Some functionals for copulas

VOL. 14 NO. (1991) 45-54

SOME FUNCTIONALS

FOR COPULAS

C.ALSINA

Dcpartament

de Matemtiques Estadistica

ETSAB,

UniversitatPoli6:cnica de Catalunya 08028 Barcelona, Spain.

and

A. DAMAS and J.J. QUESADA

Departamento

deMatemticaAplicada Universidad de Granada

18071 Granada, Spain.

(Received November 14, 1989 and in revised form April 8, 1990)

ABSTRACT. In

this paper westudysome functionals operatingonthesetof then-copulasdefined

on

[0,1 ]n.

Conditionsunder which such functionals are welldefined are determined and some counterexamplesaredescribed.The study of thefixed points

(n-copulas)

for thesefunctionalsisalso considered, and, finally,someopenproblemsarepresented.

KEY WORDS AND PIIRASES.

N-increasing[unctions, connecting copula,n-copula. 1980

AMS SUBJEC CLASSIFICATION

CODE. 62H05.

1.

INTRODUCTION.

In

1959, Abe Sklar introduced the concept ofacopula.Since

then,

theory of copulas has been playingan importantrolein Mathematicsandinparticularin Mathematical Statistics and ProbabilityTheory.

Many

applicationsof thetheoryofcopulashave beenproposedinthese parts of Mathematics.

In

any case, we think that many of the advantages and possibilities of using copulasin

ProbabilityTheoryremainstilltobe developed.

Thetheory of copulashasbeen foundtobe of greatinterestwhen describing thedependence betweentwoormorerandomvariables,which is givenbytheir joint distribution function.

Severalinterestingpapersdealingwith this

problem

and othershave been written in the last fewyears.

We

shouldmentionSchweizerandSklar

[2

3

],

SchweizerandWolff

[4-

5

],

Genestand Mackay

[6],

and Wolff

7].

In

Schweizerand

Wolff[4-5]

severalnonparametric measures ofdependence forrandom variablesaredefinedbyusing the associatedcopula. Someproperties and interesting resultsare

presented. Wolff

[7]

hasmadeadetailedstudyof these measures ofdependence.

Genest

andMackay

[6]

haveconsidered some familiesofcopulasand have studied properties forthem. Recently,Alsina and

Quesada [8]

havestudiedthe relationships betweensome pairs of copulas.

In

thispaper, we studysomefunctionalsoperating onthesetofthe copulasdefined on

[0,1 ]n

and we determine the conditions under which these functionals are well defined. Some counterexamples describing the validityof these conditions are given.

We

also study the fixedpoints

In

section2we give a brief introductiontothetheoryof copulas. Section 3 isdevotedtothe studyofthe functionals defined on thesetofcopulas.

In

section4 we consider the case

H*

H

andin section5westudythe copulasthatarefixed pointsfor thesefunctionals.Section6contains someopen problemsand conclusions.

2.

COPULAS.

A

functionCfrom

[0,1]

n_into

[0.1

_{issaidtobe n-increasing if theC-volumeofanyn-box on}

[0,1]

isnon-negative,thatistosay, if

Vc(N

)_>0 for any n-box

N =[(x,y)]

C

[0,1]

with x

(x

Xn),

Y

(Yl,

Yn),

xi

-<

Yi, 1, 2 n, where theC-volume of the n-box

N

is

givenby

Vc(N)

Y’.

e(z

zn).

C (z

z

n)

>_ 0

(2.1)

where if z x foranevennumber ofi’s.

(z

Zn)

ifz x foranodd number ofi’s. andthe sumin

(2.1)

isextendedtoallvertices of

N.

Now,

let us consider the definitionof n-copuladueto

A.

Sklar in1959:

Definition 2.1.

A

n-dimensionalcopula or,briefly, an-copulais a function

C

from

[0,1

]n

into

[0,1

thatsatisfiesthe followingconditions:

(i)

C(1

,l,xi,

,l)=x

for every suchthat _<i_<nand every

xie[0,1].

(ii)

C(x

x

n)--0

if x

i=0

forsomeiwith l_<i_<n. (iii) C is n-increasing.

Some easy examplesofn-copulasarethefunctions

II, M

and

W

respectively defined by

H(xj,

x₂

x,)

x.x

₂

x,

M(x,

x₂ x

n)

Min

(x,

x_

x,)

W(x,

x2, x

n)

Max

(x+x2+...+xn-n+l,0)

and while

H

and

M

arealwaysn-copulas,

W

is acopula only whenn 2.

It

iseasy to show that copulasareuniformlycontinuousfunctionsandthattheyare increasingineach variable.Also,any copula C satisfies that

W(x,

X₂

Xn)

_

C(Xl,

_X2

Xn)

_

M(Xl,

x2,

Xn)

for every

(XI,

X₂

Xn)

e [0,1

In.

Thefundamental resultinthetheoryofcopulasgives us an idea of the importance of acopula, becauseit isshown thatacopulaisafunctionrelating the joint distribution function ofarandom

vectorwith its marginal distribution functions. Namely, the keyresultinthe theory of copulas

assertsasfollows:

THEOREM

2.2.

Let

X,

X

2

X

n be n random variables defined on a common probability spaceandlet

FI,

F

2

F

n and

G

be the marginaldistribution functionsand thejoint distribution function respectively. Then, thereexists an-copula C such that

G(t,

2

n)

C(F(t0, F2(t

2)

Fn(tn))

(2.2)

forevery

t,

2

R.

The copula Ciscalledaconnectingcopula of

X,

X

2

X

and, if

F

1,

F

F

are continuous, then C is unique. Other’ise,

C

is uniquely determined on

(Ran

FI)

x

(Ran

F

2)

x x

(Ran

Fn).

As

aconsequence of Theorem 2.2 we canobserve thattheconnectingcopulaof n random variables is afunctionthat containsall theinformationabout the dependence of the randomvariables. Thisisone thereasons for proposingmeasuresofdependence for randomvariables intermsofthe connectingcopula.

(See

Schweizerand

Wolff[4-5]

and

Wolff[7].

THEOREM 2.3.

Let

X

l,

X

X

be n random variables defined on a common

probabilityspaceand havingcontinuonsmarginaldistributions functions.If C istheconnecting

copula,then

(i) C

II

ifand only if

X

l,

X

X

areindependent.

(ii) C

M

ifandonlyifeachrandomvariable

X

i, l,2 n isan increasing

functionoftheotherones.

(iii)

For

n=2, C

=W

ifand only if each random variable

X.(i

1,2)

is a

decreasing function of theotherone.

Thecases (ii) and (iii) correspondwith total and positive, and total and negative dependence of therandomvariables,respectively.

Moreresults and applicationscanbefoundintheabovementioned works and alsointhe book

"ProbabilisticMetricSpaces" bySchweizerand Sklar

[3],

whereawholechapterisdevotedtothe theoryofcopulas.

Now,

ourinterestis tostudysome functionals operating on thesetofthen-copulas.These functionals havetobe well defined,that istosay, wehavetodetermine the conditions under which the functionals mapcopulasintocopulas.

3.

THE FUNCTIONALS.

Let

C be anyn-copula, andlet

P

be a fixedn-copula.

We

consider the functional defined for anyn-copula C as follows

C*(Xl,

x

Xn)

h.P(Xl,

x₂

Xn)

+

(l-h)

f(xl,

x₂ x

n)

(3.1)

+

C(al(Xl,

x

Xn),

an(X

x2, x

n)

for every

(Xl,

x₂ x

n)

[0,1In,

where f is afunction definedon

[0,1In,

_{al a2} a are

functions defined from

[0,1]

into

[0,11

and

h[0,1].

Firstofall,wewant tostudyconditions for the functionsf,al,

a;

a whichassure us

that

C*

isan-copulafor anyn-copula.

Later,

wewill consider the case

II*

H

andwewillstudy the fixed pointsfor thesefunctionals,thatis, then-copulas C such that C Co.

If h 1, then

C*

P

for any n-copula

C

andit is atrivial case.

From

nowon,wewill consider h

[0,1). In

thenextresult westudyconditions onf,a1,a2 a wich ensure that

C*

satisfies conditions (i) and (ii) of definiton 2.1foranyn-copula C.

LEMMA

3.1. Let

C*

beasdefinedin

(3.1). For

anyn-copula C the followingstatement is true:

(i)

C*(l

l,xi,

l)=x

for every suchthat lgi<n andevery

xi[0,1],

and

(ii)

C*(x

x

n)=0

if x

1=0

for some with lgi<n

if andonlyif

f(xl

Xn

Xl

Xn

C(al(X

xn)

tXn(Xl,

Xn))

for any

(x

Xn)

[0,1]n

such that

x...x

=xj for somejwith lgj<n.

PROOF.

Suppose

(xi

Xn)

[0,1]

such that some xi= 0 and

C*(x

Xn)

0.

Since

P

isa copula we have

f(xi

xn)=. C(al(X

xn),

an(X

Xn)

).

Similarly,

given(l,...,

1, xi, l)

[0,1]

n,

if

C*(l

l,xi,

l)=xl

wehave that

x h.x +

(l-h).

f(l,

1, xi, 1,

1)

+

+

C(0[l(l

1,x

1)

n(l

1, xi,

1))

In

theotherdirection, letussuppose that

fix

X

n)

X X

C(Gtl(X

X

n)

Gtn(X

Xn))

for any (x x

n)

E

[0,1

]n

such that x x _xj forsomej, _< _<n.

In

this case weget

C*(

x

x

n)

..P(x

x

n)

+

(1-.).

x

x

whichcompletestheproof.

Now,

wewant to find conditions on f and al,a2 a independently of the copula

C,

whicharenecessary and sufficientfor

C*

tosatisfy conditions(i)and

(ii)

of definition 2.1.

From

nowon,wewill denoteby theunitinterval

[0,1],

and n

[0,1]

n.

_Also,

J

_{willdenote the subset}

of givenby

J={ (x

Xn)

EIn

Xl...Xn=Xi for some i, l_<i_<n

First, let usprovethefollowingtechnicalresult:

PROPOSITION

3.2. If

l-l*(xl,

x x

n)

M*(x,

x_ x

n)

then for every

n

(x,

x x

n)

J,

thereissomej,

<

<

n, such that

II oti(x,

x2 x

n)

aj(Xl,

x2

Xn).

i=l

PROOF. If

II*(x,

x

x

n)

M*(x,

x2 x

n)

then, bydefinition,

II(a(x

x

n)

0tn(X

xn))

M(a(x

x

n)

an(X

Xn))

andtheresultisproved.

LEMMA

3.3.

Let

C*

beasdefined in

(3.1).

Thenthe followingstatement istrue:

For

everyn-copula

C,

(i)

C*(I

l,xi,

l)=x

foreveryisuchthat

<

i< nandevery

xi[0,l],

and

(ii)

C*(x

x

n)=0

ifx

i--0

for someiwith l<i<n ifand only if

(iii)

f(x

Xn)

X X

fl

(Xi(X

Xn)

for every

(X

X E

J,

i=l

and

(iv) for every

(xl

x

n)

J

thereissomej, <_ <_n,such that

[-I

ai(x

1, x

n)

i=l

%(xt

x.).

PROOF.

Let

C*

besatisfying(i) and (ii),inparticular for

C

II

and by using lemma 3.1 we get(iii). Also, byhypothesis

II*.

and

M*

are in the conditionsofproposition 3.2 and(iv)is

obtained.

In

theotherdirection,let us consider

(0,

x₂ x

n)

E

J,

thenby (iii).

C*(O,

x2, x

n)

(l-i).

I

oti(O,

x2 x

n)

+

i=l

C(Otl(0

x2, x

n)

an(0,

x2, xn

))

and by(iv)wehave

C*(0,

x₂ x

n)

0. The remaining cases can similarly beproved.

Lemma3.3statesnecessary and sufficient conditions for fand

a,

a2 a in order to

have

C*

satisfy

(i)

and(ii) ofthe which definitionof n-copula

C. It

onlyremainstostudyconditions guarantee

C*

isn-increasing. The theoremprovides thesolutionforthis

problem.

THEOREM

3.4.

Let

C*

be as defined in

(3.1).

If

f(x

xn)

is n-increasing, (iii) and

(iv)

onproposition3.3aresatisfied, andeither oneof thetwofollowingconditions hold:

(v)

Every

function depends solely ofone variable, whichisdifferent for each

.

All

ai’s

aremonotone but thenumber ofthem that are decreasing, iseven.

1)

C*

isan-copula for every n-copula C.

2)

If

C

<

C2, then

C*< C2".

PROOF. Letussuppose that

(v)

issatisfied.

By

applying lemma3.3,since(iii)and(iv) hold,wehave for

C*

the conditions(i)and(ii)of definition 2.1.Itonlyremains to prove that

C*

isn-increasing.

Let

N

[(a

an),(b

bn)]

bean-box,

N

C

n. We

havetoshow that

E

t

(Z

Zn)

C*(Z

Zn)

>

0

(3.2)

where the sumisextendedtoallverticesof

N.

Suppose that, by

(v),

Gt only dependsonx for 1,2 n.Then,bytakinginto account

that

P

is an-copulaandfisn-increasing,

(3.2)

isequivalenttoproving that

Y’-

E(Z

Zn). C((Gtl(Zl)

Otn(Zn)

>

0 (3.3) Let c

(c,

c c

n)

be givenby c

Gti(ai)

if ct isincreasingand c

oti(bi)

if ai is

decreasing, and d

(d,

d_E ti

n)

givenby d

ti()

if 0t isincreasing and d

cti(a

t)

if tx isdecreasing.Clearly

N

[c,d]

is an-boxon

[0,1]

andforeveryvertex

(z,

z z of

N,

wehave

(ZI,Z

Zn)=

(GtI(ZI),

GtE(Z2)

Ctn(Zn)).

Thus

(3.3)

isreducedtoproving

Y’-

E(GtI(ZI)

Ctn(Zn)

C(Otl(Zl)

Gtn(Zn)

_> 0

whichissatisfiedbecause C isan-copula. Therefore

C*

is an-copula.

Letussupposethat(vi)holds instead of

(v). Now,

C*

willbe n-increasing ifweprove that

.

(ZI,

Z2

Zn).

C(l(Z

Zn)

n(Z

Zn)

>_ 0

Suppose,

for

dxample,

that noneofthe

ai’s

dependson

x,

then

)-’.

(ZI

Z₂ Z

n) C(GtI(Z

Z

n)

Ot.n(Z

Zn))

E

E(al,

Z2

Zn).

C(t(a,

Zn)

17.n(a

Zn))

+

+ Y

(bl,

z₂

Zn)

C(l(b

Zn)

(xn(b

Zn))

0 because

6(al,

z₂

Zn)

(bl,

z₂ z

n)

and

cti(al,

z₂

Zn)

ai(bl,

z₂ z

n)

for

i-1,2 n.

It

isobviousthat if

C

_< C2, then

C*

_<

C2".

COUNTEREXAMPLE.

It

seemsreasonabletothinkthatconditions

(v)

and(vi)intheorem 3.4 could be too strong. This is not the case, and to illustrate this, we give the following counterexamplewhichshowsus that either one ofthose hypothesis are needed.

Let

C*(x,y)

be givenby

C*(x,

y)

X.H(x, y)

+

(1-.).

f(x, y)

+

C(0t(x,y), tx2(x,y))

and let us consider

tx(x,

y)

Min

y.(x+y),

y

},

t2(x,y

)--

x and

f(x,y)

0. Then f is

2-increasingand(iii)and(iv)aresatisfied, butneither

(v)

or(vi) hold. Also,weobservethat t

and x areincreasingand continuousfunctionsandthereforewe eliminatetheideathat continuity could_{change the conditionsof ourproblem. Ifweconsiderthe values}

x

3/4, x_{2 1, Yl} 1/8,

Y2

1/2 and C

M,

thenwe get

M*(3/4,

1/8)

+

M*(I,

1/2)-

M*(3/4,

1/2)-

M*(I,

1/8)

4.

THE CASE

II*

H.

Onceweknowthe conditions for

C*

beingan-copula,we areinterestedinthestudyof the particular cases in which then-copula

H

remainsfixed, thatistosay

H*

II.

Thetheoremprovides the solution for thisproblem:

THEOREM

4.1.

Let

C*

beasdefined in

(3.1).

The followingconditions are equivalent: (i)

H

*=

H.

X _Xn

.

P

(Xl

Xn)

(ii)

f(x

x

a)

I

0ti(x

x

a)

i"l

for every x x

n)

I".

(iii)

C*(x

x

n)

x

xn +

+

(1").

C(G[I(X

I

n)

Cln(X

Xn)

1

Cli(x

Xn)

i=l

forevery X _Xll

In.

PROOF.

(i) (ii). If

FI*--FI,

then

n

.P(x

x

n)

+

(1-.).

f(x

x

n)

+

i[I.l

(Gi(X

Xn)

X X andwe get

(ii).

(ii)

(iii).

By

substituting finthe

general

expression of

C*

wehave(iii).

(iii)

(i).

Ifwe put

C

II

in

(iii),

weget

II*

H.

We

observethat

C*

in (iii) canbeseen in the general form

(3.1)

where

P

II

and fis

givenby

f(x

x

n)

x x

lal

(xt

Xn)

for every

(Xl

x

n)

n.

We

willuse this

i=l

facttoprove thefollowingresult:

THEOREM

4.2.

Let

C*

begivenby (iii) intheorem 4.1. Ifthere exists one variable

x

(1

_< _<

n)

suchthat noneof thefunctions _{i depends}on

x,

and for every

(x

x)

J

thereis

some

k,

<

k _< n, such that

I

(xt,

...,x

n)

Ctk(X

Xn),

then

i=l

1)

C*

isan-copula for every n-copula

C.

2)

II*

II.

3)

For

everyn-copula

C,

C _>

II

C*

_>

H

and C _<

H

C*

_<

II.

PROOF. Since

C*

has thegeneral form

(3.1)

where

P

II

and

f(Xl

Xn)

Xl

Xn

(Xi(Xl

Xn)

andbyusing thesecond hypothesisofthe functions0q,we can apply lemma 3.3 andtherefore

C*

satisfiesconditions

(i)

and

(ii)

ofdefinition2.1.

In

ordertobe abletousetheorem 3.4 which assures usthat

C*

is a

n-copula,

itisenoughtoshow that

f(xi

x

n)

is n-increasing.

Let

N

[(a

an),(b

bn)]

be a n-boxin

In,

andsuppose,forexample,that none of the funtions

cq

dependson xi.Then,

E

(z,

z2

Zn).

1

ai(Z,

z2

Zn)

E

(a,

z₂

Zn).

fi

ff.i(al,

Z2 Z

n)

i-I

n

+ Y"

6(bl’

_z2

zn)

"i

q(bl’

z2

Zn)

0

SOME FUNCTIONALS FOR COPULAS 51

i, g <_n.Therefore,

E

((Z

Zn)

f(z

Zn)

Y’- ((Z

Zn)

Z Z

_

0

and

C*

is an-copula.

By

theorem 4.1,it isclear that

II*

I1,andtheproofof3)isobvious.

REMARK.

We

canobservethathypothesis

(v)

oftheorem3.4 doesnotworkintheorem 4.2.Letussuppose, forexample,that cx onlydependson

x

for 1, 2 n, and the number of functions thataredecreasingiseven.

In

thiscasewehave that

E

(z

Zn). C(c(z)

n(Zn))

>-

0,

andinordertoassurethat

C*

isn-increasing we havetoprove that fisn-increasing, thatis tosay, -Y’-

((Zl,

Z

Zn). (Xl(Zl). Gt2(Z2)...

0tn(Zn)

_) 0.

But,

"

E(zI,

z2

Zn)"

1

(Zl)"

t2(z2)

n(Zn)

E

e(bl,z2,

z).

0t

(bl). 2(Z2)

n(Z

n)

+

+

"

(bl,

z2

Zn)"

l

(al). 2(z2

(n(Zn)

Y"

(bl,

z2

Zn)"

_(l

(bl)" l(al)

OrE(Z2)

n(Zn)"

Repeating the sameprocesswecan obtain

Y’.

(Zl,

Z

Zn). Gtl(Zl) 2(Z2)

n(Zn)

fi

Gti(bi)

i(ai)

_

0

i=l

because,by

(v)

all the functions Ot aremonotone and the number of them that are decreasing is

even, andtherefore

f(x,

x₂ x

n)

isnotn-increasing.

5.

FIXED

COPULAS.

Now,

weareinterested in finding thesecopulasthatremainfixedfor the functionalswe are

studying. First,we consider

X

z

(0,

l) andthenwewill see innexttheorem that the answer for this problemistheexistenceanduniqueness ofan-copula C such that

Co*

Co.

THEOREM

5.1.

Let

C*

beasdefined in

(3.1),

and

Xe(0,1).

If f(x x

n)

P(x

Xn)-

P(tXl(X

Xn)

0tn(X

Xn)),

then

C*=C

C=P.

PROOF.

Let

II

C-PII

sup

C(x

x)-P(x

xn)

I.

Then

(x Xn)(In

II

c*-PII

I-

C((ll

O[n)-P(cx,,

Gin)

-<

(x n) n

_<

x

l-

sup

C(Xl

Xn)" P(xI’

xn)

(x n)(

whichconcludestheproof.

COROLLARY

5.2.

Let

C*

be as in theorem 5.1.Then,the followingis true:

(i)

c**=c

c=P

(ii)

If

C

o=C

and

Cn+

-Cn*,

for n=0,1, 2, then

(C

n}

P.

PROOF. From

theorem 5.1.

(i)

isunisspelled.

For

proving

(ii)

it is enough to see that

]ICR- Pl[ <_11-X

.

]lc- P[I

andthereforewhen n increasesthen-copula

Cn

isasclose

aswewishtothe n-copula

P.

COROLLARY6.3. Let

C*

beasdefinedin (3.1) with %

%o

e(0,1), and let C be any n-copula.The following conditions are equivalent:

(i)

Co*=

C_o.

(ii)

C*(Xl

Xn)

Co(X

Xn)

+

(1-o)"

C(a

a

n) Co(a

a

n)

].

(iii)

Co*

C and C istheonly n-copulathat is fixed for this functional. PROOF. (i) (ii). If

Co*--C

o, then

Co(x

x,)

Xo.P(x

x

n)

+(1-%

o)

.[

f(x

x

n)

+

+

o(O(x

x,,)

n(X

x))

C*(x

x

n)

x

InI

a (x x

n)

+

C(a

a

n)

i=l i=l

If

ai(a(x

x

n)

an(X

Xn)

x for 1, 2 n, then

C**

C for every n-copula C.

PROOF. This casecorrespondswith X 0 and

H*

H.Ifwecompute

C**

weobtain

C**(x

x

n)

x.

ai(x

x

n)

+

C*(a

a

n)

C(XI

Xn).

iffil

6.

CONCLUSIONS

AND REMARKS.

The results presentedinthis paper are interesting in their own right, and lead ustoseveral openproblems.

A

briefdescriptions ofsomeof themfollows.

First, let

F

l,

F

2

F

be themarginaldistribution functionsofarandom vector andlet

F

beitsjoint distribution function andletussuppose that Cisa connectingcopulaassociatedwith F. If

C*

isthen-copulathat we obtain when applying the functional definedby

(3.1)

to

C,

then the problemistofindthe relationships between

F

and

F*,

where

F*

isthejoint distribution function

of a random vector having

FI,

F2

F

as marginal distribution functions and

C’as

a

connectingcopula.

In

short,theproblemconsists in finding the relationships between the joint distribution functions with connectingn-copulas C and

C*.

Anotherproblemis togeneralize the functionalinthesensethat the one defined here could be seen as aparticular case.

We

observe that if we consider

C(x

x

n)

X.P(x

x

n)

+

(I-X).

f(x

Xn)-

C(a

a

n)

1,

Therefore,

Co(Xl

Xn)

o"

P(xI

Xn)

f(x x

n)

Co(a

a

n)

1- X andbysubstituting in the expressionof

C*

we obtain (ii).

(ii) (iii).

We

canwrite (ii) in the following way

C*(X

Xn)

)%o.Co(Xl

Xn)

+

(1-Xo).

C(a

a

n)

+

+

Co(X

x

n)

Co(a

a

By

applyingtheorem 5.1, where

P(x

x

n)

Co(Xl

x

n)

and

f(x X

n)

Co(X

Xn)- Co(al(Xl

X

n)

an(X

Xn))

weobtain (iii). Onthe otherhand (iii) (i) istrivial.

Now

wewillstudyconditionsunderwhich the functional will have the property that

C**

C for everyn-copula C.Theresultgives us conditions for a

,

a a in particular cases of

C*

wherethatpropertyholds.

thensimilar properties andresultsforthis functional canbeshown.

In

any case the problem is to

extendthesetwocasestoamoregeneral functional.

Itwouldalso be very interestingto study the copulasthat we obtainwhenapplyingthe functionaltosomeknowncopulas.

ACKNOWLEDGEMENT.

Theauthors wouldliketothankthereferee for

helpful

comments whichimproved the presentationofthepaper.

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