Information Sciences 310 (2015) 69–76

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Quadratic constructions of copulas

http://dx.doi.org/10.1016/j.ins.2015.03.0160020-0255/� 2015 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: anna.kolesarova@stuba.sk (A. Kolesárová), gmayor@uib.es (G. Mayor), radko.mesiar@stuba.sk (R. Mesiar).

Anna Kolesárová a, Gaspar Mayor b,⇑, Radko Mesiar c,d

a Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology inBratislava, Radlinského 9, 812 37 Bratislava 1, Slovakiab Department of Mathematics and Computer Science, University of the Balearic Islands, 07122 Palma (Mallorca), Spainc Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, 81368 Bratislava 1, Slovakiad Centre of Excellence IT4Innovations, Division University of Ostrava, IRAFM, Czech Republic

a r t i c l e i n f o

Article history:Received 2 April 2013Received in revised form 19 February 2015Accepted 10 March 2015Available online 19 March 2015

Keywords:CopulaInvariant copulaPlackett copulaQuadratic construction

a b s t r a c t

In the paper we introduce and study quadratic constructions of copulas based on com-position of a copula and some quadratic polynomial. We characterize all quadraticpolynomials whose composition with an arbitrary copula always results in a copula. Dueto this result, we can assign a two-parametric class Kc;d

C

� �of copulas with parameters

ðc; dÞ in a certain subset X # R2 to each copula C. Moreover, we also determine all copulasinvariant with respect to the quadratic constructions presented. This investigation bringstwo interesting parametric classes of copulas. We show that the union of these two classesis equal to the so-called Plackett family of copulas. We add some properties of these copu-las and their statistical consequences.

� 2015 Elsevier Inc. All rights reserved.

1. Introduction

Copulas [23,19,9] play an important role in all areas dealing with random vectors. Fitting an appropriate copula to givenexperimental data requires a large buffer of copulas, and therefore any new construction method of copulas extends thepossibility of their applications. There are several construction methods determining new copulas by means of two or moregiven copulas. For example, we recall the standard convex combinations of copulas, several types of ordinal sums, or otherpatchwork techniques; see e.g., [1,2,4,5,7,8,6,10,14–17,19,20,22]. A new copula can also be obtained by a transformation ofthe given copula C [18], or by its flipping, or by related techniques based on measure preserving transformations [9,19,12]. Inall these cases, the knowledge of the values x; y and Cðx; yÞ is not sufficient for determining the value Dðx; yÞ of a new copulaD constructed by means of the copula C. One of a few known methods which determine the values Dðx; yÞ directly by meansof x; y and Cðx; yÞ, is the method recently introduced in [3], and also discussed in [13]. By the result proved there, for any

copula C : ½0;1�2 ! ½0;1�, the function DC defined on ½0;1�2 by

DCðx; yÞ ¼ Cðx; yÞðxþ y� Cðx; yÞÞ ð1Þ

is a copula. Note, that if we consider a quadratic polynomial P of three variables, Pðx; y; zÞ ¼ xzþ yz� z2, then the construc-tion (1) can be seen as the composite function

DCðx; yÞ ¼ Pðx; y;Cðx; yÞÞ: ð2Þ

70 A. Kolesárová et al. / Information Sciences 310 (2015) 69–76

As already mentioned, copulas can also be constructed by flipping [19]. For example, it is known that for any copula

C : ½0;1�2 ! ½0;1�, the function C0;1 : ½0;1�2 ! ½0;1� constructed by y-flipping of C, i.e., given by

C0;1ðx; yÞ ¼ x� Cðx;1� yÞ; ð3Þ

is a copula. Combining (1) with (3) yields for each copula C : ½0;1�2 ! ½0;1� the function HC : ½0;1�2 ! ½0;1�,

HCðx; yÞ ¼ DC0;1

� �0;1ðx; yÞ;

that is also a copula. It can easily be shown that for each ðx; yÞ 2 ½0;1�2,

HCðx; yÞ ¼ C2ðx; yÞ þ xy� xCðx; yÞ � yCðx; yÞ þ Cðx; yÞ: ð4Þ

The function HC can be written as a composite function,

HCðx; yÞ ¼ Pðx; y;Cðx; yÞÞ;

where P is the polynomial Pðx; y; zÞ ¼ z2 þ xy� xz� yzþ z. Both mentioned functions DC and HC are special instances of acomposite function of the form Pðx; y;Cðx; yÞÞ, where P is a quadratic function

Pðx; y; zÞ ¼ ax2 þ by2 þ cz2 þ dxyþ exzþ fyzþ gxþ hyþ izþ j; ð5Þ

with real coefficients a; . . . ; j and C is a copula.The aim of the paper is to investigate in general for which coefficients a; . . . ; j 2 R, the functions

KCðx; yÞ ¼ Pðx; y;Cðx; yÞÞ; ðx; yÞ 2 ½0;1�2;

are copulas for each copula C. We emphasize that the construction we intend to study only requires, for determining thevalues KCðx; yÞ of a new function KC , the knowledge of x; y and Cðx; yÞ. Throughout the paper we will only consider two-di-mensional copulas, which will be called copulas in short. The paper is organized as follows: In the next section, we character-ize all quadratic polynomials Pðx; y; zÞ of the form (5) for which the composition Pðx; y;Cðx; yÞÞ is a copula for each copula C. InSection 3, we study copulas invariant with respect to the quadratic constructions presented in Section 2. As a by-product,two parametric families of copulas are introduced and properties of their members are examined. Surprisingly, these twofamilies together make up a new parametrization of the well-known Plackett family of copulas. Finally, some concludingremarks are added.

2. Copulas obtained by a quadratic construction

A copula is a function C : ½0;1�2 ! ½0;1� satisfying the properties.

(C1) for all x; y 2 ½0;1�, Cðx;0Þ ¼ 0; Cð0; yÞ ¼ 0,(C2) for all x; y 2 ½0;1�, Cðx;1Þ ¼ x; Cð1; yÞ ¼ y,(C3) for all x1; x2; y1; y2 2 ½0;1�; x1 6 x2; y1 6 y2,

Cðx2; y2Þ � Cðx1; y2Þ þ Cðx1; y1Þ � Cðx2; y1ÞP 0:

By property (C1), 0 is the annihilator of each copula C, and by (C2), 1 is its neutral element. The third property is the 2-increasing property of C. The expression on the left-hand side in (C3) defines the so-called C-volume VC of the rectangle½x1; x2� � ½y1; y2�.

As distinguished examples of copulas we recall the functions W; P and M, given by

Wðx; yÞ ¼maxf0; xþ y� 1g; Pðx; yÞ ¼ xy; Mðx; yÞ ¼minfx; yg;

which are called the Fréchet–Hoeffding lower bound, the product copula and the Fréchet–Hoeffding upper bound or theminimum copula, respectively. Note that for each copula C it holds

W 6 C 6 M:

For more details on copulas, refer to monographs [9,19].

Referring to the previous discussion, to any copula C we assign the function KC defined on ½0;1�2 by

KCðx; yÞ ¼ Pðx; y;Cðx; yÞÞ; ð6Þ

where P is a quadratic polynomial of the form (5) with real coefficients. We will say that the function KC is obtained by a quad-ratic construction from the copula C. The function KC can be written as

KCðx; yÞ ¼ ax2 þ by2 þ cz2 þ dxyþ exzþ fyzþ gxþ hyþ izþ j; ðx; yÞ 2 ½0;1�2; ð7Þ

where z ¼ Cðx; yÞ and a; . . . ; j 2 R.

A. Kolesárová et al. / Information Sciences 310 (2015) 69–76 71

In general, the functions KC need not be copulas. In the previous section, we mentioned two positive examples in whichcomposition of the considered polynomials with any copula C led to a copula. But, for example, using the polynomialPðx; y; zÞ ¼ z2, we never obtain a copula by composition (6). There are also partially positive examples, leading only somecopulas to new copulas. For example, composition KW of the polynomial Pðx; y; zÞ ¼ z2 � xz� yzþ 2z with copula W is equalto W, while the function KM obtained by composition of P and the minimum copula M, is not a copula. Indeed,

KMðx; yÞ ¼xð2� yÞ if ðx; yÞ 2 ½0;1�2; y P x;

yð2� xÞ if ðx; yÞ 2 ½0;1�2; y < x;

(

which can also be written as KMðx; yÞ ¼minfx; ygð2�maxfx; ygÞ, i.e., KM > M, and thus KM cannot be a copula. Note, thatcomposition of the considered polynomial P with the product copula P gives the function KP; KPðx; yÞ ¼xyþ xyð1� xÞð1� yÞ, i.e., a member of the Farlie–Gumbel–Morgenstern family of copulas. It is important to find conditionsunder which the function KC defined by (6) is a copula for each copula C.

We start our investigation with looking for restrictions of coefficients a; . . . ; j following from the boundary conditions(C1). They lead to the equations

ax2 þ gxþ j ¼ 0; by2 þ hyþ j ¼ 0; ð8Þ

which have to be valid for each x; y 2 ½0;1�. Thus, necessarily j ¼ 0 and also a ¼ g ¼ 0; b ¼ h ¼ 0. Taking into account theobtained zero coefficients, we can simplify the formula (7) and write it as:

KCðx; yÞ ¼ cz2 þ dxyþ exzþ fyzþ iz; z ¼ Cðx; yÞ: ð9Þ

Next, conditions (C2) lead to the equations

c þ e ¼ 0dþ f þ i ¼ 1c þ f ¼ 0dþ eþ i ¼ 1; ð10Þ

determining the following relationships between coefficients:

c;d 2 R; e ¼ �c; f ¼ �c; i ¼ 1þ c � d: ð11Þ

Consequently,

KCðx; yÞ ¼ cC2ðx; yÞ þ dxy� cxCðx; yÞ � cyCðx; yÞ þ ð1þ c � dÞCðx; yÞ; c;d 2 R: ð12Þ

Some additional conditions for coefficients c and d can be found by analyzing the properties of KC for distinguished copu-las C ¼ M; W and P.

Let us first consider that C ¼ M and investigate the 2-increasing property of KM . As

KMðx; yÞ ¼ ðd� cÞxyþ ð1þ c � dÞminfx; yg;

the KM-volume of any rectangle R ¼ ½a; b� � ½c; d�, such that R � ID ¼ fðx; yÞ 2 ½0;1�2jx 6 yg or R � ID ¼ fðx; yÞ 2 ½0;1�2jx P yg,is

VKM ðRÞ ¼ ðd� cÞðb� aÞðd� cÞ:

A necessary condition for VKM ðRÞP 0 is d P c. Moreover, for all 0 6 a 6 b 6 1 we have

VKM ð½a; b�2Þ ¼ ðb� aÞð1� ðd� cÞð1� bþ aÞÞ;

and thus VKM ð½a; b�2ÞP 0 for all mentioned a; b if and only if d� c 6 1.

In summary, the necessary conditions for KM to be a copula are

0 6 d� c 6 1: ð13Þ

Note, that for any such coefficients c; d; KM is a copula, because it is a convex combination of copulas P and M,

KM ¼ ðd� cÞPþ ð1þ c � dÞM:

Now, assume that C ¼W . As

KWðx; yÞ ¼ dxyþ ð1� dÞmaxf0; xþ y� 1g;

necessarily d P 0 and the requirement of the monotonicity of KW leads to the restriction d 6 1. Thus, we have to require

0 6 d 6 1: ð14Þ

Note, that for any such d, the function KW ¼ dPþ ð1� dÞW is a convex combination of copulas P and W, thus, a copula.

72 A. Kolesárová et al. / Information Sciences 310 (2015) 69–76

Finally, for the product copula P we obtain the function KP,

KPðx; yÞ ¼ xyþ cxyð1� xÞð1� yÞ:

It is known that KP is a copula if and only if c 2 ½�1;1�, see, e.g., [19]. Recall that the parametric family of copulas ðKcÞc2½�1;1�,with Kcðx; yÞ ¼ xyþ cxyð1� xÞð1� yÞ, is known as the Farlie–Gumbel–Morgenstern family, containing all copulas with quad-ratic sections in both variables [19]. Note, that if the conditions (14) and (13) are satisfied, then necessarily �1 6 c 6 1, sothis condition need not be considered explicitly.

As we require the function KC given by (12) to be a copula for each copula C, in general, the obtained conditions

0 6 d 6 1; 0 6 d� c 6 1; ð15Þ

are necessary conditions only. However, in the next theorem we show that they are also sufficient for KC to be a copula forany copula C.

Theorem 1. For a copula C : ½0;1�2 ! ½0;1�, let KC be a function defined on ½0;1�2 by (7). Then the following are equivalent.

(i) For any copula C;KC is a copula.(ii) KC is given by (12), i.e.,

KCðx; yÞ ¼ cC2ðx; yÞ þ dxy� cxCðx; yÞ � cyCðx; yÞ þ ð1þ c � dÞCðx; yÞ;

with coefficients c; d satisfying conditions (15).

Proof. The implication (i)) (ii) has already been proved in the previous discussion.(ii)) (i): Let C be a copula. Denote by F C the set of all functions assigned to the copula C by formula (12), whose

coefficients c; d satisfy (15). The set of all possible coefficients will be denoted by X, i.e.,

X ¼ fðc;dÞ 2 R2j0 6 d 6 1; 0 6 d� c 6 1g: ð16Þ

There is a one-to-one correspondence between F C and X. The function in F C corresponding to a pair of coefficients ðc; dÞ 2 X

will be denoted by Kc;dC . X is a convex subset of R2 with vertices U ¼ ð�1;0Þ; V ¼ ð0;0Þ; X ¼ ð1;1Þ and Y ¼ ð0;1Þ, see Fig. 1.

The vertex U ¼ ð�1;0Þ corresponds to the function

K�1;0C ðx; yÞ ¼ �C2ðx; yÞ þ xCðx; yÞ þ yCðx; yÞ ¼ Cðx; yÞðxþ y� Cðx; yÞÞ

¼ DCðx; yÞ:

As DC is a copula for each C, see (1), so is K�1;0C . The vertex V ¼ ð0;0Þ corresponds to the function

K0;0C ðx; yÞ ¼ Cðx; yÞ;

i.e., K0;0C is always a copula. The vertex Y ¼ ð0;1Þ corresponds to the function

Fig. 1. The domain X of possible pairs of coefficients ðc;dÞ.

A. Kolesárová et al. / Information Sciences 310 (2015) 69–76 73

K0;1C ðx; yÞ ¼ xy ¼ Pðx; yÞ;

thus, K0;1C is a copula for each C. Finally, the vertex X ¼ ð1;1Þ corresponds to K1;1

C given by

K1;1C ðx; yÞ ¼ C2ðx; yÞ þ xy� xCðx; yÞ � yCðx; yÞ þ Cðx; yÞ;

i.e., K1;1C ¼ HC , see (4), which means that K1;1

C is a copula for any C.Due to the convexity of the set X, each point ðc; dÞ 2 X is a convex combination of its vertices and, consequently, the

function Kc;dC 2 F C is the same convex combination of copulas K�1;0

C ; K0;0C ; K1;1

C and K0;1C corresponding to vertices of X, and

therefore Kc;dC is a copula. h

Note, that due to the previous result, to any copula C, we can assign a two-parametric class of copulas, namely the class

Kc;dC

� �ðc;dÞ2X

. For each copula C, it is a convex class of copulas, always containing copulas C and P.

Remark 1. Note, that the copulas of the type Kc;0C and Kc;1

C already appeared in [3]. However, the results given there wereobtained by probabilistic methods.

Quadratic constructions given by (12) with coefficients ðc; dÞ 2 X will be simply called universal quadratic constructions.The following example illustrates Theorem 1.

Example 1. Let C be a copula. Considering the pair of coefficients ðc; dÞ ¼ ð1=2;3=4Þ, we obtain the function

K12;

34

C ðx; yÞ ¼12

C2ðx; yÞ þ 34

xy� 12

xCðx; yÞ � 12

yCðx; yÞ þ 34

Cðx; yÞ; ðx; yÞ 2 ½0;1�2:

By Theorem 1, it is a copula. As ð1=2;3=4Þ ¼ ð1=4Þð0;0Þ þ ð1=4Þð0;1Þ þ ð1=2Þð1;1Þ; K12;

34

C can be expressed as the same linearcombination of copulas C; P and HC corresponding to vertices V ¼ ð0;0Þ; Y ¼ ð0;1Þ and X ¼ ð1;1Þ, respectively, i.e.,

K12;

34

C ðx; yÞ ¼14

Cðx; yÞ þ 14

Pðx; yÞ þ 12

HCðx; yÞ; ðx; yÞ 2 ½0;1�2:

It can easily be checked that both formulas for K12;

34

C ðx; yÞ give the same values. For example, for C ¼ M, as HM ¼ M, see (4), we

obtain K12;

34

M ¼ ð3=4ÞM þ ð1=4ÞP, which agrees with the previously derived formula for KM .

Remark 2. Some pairs ðc; dÞ 2 X can be represented as convex combinations of vertices U; V ; X and Y in different ways. Forexample, ð0;1=2Þ ¼ ð1=2ÞV þ ð1=2ÞY but also ð0;1=2Þ ¼ ð1=2ÞX þ ð1=2ÞU. By the proof of Theorem 1,

K0;12C ¼ ð1=2ÞðC þPÞ ¼ ð1=2ÞðHC þ DCÞ;

which yields HC ¼ C þP� DC . The last equality can easily be verified by using (4) and (1). It says that functions HC ; C; P and

DC are not linearly independent. Note, that each copula Kc;dC can be expressed as a linear combination of copulas P; C and DC ,

namely, Kc;dC ¼ dPþ ð1þ c � dÞC � cDC .

Quadratic constructions of copulas are related to flipping and survival constructions of copulas. Recall that for a copula C,the x-flipped copula C1;0; y-flipped copula C0;1 and the survival copula C1;1 are given by formulas

C1;0ðx; yÞ ¼ y� Cð1� x; yÞ; C0;1ðx; yÞ ¼ x� Cðx;1� yÞ;C1;1ðx; yÞ ¼ xþ y� 1þ Cð1� x;1� yÞ;

see [19].

Proposition 1. Let ðc; dÞ 2 X. Then also ð�c; d� cÞ 2 X, and for each copula C : ½0;1�2 ! ½0;1� it holds

Kc;dC

� �0;1¼ K�c;d�c

C0;1; Kc;d

C

� �1;0¼ K�c;d�c

C1;0; Kc;d

C

� �1;1¼ Kc;d

C1;1:

Proof. The claims can be proved directly using the inequalities characterizing X, and definitions of the concerned copulas. h

Note, that the last equality in Proposition 1 says that all universal quadratic constructions commute with survivalconstruction.

3. Copulas invariant with respect to quadratic constructions

In this section, we are interested in finding all copulas C for which it holds

Kc;dC ¼ C; ð17Þ

74 A. Kolesárová et al. / Information Sciences 310 (2015) 69–76

where ðc; dÞ 2 X is a fixed pair of coefficients and Kc;dC is a copula given by (12), corresponding to the pair ðc; dÞ and the copula

C. In other words, we are looking for all copulas C, for which the equation

cC2ðx; yÞ þ dxy� cxCðx; yÞ � cyCðx; yÞ þ ð1þ c � dÞCðx; yÞ ¼ Cðx; yÞ

is valid for all ðx; yÞ 2 ½0;1�2. We will say that such copulas C are invariant with respect to the quadratic construction with coeffi-cients ðc; dÞ.

The previous equation is equivalent to the equation

cC2ðx; yÞ � ðcxþ cy� c þ dÞCðx; yÞ þ dxy ¼ 0: ð18Þ

Let c ¼ 0. Then (18) reduces to the equation

�dCðx; yÞ þ dxy ¼ 0;

which is satisfied for all C if d ¼ 0, and if d – 0, the only copula satisfying the equation, is the product copula C ¼ P.Let us now suppose that c – 0. Then (18) is equivalent to

C2ðx; yÞ � xþ y� 1þ dc

� �Cðx; yÞ þ d

cxy ¼ 0: ð19Þ

If we put dc ¼ h, the previous quadratic equation can be written in the form

C2ðx; yÞ � ðxþ y� 1þ hÞCðx; yÞ þ hxy ¼ 0: ð20Þ

As c 2 ½�1;0½[ �0;1�, d 2 ½0;1� and d P c, we can conclude that h 2� �1;0� [ ½1;1½. The discriminant of the Eq. (20) is

D ¼ ðxþ y� 1þ hÞ2 � 4hxy. We show that D is non-negative for all ðx; yÞ 2 ½0;1�2 and all considered values h.

If h 6 0, the claim is clearly true. Assume that h P 1. Let k ¼ xþ y. As for all ðx; yÞ 2 ½0;1�2; ffiffiffiffiffixyp

6xþy

2 , it also holds

4xy 6 k2, and we can estimate

DP ðk� 1þ hÞ2 � hk2 ¼ k2 þ 2ðh� 1Þkþ ðh� 1Þ2 � hk2

¼ ðh� 1Þðh� k2 þ 2k� 1Þ ¼ ðh� 1Þðh� ðk� 1Þ2Þ:

As 0 6 k 6 2, i.e., ðk� 1Þ2 6 1, and h P 1, we can conclude that DP 0 in this case, too.The requirement Cðx; yÞ 2 ½0;1� leads to the following solutions C ¼ Ch of Eq. (20):

Chðx; yÞ ¼12

xþ y� 1þ hþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxþ y� 1þ hÞ2 � 4hxy

q� �if h 6 0; ð21Þ

Chðx; yÞ ¼12

xþ y� 1þ h�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxþ y� 1þ hÞ2 � 4hxy

q� �if h P 1: ð22Þ

We have obtained two classes of symmetric functions, Chð Þh60, with members defined by (21), and ðChÞhP1, whose membersare defined by (22). To distinguish their members we will write Ch60 or ChP1, if necessary. A function Ch satisfies the Eq. (17)for all pairs ðc; dÞ whose ratio d

c is equal to h.

Theorem 2. Let ðc; dÞ be in X; c – 0. Then the only copula invariant with respect to the quadratic construction with coefficientsðc; dÞ is the copula Ch with h ¼ d

c, given by (21) if h 6 0, or by (22) if h P 1.

On the other hand, each copula Ch; h 2� �1;0� [ ½1;1½, is invariant with respect to the quadratic construction with anarbitrary pair of coefficients ðc; dÞ 2 X satisfying d

c ¼ h.

Proof. We only have to show that functions Ch; h 2� �1;0� [ ½1;1½, are copulas. Based on a statistical argumentation, in[21], a family of copulas, which is now known as the Plackett family, was introduced. For parameters a 2�0;1½, the members

CPla of this family are defined by (see [19])

CPla ðx; yÞ ¼

½1þða�1ÞðxþyÞ��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½1þða�1ÞðxþyÞ�2�4aða�1Þxyp

2ða�1Þ ; a P 0;a – 1;

Pðx; yÞ; a ¼ 1:

(

Substitution h ¼ aa�1 transforms the family Chð Þh2��1;0� [ ½1;1½ into the Plackett family CPl

a

� �a2�0;1½

. For the limit cases it holds

lima!0þ

CPla ¼W ¼ C0; lim

a!1CPl

a ¼ M ¼ C1:

It is not difficult to check that limh!�1Ch ¼ P ¼ CPl1 . h

Fig. 2. All pairs ðc; dÞ 2 X with dc ¼ h have the same invariant copula Ch .

A. Kolesárová et al. / Information Sciences 310 (2015) 69–76 75

Theorem 2 is illustrated in Fig. 2.Our method has shown a new interesting way how the Plackett family of copulas can be introduced. Moreover, we can

observe that each Plackett’s copula can be obtained as a limit of a sequence of copulas:

If for any copula C and any pair ðc; dÞ 2 X n fð0;0Þg, we introduce a family of copulas Cc;dn

� �n2N

, putting Cc;d1 ¼ Kc;d

C , and for

each n 2 N; Cc;dnþ1 ¼ Kc;d

Cc;dn

, then this family converges to the Placket copula Ch (in our parametrization), i.e.,

limn!1

Cc;dn ¼ Ch; where h ¼ d

c:

Clearly, for each h 2� �1;0�; Ch 6 P, and thus copulas Ch60 are negative quadrant dependent. Similarly, for eachh 2 ½1;1½; Ch P P, thus copulas ChP1 are positive quadrant dependent, see [19]. Even stronger results can be shown, seeExample 5.16 in [19]. Namely,

(i) Copulas Ch60 given by (21) have convex one-dimensional (horizontal and vertical) sections, i.e., they are ultramodular.(ii) Copulas ChP1 given by (22) have concave one-dimensional sections.

The above mentioned properties have an important statistical interpretation. Indeed, the ultramodularity [11] of a copulaC which links random variables X and Y, is equivalent to the stochastic decreasingness of the random variable Y wrt. X.Similarly, the concavity of one-dimensional horizontal and vertical sections of a copula C linking random variables X andY, is equivalent to the stochastic increasingness of the random variable Y wrt. X.

4. Conclusion

We have introduced and discussed quadratic constructions of binary copulas. We have shown that each such constructioncan be seen as a convex combination of four basic quadratic constructions, i.e., quadratic constructions assigning to the givencopula C, the copulas DC ; HC ; C and P. In addition to the complete characterization of universal quadratic constructions, wehave shown that the Plackett copulas are the only copulas invariant wrt. quadratic constructions related to their parameter h.We have also shown that each Plackett’s copula can be obtained as a limit of an iterated sequence of copulas. For a givenparameter h, we can start from any copula C, applying then consecutively the same quadratic construction with any coeffi-cients ðc; dÞ 2 X n fð0;0Þg, such that h ¼ d

c.Moreover, we have shown the relationship between universal quadratic and flipping constructions, as well as the fact that

the survival construction commutes with any of these quadratic constructions. The discussed quadratic constructions pre-serve several distinguished properties of original copulas as, e.g., symmetry or absolute continuity. While in some cases ourapproach brings constructions (the case of quadratic constructions for pairs ðc;0Þ and ðc;1Þ, see [3]), or copulas (Plackett’scopulas [19]) which are already known, our functional approach offers a new look at them and supports their betterunderstanding.

Acknowledgments

The first author acknowledges the support of the Grant VEGA 1/0419/13. The second author acknowledges the support ofthe Spanish Government Grant MTM2009-10962. The third author acknowledges the support of the Grant VEGA 1/0171/12

76 A. Kolesárová et al. / Information Sciences 310 (2015) 69–76

and the project of Science and Technology Assistance Agency under the Contract No. APVV-0496-10. Moreover, the work ofthe third author on this paper has been done in connection with the project IT4Innovations Centre of Excellence, Reg. No.CZ.1.05/1.1.00/02.0070 supported by Research and Development for Innovations Operational Programme financed byStructural Founds of Europe Union and from the means of state budget of the Czech Republic.

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