 # Copulas and Preserver Problems

Preserver problems concern the characterization of operators on general spaces that leave invariant some categories of subsets or ratios. The most known in the mathematical literature are those of linear preserver problems (LPP) which date back to the ninth century. Here, we treat the preserver problem in the recent and emerging field of copulas. Precisely, we prove that copula property is preserved uniquely under increasing transformations.

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## 1. Introduction

The study of Preserver problems and linear preserver problems (LPPs for short) is a vast field and an interesting topic for mathematical researches. For a historical overview on preserver problems and techniques used in their treatment, the important article of Chi-Kwong Li and Nam-Kiu Tsingt  is recommended and advisable. Recently the linear case has been intensively studied. We start by making precise this notion of linear preservation. By ’linear’ we mean here linear or at least additive maps on Banach algebras that leave invariant certain subsets. It is a kind of invariance under linear operations. The mappings defined on Banach spaces which preserve some properties such as invertibility are widely studied mainly in ,  and  and in another context by  where surjective linear maps preserving the set of Fredholm operators are discussed. In the same spirit, M. Mbkhta et.al  pursued in a strong work the description of all additive and surjective continuous maps in the algebra of all bounded linear operators acting on a complex separable infinite-dimensional Hilbert space, preserving some interesting classes of operators acting such as the operators of finite ascent, the operators of finite descent or Drazin invertible ones. As a pedagogical illustration, one may consider in the elementary algebra the preservation of determinant function on by the linear mappings where and are two fixed square matrices. The spectacular result in this direction was the converse characterization by Frobenus .
Recently, the attention is paid to non linear preserver problems since they do not restrict the study to linear operators and allow the preservation treatment of some emergent stochastic and statistical properties. For example, in , some unnecessary linear continuous bijections on the set of all self adjoint operators maps which preserve the star order are characterized.
In this paper we are concerned with preserver problem of copula property. For the best of our knowledge, this problem is less investigated in the literature. The sense of preservation as well as necessary reminders and complements on copulas will be made precise bellow. The introductory note on copulas is based on the unsurmountable book of R. Nelson .
The term copula appeared for first time in the edifying paper of Abe Sklar at 1959 

as an answer to Maurice Fréchet’s probability problem. It became henceforth an efficient tool when statisticians deal with ex-changeability, dependence or symmetry and asymmetry problems (see

 for a functional point of view of these concepts). It is well known that if is a copula then for every nondecreasing functions and on the function defines a new copula. To describe this situation, we say that copula property is preserved by nondecreasing functions. The present paper clarifies and discusses whether only this type of mappings have this property or not.
After this brief preamble, the next section is devoted to recall important notions on copulas and gives a general reminder of most important results on preserving problems.
The paper is organized as follows: After the introduction, some reminders on mathematical concepts used here are reviewed as preliminaries. The second section reshuffles essential results on increasing functions in a general framework before specifying the case of functions on which we limit our work. In the third section, some properties which are preserved by increasing transformations are discussed. Classical results on this topic will be overflown. At the last section, some examples are given as illustration.

## 2. Preliminaries

We start by giving an overview on copulas. One may restrict the study on the bivariate case without affecting the generality of the problem. Let denote from now on the interval . We assume in addition that all copulas treated here are supported on which means in probabilistic terms that the laws are diffuse.

###### Definition 2.1.

A copula is a function on into which satisfies the following conditions for all :

1. and

2. the 2-increasing property:

Let and

be two continuous random variables with distribution functions

and . Let

be their joint distribution function. Then according to Sklar theorem(see

), there exists a unique copula such that The continuity hypothesis on and is assumed just to guarantee uniqueness of

Copulas are popular in high-dimensional statistical applications as they allow us to model easily the distribution of random vectors by estimating marginals and copula separately. There are many

parametric copula families in the literature, which usually have parameters that control the strength of dependence or symmetry.

Question: What are transformations and of random variables and which preserve properties (1)-(3) in Definition 2.1 for the new mapping ?
A historical answer to this important question of preservation was given by R. Nelsen in . Precisely, the following theorem was proved therein

###### Theorem 2.2.

Let and be continuous random variables with copula . If and are strictly increasing on their respective ranks and then

 Cϕ(X)ψ(Y)=CXY

For convenience, we will rewrite (see section 4 below) the original proof of this important theorem which gives a first bridge between linear and non linear preservation problems. Nelson comments on this result saying that copula property is preserved.
The discussion about correlation coefficient given so far in the current paper (see section 3 below) gives an elementary example for a quick familiarization of our concern.
To clarify how to preserve the copula property, a deep understanding of increasing behavior of real functions defined on a given area of is necessary. In the following section we give enough tools to describe -increasing functions and understand why conditions (1) and (2) in definition (2.1) are necessary to serve as a bridge between monotonicity of one and two variables.

## 3. On 2-increasing functions

The interest of monotone functions does not need to be clarified. For our purposes, it suffices to refer to any document tracing the concordance problems in statistics. Mathematically speaking, the suitable framework to a general treatment is the lattice Banach spaces by which one understands ordered Banach spaces with a positive cone A lot of topics in classical fields of mathematics used the notion of monotonicity to prove interesting results mainly on fixed points. One may see for example  where the existence of maximal and minimal fixed points are proved. Anyhow for application to copulas, the natural framework should be functions defined on It seems obvious that for a given function the simultaneous monotony of with respect to will be equivalent to ordinary monotony of partial function and with respect to and respectively. Far from being so, counterexamples prove a negative conclusion. Let us begin by clarifying this unpredictable statement

###### Definition 3.3.

A function of two variables defined on is increasing in volume or increasing, if for all we have

 (3.1) [f(x2,y2)−f(x1,y2)]−[f(x2,y1)−f(x1,y1)]≥0.

As mentionned above, not any 2-increasing function in volume is nondecreasing in each argument. The examples are not lacking. It suffices to consider the following classical one:

###### Example 3.4.

let be the function defined on by: Then is increasing but not nondecreasing of the first component for arbitrary values of nor of the second one for arbitrary values of On the other hand the function is nondecreasing on each argument and but is far to be increasing since

Even if these negative results occur, it is easy to verify that partial variations of every increasing function are nondecreasing one-valued mappings. More precisely, we have

###### Proposition 3.5.

Let be a increasing function. Then for all such that , the partial function is a non decreasing function. Analogously, for every such that the function is nondecreasing on

The proof is a direct consequence of volume monotony (3.1). At this point, one may see easily that condition (1) in definition (2.1) is fundamental to ensure that partial functions and are nondecreasing ones. In probabilistic words, these conditions explain that the margins and of the copula

(which are uniform distributions) are diffuse and supported entirely by the interval

Nelson  took advantage of this quality to prove an interesting analytic property of copulas seen as functions of two variables. He proved that they are Lipschitz continuous as the following proposition states

###### Proposition 3.6.

Let be a copula as given by definition (2.1). For all , in we have

 (3.2) |C(x2,y2)−C(x1,y1)|≤|x2−x1|+|y2−y1|

The fact that a copula enjoys this property is of great interest. One may see [Sa18] for some clarifications and applications to asymmetry problem of copulas. The reason that we recall this result lies in its importance in computing ratios (coefficients) like of Kendall or of Spearman given by formulas

 τ=2n(n−1)∑i

where is the sign function and is the difference between two ranks of each observation of statistical collected data .
It is known that 2-increasing functions preserve such ratios, this yields the interest of such functions.

## 4. Improvement of Nelson classical result

The historical and simple Nelson’s reproach to linear regression ratios was the instability with usual non linear transformations like square and/or exponential functions. This kind of mappings perturbs widely regression dependence given by the correlation coefficient

This latter coefficient is just preserved under linear transformations. Indeed, it is elementary to check that for any two linear scalar transformations and (i.e and for certain ), one has

 (4.3) ρ(ϕ(X),ψ(Y)) = Cov(ϕ(X),ψ(Y))σ(ϕ(X))σ(ψ(Y)) (4.4) = Cov(aX,bY)abσ(X)σ(Y)) (4.5) = Cov(X,Y)σ(X)σ(Y) (4.6) = ρ(X,Y).

The natural question in this case and in our framework is whether this kind of transformation is the only one which ensures the preservation of the linear regression dependence. An immediate answer to the fact of failure of preservation under non linear transformations follows from the simple consideration of So for a standard normal random variable , one has
Assume now that the coefficient is invariant under all convex transformations and Without loss of generality and thanks to usual rescaling procedure, one may suppose that the random variables and are centred and reduced. In such case, the assumed preserving property will be expressed namely as:

 (4.7) Cov(ϕ(X),ψ(Y))σ(ϕ(X))σ(ψ(Y))=Cov(X,Y) % for all convex functions ϕ and ψ.

Question: Under the hypothesis (4.7), what will be the general form of functions and ? More precisely, if we assume that (4.7) holds for every choice of random variables and , what should be the expressions of and when and vary on ?
The elementary calculus in (4.3)-(4.6) says in other words that all linear functions belong to the desired class. Are they alone?

The following proposition gives a positive answer

###### Proposition 4.7.

Let and real valued functions such that (4.7) holds. Then

 ∀(x,y)∈\tenBbR2:  ϕ(x)=ax+b and ψ(y)=a′y+b′

for some real constants , , and

###### Proof.

All random variables are considered on the same probability space . This is possible thanks to Skohorod theorem (see  among others) stated for other concerns that we do not develop here (mainly types of stochastic convergence). After reducing and centering the random variables and the condition (4.7) is equivalent to

 E[(ϕ(X)−E(ϕ(X))).(ψ(Y)−E(ψ(Y)))]=σ(ϕ(X))σ(ψ(Y))E(XY).

For the sake of simplicity and readability, let us denote In order to dodge uniqueness of antecedent we assume in addition that is invertible. For a particular choice instead of , and taking into account that this yields

 E[(ϕ(X)−E(ϕ(X))).Y]=λE(XY).

Since this last equality occurs for all random variables , classical results on Lebesgue integration theory lead to

 (ϕ(X)−E(ϕ(X)))=λX, P−a.e.

On the set that we may assume with null measure, one has The claim follows by taking and as long as this latter quantity is well defined.
The manifest symmetry in (4.7) ensures that an analogous argument prevails for the function . ∎

We hope now to give a similar argument for the copula preserver problem. To do so, let’s go back to copulas and recall the classical result of Nelson [3, pages 25 and 26] concerning the preservation of copula property under strictly increasing transformations

###### Proposition 4.8.

Let and be continuous random variables associated with the copula . If and are respectively strictly increasing on and , then .

A proof may be found in . As promised and for convenience we rewrite briefly the argument given therein.

###### Proof.

In the sequel if is a random variable, we denote from now on its distribution function.
As and are assumed continuous, the range problem is excluded. On the other hand, for all in and all strictly increasing functions and , the random variables and are continuous and the direct computation leads to

 C(ϕ(X),ψ(Y)(Fϕ(X)(a),Fψ(Y)(b)) = P((ϕ(X)≤a,ψ(Y)≤b) = P(X≤ϕ−1(a),Y≤ψ−1(b)) = C(X,Y)(FXoϕ−1(a),FYoψ−1(b)) = C(X,Y)(Fϕ(X)(a),Fψ(Y)(b)).

The continuity assumption ensures that every couple may be written for some convenient This yields

 ∀(u,v)∈[0,1]2: C(ϕ(X),ψ(Y)(u,v)=C(X,Y)(u,v).

A real challenge consists in proving the converse in the following sense: If the copula property is preserved by the couple for all continuous random variables and , the functions and are automatically increasing? if not what should be their general form?

A first and simple analysis leads to the following remark: not every real value linear functions and warrant the preservation of copula property. This is obvious since if one or both of and are not increasing, the preservation property fails according to [3, Theorem 2.4.4] (choose or strictly negative in proposition 4.7). Hence the problem is far from being LPP one.
The following proposition gives a partial answer that we hope to improve in the future.

###### Proposition 4.9.

Let and be two real value functions such that for a given random vector , the identity holds. Then and are increasing.

###### Proof.

Consider two real functions and such that for all random variables and we have for all couple
Assume first that is continuously differentiable and . According to Definition (2.1) and Proposition (3.5), one may write

 ∂C(ϕ(X),ψ(Y))(u,v)∂u = ∂C(ϕ(X),ψ(Y))(u,v)−C(ϕ(X),ψ(Y))(u,0)∂u ∂C(ϕ(X),ψ(Y))(u,v)∂u = ∂C(X,Y)((ϕ(u),ψ(v)))−C(X,Y)((ϕ(u),ψ(0)))∂x = ϕ′(u)∂C(X,Y)((ϕ(u),ψ(v)))∂x

The assumption on and ensures that for all , one has In equivalent words, the mapping is nondecreasing. For symmetry reasons in the third item of (2.1) an analogous argument is valid to prove that is also an increasing function.
For function without nonzero value at the origin, one may consider which shares the same monotony with and verifies
For general statement ( is not necessarily differentiable), one may argue as follows:

1. Almost every real number is Lebesgue point of .

2. For such point we have

To complete the claim, a topological argument and density of regular functions on allows the conclusion.

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