.We haveδ∈D⁰if and only if for all2≤i≤d,a.e.:

δ_(i₋₁₎1₍δ

i)^c=δ_(i) 1₍δ

i)^c=0. (31)

Furthermore,we have thatJ(δ) <+∞impliesδ∈D⁰.

Proof. LetJ be a function defined onI, Lipschitz and non-decreasing. LetAbe a Borel subset ofI. We have:

J (A)=

1J (A)(t) dt= ₁

0

1_{_s_∈_J−1◦J (A)}J(s) ds= ₁

0

1A(s)J(s) ds,

where we used (3) and (5) for the last equality. This gives that |J (A)| =0 if and only if a.e.

J1A=0. Then use that δ∈D⁰ if and only if |δ(i)((_i^δ)^c)| =0 for all 1≤i≤d and that δ_(i₋₁₎((_i^δ)^c)=δ_(i)((_i^δ)^c)to conclude thatδ∈D⁰if and only if (31) holds for all 2≤i≤d.

The last part of the lemma is clear.

Definition 3.17. LetF_d⁰⊂Fdbe the subset of marginalsFsuch that there exists an absolutely continuous c.d.f.of order statistics with marginalsF,that isL^OS_d (F)∩L⁰_d=∅.

In particular, we haveD⁰⊂F_d⁰. The next lemma gives a characterization of the setF_d⁰.