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Stochastic ordering of risks, influence of dependence and a.s. constructions

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Stochastic ordering of risks, in?uence of dependence and a.s. constructions
Ludger R¨ uschendorf University of Freiburg
Abstract In this paper we review and extend some key results on the stochastic ordering of risks and on bounding the in?uence of stochastic dependence on risk functionals. The ?rst part of the paper is concerned with a.s. constructions of random vectors and with di?usion kernel type comparisons which are of importance for various comparison results. In the second part we consider generalizations of the classical Fr? echet-bounds, in particular for the distribution of sums and maxima and for more general monotonic functionals of the risk vector. In the ?nal part we discuss three important orderings of risks which arise from ?-monotone, supermodular, and directionally convex functions. We give some new criteria for these orderings. For the basic results we also take care to give references to “original sources” of these results.

1

Introduction

It has been recognized in recent years that the methods and tools of stochastic ordering and construction of probabilities with given marginals are of essential relevance for the problem of modeling multivariate portfolios and bounding functions of dependent risks like the value at risk, the expected excess of loss and other ?nancial derivatives and risk measures Even if many results on stochastic ordering and dependent risks have been developed in early years, a new impetus on reconsidering this ?eld came recently from ?nancial modelling and risk management and a lot of papers in economics and insurance journals is devoted to this subject (see e.g. the recent article of Embrechts, H¨ oing and Juri (2003) and the references therein). Stochastic ordering and marginal modelling has a long history and several books and conference proceedings to this subject have appeared. To mention are in particular proceedings of conferences on marginal modelling and stochastic ordering ([11], [41], [71], [3], [9]) as well as the comprehensive volumes of Stoyan (1977), Marshall and Olkin (1979), Tong (1980), Mosler (1982), Shaked and Shantikumar (1994), Joe (1997), Nelsen (1999), and M¨ uller and Stoyan (2002). The main purpose of this paper is to point out and partially extend some of the orderings and results on orderings which seem to be of particular importance for bounding risks and the in?uence of dependence on functionals. For several of the key results we also want to give some of the early and original references. The ?eld of stochastic orders is very diversi?ed. But some of the recent work and results has already been stated and established in early papers on stochastic ordering.

2

L. R¨ uschendorf

We essentially restrict to “integral orders ?F ” on the probability measures induced by some function class F and de?ned by P ?F Q if f dP ≤ f dQ for all f ∈ F, (1.1)

such that f is integrable w.r.t. P and Q. Some natural questions for the analysis of a stochastic order ? are to ?nd simple and maximal generators F of ? such that ? and ?F are equivalent orderings (or to ?nd at least large classes F such that P ? Q implies P ?F Q). This aspect is discussed in most of the books mentioned above. Additional particular references to the subject of integral stochastic orders are R¨ uschendorf (1979), Reuter and Riedrich (1981), Mosler and Scarsini (1991b) Marshall (1991), M¨ uller (1997), and Denuit and M¨ uller (2001). The plan of this paper is to discuss at ?rst a.s. construction of random vectors which lie at the core of several ordering results. Related are kernel representation results which give ‘pointwise’ characterization of stochastic orders by di?usion kernels. Each ordering generates a notion of positive resp. negative dependence by comparing a probability measure P ∈ M (P1 , . . . , Pn ) (1.2)

– the class of all probability measures with marginals P1 , . . . , Pn – to the product n ?n i=1 Pi of its marginals. If ?i=1 Pi ? P , then we speak of positive dependence of n P , if P ? ?i=1 Pi , then of negative dependence. Related is the problem to describe the maximal in?uence of dependence on a function f (or class of functionals F ), M (f ) = sup resp. m(f ) = inf f dP ; P ∈ M (P1 , . . . , Pn ) f dP ; P ∈ M (P1 , . . . , Pn ) (1.3)

which we call the problem of (generalized) Fr? echet-bounds. One of the most prominent results in stochastic ordering are the classical Fr? echet-bounds due to Hoe?ding (1940) and Fr? echet (1951), a) For a n-dimensional df F holds: F ∈ F (F1 , . . . , Fn ) – the Fr? echet class of n-dimensional df ’s with marginals F1 , . . . , Fn – if and only if F? ≤ F ≤ F+ , where F+ (x) := min1≤i≤n {Fi (xi )} and F? (x) ? (n ? 1)} are the upper and lower Fr? echet-bound. := max{0, (1.4)
n i=1 Fi (xi )

b) Moreover, F+ ∈ Fn is a n-dimensional df , while F? ∈ Fn if and only if n = 2 or
n

for n > 2 either
i=1 n

Fi (xi ) ≤ 1 for all x with Fj (xj ) < 1, ? j (1.5) Fi (xi ) ≥ n ? 1
i=1

or

for all x with Fj (xj ) > 0, ? j.

Ordering of Risks

3

The important characterization in b) of the cases where F? ∈ Fn is due to Dall’Aglio (1972). For a review on various aspects of Fr? echet-bounds see R¨ uschendorf (1991b) (in the following abbreviated by Ru (1991b)). The most important general technique to determine generalized Fr? echet-bounds is duality theory. A comprehensive survey of this theory and its applications is given in Rachev and Ru (1998, Vol. I/II). We will describe some interesting aspects of the problem of Fr? echet-bounds on the in?uence of dependence in chapter 3. For the application to the comparison of risks it has turned out (see the interesting recent book of M¨ uller and Stoyan (2002)) that of particular importance are the classes of supermodular (quasi-monotone), directionally convex and ?monotone functions F sm , F dcx , F ? together with the induced orderings and some variants like F idcx , the increasing directionally convex functions. In section four we describe and extend some of the basic comparison criteria for these functions. We use some standard notation throughout. X ? P means that the random variable X has distribution P . We write for some ordering ?, X ? Y synonymously for P ? Q or F ? G, where F , G are the df s of X , Y and P , Q are the distributions. ≤st denotes the usual stochastic order w.r.t. nondecreasing functions.

2

Stochastic Ordering and A.S. Construction of Random Variables

For the comparison of distributions P , Q w.r.t. some stochastic order it is in some cases useful to compare explicit a.s. constructions of rv s X , Y where X ? P and Y ? Q. A general useful construction for P, Q ∈ M 1 (IRn ), the set of probability measures on IRn is the following ‘standard construction’ : Let F ∈ Fn be a n-dimensional df and let V1 , . . . , Vn be independent rv s uniformly distributed on [0, 1], independent of X ? F . Let V = (V1 , . . . , Vn ) and let Fi|1,...,i?1 (xi |x1 , . . . , xi?1 ) denote the conditional df s of Xi given Xj = xj , j ≤ i ? 1. We de?ne τF : IRn × [0, 1]n → IRn by τF (x, λ) = (F1 (x1 , λ1 ), F2|1 (x2 , λ2 |x1 ), . . . , Fn|1,...,n?1 (xn , λn |x1 , . . . , xn?1 )) where Fi|1,...,i?1 (xi |x1 , . . . , xi?1 ) = P (Xi < xi |Xj = xj , j ≤ i ? 1) + λi P (Xi = ?1 xi |Xj = xj , j ≤ i ? 1). We de?ne the ‘inverse’ transformation τF recursivly
?1 τF (u) = z = (z1 , . . . , zn ),

(2.1)

(2.2)

?1 Fn |1,...,n?1 (un |z1 , . . . , zn?1 ).

?1 ?1 (u1 ), z2 = inf {y : F2|1 (y |z1 ) ≥ u2 } = F2 with z1 = F1 |1 (u2 |z1 ), . . ., zn =

Theorem 2.1 (Regression construction) Let X be a n-dimensional random vector with df F , then: a) U := τF (X, V ) has independent components, uniformly distributed on [0, 1].
?1 b) Z = τF (V ) is a rv with df F ; Z is called the “regression construction” of F .

4 c)

L. R¨ uschendorf

?1 X = τF (τF (X, U ))

a.s.

(2.3)

Remark 2.2 1) Part a) of (2.1) is due in the case of absolutely continuous conditional df ’s to Rosenblatt (1952). a) and b) were stated in this form in Ru (1981b). b) had been given before in an equivalent form in O’Brien (1975), while c) is from Ru and de Valk (1993). The one dimensional case was used since long time for the simulation of rv ’s. 2) By the recursive de?nition in (2.2) one obtains Z also as a function of (V1 , . . . , Vn ) ? which we denote by τF
? Z = τF (V ) ? F

(2.4)

? (V ) = (h (V ), h (V , V ), . . . , h (V , . . . , V )). In this functional form where τF 1 1 2 1 2 n 1 n the construction is called “standard representation” of F . It gives a construction of a random vector with df F as function of independent uniforms. The functions hi represent conditional df ’s.

3) A “copula” of X (resp. F ) is any df C with uniform marginals such that C (F1 , . . . , Fn ) = F (2.5)

where Fi are the marginal df ’s. If U is a random vector with U ? C then
?1 ?1 (F1 (U1 ), . . . , Fn (Un )) ? F.

(2.6)

U represents some aspects of the dependence structure of F (resp.X ). To obtain a copula one can apply Theorem 2.1 in the one-dimensional case and consider U := (τF1 (X1 , V1 ), . . . , τFn (Xn , Vn )). Then the df C of U is a copula and X = (Fi?1 (U i )) a.s. We next give some applications of the standard resp. regression construction. Corollary 2.3 (Stochastic ordering) If F, G ∈ Fn and V = (V1 , . . . , Vn ) is an iid uniform sequence. Then
?1 ?1 τF (U ) ≤ τG (V ) implies F ≤st G,

(2.7)

where ≤st denotes the usual stochastic ordering w.r.t. F m the class of monotonically nondeceasing functions. Remark 2.4 Condition (2.7) is stated in Ru (1981b). It implies various su?cient conditions for stochastic ordering going back to classical results of Veinott (1965), Kalmykov (1962) and Stoyan (1972) in the context of markov chains. The regression and standard construction is used essentially in various papers on stochastic ordering. The positive dependence ordering ‘conditional increasing in ? are monotonically nondecreassequence CIS’ just says that the components hi of τF ing. This is used essentially in many papers e.g. in M¨ uller and Scarsini (2001) to state su?cient conditions for the supermodular ordering of positive dependent

Ordering of Risks

5

sequences (see also section 4). An application of the standard construction to convex ordering analogously to (2.7) is given in Shaked and Shantikumar (1994). An alternative application to positive regression dependence ordering of rank statistics as well as to further statistical ordering results is given in Ru (1986). As second application we consider the following optimal coupling problem: Determine for some P, Q ∈ M 1 (IRn ) and with Sn (X ) = n i=1 Xi : inf E |Sn (X ) ? Sn (Y )|2 : X ? P, Y ? Q , (2.8) i.e. the problem is to construct two n-dimensional random vectors X , Y with n distributions P , Q such that the sums Sn (X ) = n i=1 Xi , Sn (Y ) = i=1 Yi are as 2 close as possible in L -distance. The answer to this problem is : Corollary 2.5 (Optimal coupling of sums, Ru (1986)) For P, Q ∈ M 1 (IRn ) let P1 , P2 denote the distributions of the sums resp., then
? ? ?
n n 2

Xi ,

Yi

? ? ?

inf where

E
i=1

Xi ?
i=1

Yi

: Y ? P, Y ? Q

=

2 2 (P1 , P2 )

(2.9)
2 -metric.

2 (P , P ) 2 2 1

=

1 ?1 0 (F1 (u)

?1 ? F2 (u))2 du is the squared minimal

For the proof one applies the regression construction to the extended random n vectors ( n i=1 Xi , X ) resp. ( i=1 Yi , Y ) and obtains directly (2.9). Of course similar results hold for the coupling of other functionals like n ( n i=1 Xi , maxi≤n Xi ) simultaneously to ( i=1 Yi , maxi≤n Yi ) etc. The standard construction does not in general give a pointwise a.s. construction of random vectors X ? P , Y ? Q such that X ≤ Y a.s. if P ≤st Q. But Strassen’s comparison theorem implies the existence of such a.s. representations. This result was extended to closed partial order’s ? on a polish space S . The order ? on S induces the stochastic order ?st on M 1 (S ) the set of probability measures on S for the corresponding class F m of monotonically nondecreasing functions w.r.t. ?. Theorem 2.6 (Strassen’s theorem) Let ? be a closed partial order on a polish space S and P , Q probability measures on S. Then: P ?st Q if and only if there exist rv ’s X ? P , Y ? Q (on some space (?, A, R)) such that X?Y a.s. (2.10)

Remark 2.7 (2.10) was introduced in Strassen (1965) and extended in various ways in Kamae, Krengel and O’Brien (1978), Kellerer (1984) and Ramachandran and Ru (1995). A proof by means of Strassen’s abstract set representation theorem is given in Ru (1980b, Theorem 1) in the case of the Schur-ordering ?S on IRn de?ned by:
k k

a ?S b

if
i=1 n

a(i) ≤
i=1 n

b(i) , bi ,
i=1

i≤k ≤n?1 (2.11)

and
i=1

ai

=

6

L. R¨ uschendorf

where a(1) ≥ . . . ≥ a(n) are the conponents arranged in decreasing order. The monotone functions w.r.t. ?S are the Schur-convex functions and (2.10) implies: Corollary 2.8 (Schur-convex ordering) Let P, Q ∈ M 1 (IRn ), then: P ?S Q ?? ?? There exist X ? P, Y ? Q with X ?S Y There exist X ? P and a random doubly stochastic matrix Π, such that ΠX ? Q A more general comparison result for integral stochastic orders ?F is based on ?F -di?usions. De?nition 2.9 (?F -di?usions) Let F be a class of functions on some space (E, A). A markov kernel K on E is called a F -di?usion if εx ?F K (x, .) for all x ∈ E. (2.13) (2.12)

A ?F -di?usion kernel K ‘di?uses’ locally in any point x mass w.r.t. ?F . The composition KP is de?ned by KP (A) = K (x, A) P (dx). Proposition 2.10 Let K be a ?F -di?usion then P ?F KP for all P ∈ M 1 (E ). (2.14)

Proof: The proof is obvious from the idea of di?usions. For f ∈ F holds f dKP = ≥ f (y ) K (x, dy ) f (x) d(P (x) dP (x)

2 From a general kernel representation result in Strassen (1965) one can obtain a converse of (2.14) and characterize several stochastic integral orders by corresponding F -di?usions.A result of this type was stated in Ru (1980b) for several examples including the stochastic order, the convex and convex increasing order which were well established before and are related to famous results of Blackwell, Stein, Sherman, Cartier, Meyer and Strassen. It also included the class of symmetric convex functions and the class of norm increasing functions. The list of examples was further extended in Mosler and Scarsini (1991b). The proof in Ru (1980b) uses an idea from the theory of balagage (Meyer (1966, Theorem 53)) and Strassen’s kernel representation theorem (Strassen (1965, Theorem 3)). For a general formulation of this result we de?ne hf (x) = sup f dP ; εx ?F P (2.15)

for f ∈ Cb (E ) and assume that F ? Cb (E ) is some order generating class. Let F ? denote the convex maximal generator of the order dual to ?F such that P ?F Q P,Q is equivalent to Q ?F ? P . Let F ? be the set of all pointwise limits of sequences in F ? which are uniformly integrable w.r.t. P , Q.

Ordering of Risks

7

Theorem 2.11 (F -di?usions) Let E be a polish space, F ? Cb (E ) with dual P,Q convex cone F ? and P, Q ∈ M 1 (E ). Assume that for f ∈ Cb (E ), hf ∈ F ? . Then P ?F Q if and only if there exists a F -di?usion K such that Q = KP. (2.16)

Proof: De?ne Πx := {P ∈ M 1 (E ) : εx ?F P }. Then Πx is convex, weakly closed and hf (x) = sup{ f dP ; P ∈ Πx }. For f ∈ Cb (E ) holds f (x) ≤ hf (x) and thus P,Q since hf ∈ F ? f dQ ≤ hf dQ ≤ hf dP (2.17)

This implies by Strassens kernel representation theorem (Strassen (1965, Theorem 3)) the existence of a kernel K on E with Q = KP and K (x, .) ∈ Πx for all x ∈ E , i.e. K is a F -di?usion. 2

Remark 2.12 a) The ?rst general formulation of the di?usion characterization theorem (2.16) is due to Meyer (1966, Theorem 53). It is formulated in the context of integral stochastic orders ?F in M¨ uller and Stoyan (2002, Theorem 2.6.1): Suppose that for any f, g ∈ RF – the maximal generator of ?F – holds max(f, g ) ∈ RF then the equivalence in (2.16) holds true. b) We consider the following examples of applications of (2.16) 1) If F = F sym,cx is the set of symmetric convex functions on IRn . Then hf is symmetric and concave; so it lies in the closure of the dual cone F ? . From (2.16) we obtain: P ?sym,cx Q if and only if ? X ? P, Y ? Q such that X ?S E (Y( ) |X ), (2.19) (2.18)

where ?S is the Schur order, Y( ) is the ordered vector (see Ru (1981)). It is interesting that there is a di?erence to the condition for stochastic Schur-ordering in (2.12) 2) If F || || is the class of norm increasing then εx ?F P i? P has support in {y : holds hf (x) = sup { f dP : εx ?F || || P,Q implies hf (y ) ≤ hf (x). So hf ∈ F ? P ?F ||
||

functions f (x) = g (||x||) in Cb (IRn ), ||y || ≥ ||x||}. Further for any f ∈ Cb P } is norm decreasing, ||x|| ≤ ||y || and we obtain: (2.20)

Q i? there exist X ? P, Y ? Q

such that ||X || ≤ ||Y || a.s. (see Ru (1980b).

8

L. R¨ uschendorf

3

Fr? echet-Bounds – Extremal Risk

As mentioned in the introduction Fr? echet-bounds deal with the basic problem in risk theory to describe the maximal in?uence of stochastic dependence on the expectation of a functional ?(x1 , . . . , xn ). Examples of interest are e.g. convex functionals of the joint position x1 + · · · + xn , where xi are risks with distributions + Pi . A typical case is ?(x) = ( n i=1 xi ? k ) , the excess of loss function. A nonconvex function of interest is ?t (x) = 1[t,∞) ( n i=1 xi ) which yields just the inverse of the value at risk functional. Of interest is also the maximal risk of the components maxi≤n xi and variants hereof. For a detailed introduction to this kind of questions we refer to Embrechts et al. (2003). An extension of the classical Fr? echet-bounds in (1.4) is the following result which in particular implies sharpness of the classical Fr? echet-bounds. Theorem 3.1 (Sharpness of Fr? echet-bounds, Ru (1981a)) Let (Ei , Ai ) be polish spaces, Pi ∈ M 1 (Ei , Ai ) and Ai ∈ Ai , 1 ≤ i ≤ n, then for any P ∈ M (P1 , . . . , Pn ) holds
n

Pi (Ai ) ? (n ? 1)
i=1 +

≤ P (A1 × · · · × An ) ≤ min {Pi (Ai )}

(3.1)

and the upper and lower bounds in (3.1) are attained. As consequence we get sharp bounds for the in?uence of dependence. As a ?rst example we consider the maximal risk of the components. Let X = (X1 , . . . , Xn ) be a random vector, with Xi ? Pi being real rv ’s with df ’s Fi . Then with Ai = (?∞, t] (3.1) implies sharp bounds for the maxima Mn = maxi≤i≤n Xi . Corollary 3.2 (Maximally dependent rv ’s) H ? (t) =
n

Fi (t) ? (n ? 1)
i=1 +

≤ P max Xi ≤ t
i≤n

(3.2)



1≤i≤n

min Fi (t) = H + (t)

Remark 3.3 a) Corollary 3.2 is due for F1 = · · · = Fn to Lai and Robbins (1976). The general case is from Lai and Robbins (1978) and by di?erent methods from Meilijson and Nadas (1979), Tchen (1980) and Ru (1980a). Also a random vector X = (X1 , . . . , Xn ) is constructed with Xi ? Fi and Mn (X ) = max Xi ? H ? .
i≤n

(3.3)

X yields the lower bound in (3.2). It is called maximally dependent random vector in Lai and Robbins (1976). The upper bound H + (t) is attained by the ?1 ?1 (U )), where U is uniform on [0, 1]. (U ), . . . , Fn comonotonic vector X ? = (F1 In stochastic ordering terms (3.2), is equivalent to Mn (X ? ) ≤st Mn (X ) = max Xi ≤st Mn (X )
i≤n

(3.4)

Strongly positive dependent rv ’s have in stochastic order small maxima i.e. they have small maximal risks of the components.

Ordering of Risks

9

b) The most simple way to explain the upper bound in (3.4) is the following argument due to Lai and Robbins (1976). In fact this is a typical argument for the duality approach to problems of this kind (see Rachev and Ru (1998)). Note that for any real v ∈ IR :
n

Mn (X ) = max Xi ≤ v +
i≤n i=1

(Xi ? v )+

(3.5)

Equality holds in (3.5) i? for some ‘splitting point’ v ? the sets {Xi ≥ v ? } are ? pairwise disjoint and n i=1 {Xi ≥ v } = ?. The maximally dependent random vector X is constructed such that there is a splitting point v ? as above. In the case F1 = · · · = Fn , Lai and Robbins (1976) proved the extremely interesting result that the maximally dependent case is close to the independent case in the following asymptotic sense under the usual domain of attraction conditions for maxima (? denote here asymptotic equivalence). EMn X ⊥ ? EMn X ? F ?1 1 ? 1 , n (3.6)

1 ) is the usual where X ⊥ is an iid sequence with df F and an = F ?1 (1 ? n normalization for the maximum law.

There are many alternative applications of (3.1) e.g. to get sharp bounds for the concentration probabilities or to get sharp multivariate Fr? echet-bounds (see Ru (2003)). Corollary 3.4 a) Maximal concentration.
n

(Fi (bi ) ? Fi (ai )) ? (n ? 1)
i=1 +

≤ P (Xi ∈ [ai , bi ], 1 ≤ i ≤ n) ≤
i≤i≤n

min (Fi (bi ) ? Fi (ai ))

(3.7)

The bounds in (3.7) are sharp. b) (Sharp) multivariate Fr? echet-bounds. If Xi are ki -dimensional random vectors with df ’s Fi , 1 ≤ i ≤ n and F is the df of X = (X1 , . . . , Xn ) then for any xi ∈ IRki , 1 ≤ i ≤ n:
n

Fi (xi ) ? (n ? 1)
i=1 +

≤ F (x1 , . . . , xn ) ≤ min (Fi (xi ))
1≤i≤n

(3.8)

and the multivariate Fr? echet-bounds are sharp. Of particular interest in risk theory is the distribution and risk of the combined portfolio given by the sum Sn (X ) = n i=1 Xi . The following basic ordering result for the ordering of sums has been stated ?rst in Meilijson and Nadas (1979) for the convex increasing order and in Ru (1983) for the convex order.

10

L. R¨ uschendorf

Theorem 3.5 (Maximal sums w.r.t. convex order) Let X be a random vector with marginal df ’s F1 , . . . , Fn , then: a) Convex increasing order
n

E
i=1

Xi ? t
+


n

ψ+ (t) :=

v =(v1 ,...,vn )

inf

vi ? t

+

+
i=1

E (Xi ? vi )+

(3.9)

The bound in (3.9) is sharp. b) Convex order
n n

Xi ?cx
i=1 i=1

Fi?1 (U ) ? t)+ = ψ+ (t)

(3.10)

and E (

?1 n i=1 Fi (U )

Remark 3.6 1) The proof of a) was given by Meilijson and Nadas (1979) by a duality argument similar to that in (3.5). Also a construction of a rv attaining the upper bound is given there (for an even more general situation). The result of Meilijson and Nadas (1979) describes a sharp upper bound for the ordering ?icx w.r.t. increasing convex functions which is also called stopp-loss ordering, ?sl (in particular in the economics and insurance literature). That the comonotone case yields the maximum w.r.t. the convex order in (3.10) was stated in Ru (1983) as consequence of a more general result for supermodular functions and based on the rearrangement method which in the discrete case goes back to inequalities of Lorentz (1953). Implicitly this result is also contained in the ‘Lorentz Theorem’ of Tchen (1980, Theorem 5), observing that for ? convex, ?(x1 + · · · + xn ) is quasimonotone (in Tchens terminology) or supermodular in the now more common terminology. This convex ordering result for sums of random variables and also the simple duality proof have been detected and rederived again several times in the literature. 2) The sharpness of the bound in (3.9) resp. in (3.10) also implies that
n

E
i=1

Fi?1 (U ) ? t
+

= ψ+ (t)
n n

= inf
i=1

E (Xi ? vi )+ ;
i=1

vi = t .

(3.11)

If X ? , v ? attain the upper bound in (3.9) then
n i=1

Xi? ? t
+

n

=
i=1

? (Xi? ? vi )+

a.s.

(3.12)

Ordering of Risks

11

This equality has a simple geometric meaning and is ful?lled only for the comontonic vector (Fi?1 (U ))1≤ i≤n (see Meilijson and Nadas (1979) or the recent paper by Kaas, Dhaene, Vyncke, Goovaerts and Denuit (2001)) 3) Meilijson and Nadas (1979) in fact gave sharp bounds for more general functionals. Let Ij ? {1, . . . , n}, 1 ≤ j ≤ k be subsets with k j =1 Ij = {1, . . . , n} and consider M = max1≤i≤k j ∈Ii Xj . Then for all x: E (M ? x)+ ≤
?? ? ψ+ (x) := inf ? max v ? 1≤i≤k ? ? ?

n

vj ? x? +
j ∈Ii + i=1

E (Xi ? vi )+

?

(3.13)

and the upper bound in (3.13) is pointwise sharp. Furthermore for cyclic directed networks the bound is attained stochastically for some P ∈ M (P1 , . . . , Pn ). 4) Comonotonic vectors, multivariate marginals. While for one dimensional marginals comonotonic vectors maximize the risk for many convex functionals (like in (3.9), (3.10)) for ?( n i=1 xi ), ? convex) this is no longer the case for multivariate marginals, where they can even minimize the risk. For some illustrative examples see Ru (2003). The reason for this is the possible negative dependence in the components of the marginals. In the recent paper of Denuit, Dhaene and Ribas (2001) the following interesting result related to (3.9), (3.10) was proved by a simple induction argument: Theorem 3.7 (Positive dependence increases risk) If X is associated then
n i=1

Xi⊥

n

≤sl
i=1

Xi .

(3.14)

⊥ , . . . , X ⊥ ) has independent components and X ⊥ ? X . Here X ⊥ = (X1 i n i

So positive dependence (association) leads to riskier portfolios. In Christo?des and Vaggelatou (2004) and Ru (2003) a general version of this result has been given stating that positive dependence leads to higher risk for a general class of proper risk functions f (X1 , . . . , Xn ). Of particular interest in risk theory is to describe the in?uence of dependence on the value at risk functional of the combined portfolio VaRα (X1 + · · · + Xn ) which is de?ned as the α-quantile of the combined portfolio X1 + · · · + Xn . For the description of the maximal in?uence the following functionals are of interest: Given n df ’s F1 , . . . , Fn consider:
n

Mn (t) = sup P
i=1 n

Xi ≤ t ; Xi ? Fi , 1 ≤ i ≤ n Xi < t ; Xi ? Fi , 1 ≤ i ≤ n
i=1

mn (t) = inf P

(3.15)

12 Then 1 ? mn (t) = sup{P ( Xi ≥ t); Xi ? Fi , 1 ≤ i ≤ n}

L. R¨ uschendorf

(3.16)

and one obtains the sharp upper bound VaRα (X1 + · · · + Xn ) ≤ (1 ? mn )?1 (α). (3.17)

For the case n = 2 the following bounds were ?rst established in Sklar (1973) and in more general form in Moynihan, Schweizer and Sklar (1978). The bounds and also their sharpness were independently established in Makarov (1981) and Ru (1982) (For the history of this result see also Schweizer (1991)). Theorem 3.8 (Maximal sum risks, n=2, Makarov (1981), Ru (1982), Sklar (1973)) Let X be a random vector with marginal df ’s F1 , . . . , Fn , then for n = 2 holds: P (X1 + X2 ≤ t) ≤ M2 (t) = F1 ∧ F2 (t) P (X1 + X2 < t) ≥ m2 (t) = F1 ∨ F2 (t) ? 1, (3.18)

where F1 ∧ F2 (t) = inf x (F1 (x? ) + F2 (t ? x)) is the in?mal convolution function and F1 ∨ F2 (t) = supx (F1 (x? ) + F2 (t ? x)) is the supremal convolution function. (3.16) is derived in Ru (1982) as consequence of the following general representation of the upper Fr? echet-bounds for ? = 1A , P1 , P2 ∈ M 1 (IRn ) and A ? IR2n closed: M (A) = sup {P (A); P ∈ M (P1 , P2 )}
n n

(3.19)

= 1 ? sup {P2 (O) ? P1 (π1 (A ∩ (IR × O))); O ? IR open} , where π1 is the projection on the ?rst component. (3.19) is a consequence of Strassen (1965, Theorem 11) (see Ru (1982,1986)). The type of bounds in (3.18) extends easily to n ≥ 3 (see Frank, Nelsen and Schweizer (1987) and Denuit, Genest and Marceau (1999)). Proposition 3.9 Let X be a random vector with marginal df ’s F1 , . . . , Fn . Then for any t ∈ IR1 holds:
n n n

Fi (t) ? (n ? 1)
i=1 +

≤P
i=1

Xi ≤ t

≤ min
i=1

Fi (t), 1

(3.20)

n

n

n

where
i=1 n

Fi (t) := inf
i=1 n

Fi (ui );
i=1 n

ui = t ui = t
i=1

and
i=1

Fi (t) := sup
i=1

Fi (ui );

Ordering of Risks

13

Proof: The proof of (3.20) follows by induction from the case n = 2; alternatively by the following simple argument. For any u1 , . . . , un with n i=1 ui = t holds:
n n

P
i=1

Xi ≤ t

≤ P
i=1 n

{Xi ≤ ui } Fi (ui ),

(3.21)


i=1

which gives the upper bound. Similarly, using the Fr? echet lower bound in (1.4) we obtain
n

P
i=1

Xi ≤ t

≥ P (X1 ≤ u1 , . . . , Xn ≤ un )
n

(3.22)


i=1

Fi (ui ) ? (n ? 1)
+

. 2

Remark 3.10 The bounds in (3.17) are however in contrast to the case n = 2 not sharp. If n = 3, F1 = F2 = F3 are the df of the uniform distribution on [0, 1] then M3 (t) = 1
2 3t

, 0≤t≤ , t> 3 2

3 2

,

m3 (t) =

1

2 3t

?1 , 0≤t≤3 , t≥3

(3.23)

(see Ru (1982)). The bounds in (3.20),(3.21) are in this case more crude.
3 3

min 1,
i=1

Fi (t) = min (1, t) and
i=1

Fi (t) ? 2
+

= (t ? 2)+

(3.24)

For some examples sharp bounds for n ≥ 3 have been given in Ru (1982) and Rachev and Ru (1998). The simple method of bounding the risk probability in (3.20) has been given and extended in Frank, Nelsen and Schweizer (1987) to general monotonically nondecreasing functions ψ (x1 , . . . , xn ). The resulting bounds are of interest and markable relevance if further information on the underlying df ’s can be used. The following is essentially a reformulation of corresponding results in Moynihan, Schweizer and Sklar (1978), Frank, Nelsen and Schweizer (1987), Denuit, Genest and Marceau (1999) and Embrechts H¨ oing and Juri (2003). For a df H let H denote the corresponding multivariate survival function H (x) = PH ([x, ∞)). For n t ∈ IR let A+ ψ (t) := {u = (u1 , . . . , un ) : u a maximal point in IR with ψ (u) ≤ t} Theorem 3.11 (Bounds for monotonic functionals) Let X = (X1 , . . . , Xn ) be a random vector with df F ∈ F (F1 , . . . , Fn ) and let ψ (x) be monotonically nondecreasing and lower semicontinuous. Then

14 a) General bounds.
? ? sup
u∈A+ (t) ψ i=1 n

L. R¨ uschendorf

?

Fi (ui ) ? (n ? 1)?
+

≤ P (ψ (X ) ≤ t)
n

(3.25)



u∈A+ (t) i=1 ψ

inf

Fi (ui )

b) Improved bounds. If G, H are df ’s, then 1) F ≥ G implies P (ψ (X ) ≤ t) ≥ 2) If F ≥ H , then P (ψ (X ) < t) ≤ 1 ? sup H (u),
u∈A? (t) ψ n where A? ψ (t) := {u ∈ IR : ψ (u) ≥ t}

sup G(u).
u∈A+ (t) ψ

(3.26)

(3.27)

Proof: a) For any u ∈ A+ ψ (t) holds, using maximality of u,
n n

P (ψ (X ) ≤ t) ≤ P

{Xi ≤ ui }
i=1


i=1

Fi (ui ).

This implies the upper bound in a). Further, P (ψ (X ) ≤ t) ≥ P (X1 ≤ u1 , . . . , Xn ≤ un )
n

(3.28)


i=1

Fi (ui ) ? (n ? 1)
+

by the lower Fr? echet-bound. b) If F ≥ G, then in (3.28) we get
?
u∈A+ (t) ψ

?

P (ψ (X ) ≤ t) ≥ ? sup G(u) ? (n ? 1)? .
+

If F ≥ H then for u ∈ A? ψ (t) P (ψ (X ) < t) = 1 ? P (ψ (X ) ≥ t) ≤ 1 ? P (X1 ≥ u1 , . . . , Xn ≥ un ) = 1 ? F (u) ≤ 1 ? H (u), which implies 2).

Ordering of Risks

15 2

Remark 3.12 a) For the case n = 2 one gets sharp upper and lower bounds for P (ψ (X ) ≥ t) by applying (3.20) to the set A = {x = (x1 , x2 ) : ψ (x1 , x2 ) ≥ t} for any function ψ , in particular for monotonically nondecreasing functions. b) Theorem 3.2 in Embrechts, H¨ oing and Juri (2003) state sharpness of the bounds in (3.26) and (3.27). In comparison to Embrechts, H¨ oing and Juri (2003) we omit some continuity assumption on ψ and omit the language of copulas which is not necessary here. The corresponding bound for the value at risk functionals are in consequence of the monotonicity of ψ easy to achieve (see Embrechts et al. (2003, Theorem 4.1)). There is still a lot of open problems in this area. In particular how to obtain applicable and good bounds under additional information on the model. c) An extension of the bounds in (3.14), (3.25) to increasing functions ψ (X, Y ) of k -dimensional vectors has been given in Li, Scarsini and Shaked (1996). For n = 2 one gets sharpness by Strassens theorem as in (3.17). Also the partial integration argument from Ru (1980a) can be applied to obtain bounds for Eg (X + Y ) for increasing di?erentiable functions g (see Li, Scarsini and Shaked (1996, Theorem 4.2)) and more general to any ?-monotone functions f (X1 , . . . , Xn ) for ki -dimensional random vectors Xi (see Ru (2003)). Several further bounds and techniques for obtaining bounds are discussed in Ru (1991a). One rich source of such bounds are Bonferroni inequalities which in many cases can be proved by a general reduction principle to be sharp. Let e.g. A1 , . . . , An ∈ A where (E, A) is any measure space and Pi ∈ M 1 (E, A). Let X = (X1 , . . . , Xn ) be a random vector with Xj ? Pj , 1 ≤ j ≤ n and de?ne the set that at least k of the events {Xj ∈ Aj } holde true, Lk :=
J ?{1,...,n}, |J |=k

{Xj ∈ Aj , j ∈ J } .

(3.29)

Then P (Lk ) ≤ bk := min 1, 1 k?r
n?r

0≤r≤k?1

p(i)
i=1

(3.30)
?

?

P (Lk )

p(i) ? (k ? 1) ? ? ? i=r+1 ? ≥ ak := max ?0, ? ? n ? r ? (k ? 1) ?

n

where pi = P (Xi ∈ Ai ) and p(1) ≤ · · · ≤ p(n) (see Ru (1991a)). The bounds in (3.30) are sharp. They are consequences of a Bonferroni type result in R¨ uger (1979) and a general reduction principle (see Ru (1991a)). In

16

L. R¨ uschendorf particular for real random variables and Ai = [t, ∞) one gets sharp upper and lower bounds for the tail of the k -th order statistic P X(k) ≥ t ≤ bk , with pi = P (Xi ≥ t) . ≥ ak (3.31)

Also extensions to higher order Bonferroni bounds are given in Ru (1991) and to improved bounds in the case that one can use some of the higher order joint marginal distributions.

4

?-Monotone, Supermodular and Directionally Convex Function Classes

In various applications of comparing risks it has turned out that ?-monotone, supermodular and directionally convex functions and variants of them play an eminent role (see M¨ uller and Stoyan (2002)). For the de?nition we introduce for n f : IR → IR, ε > 0, the di?erence operator ?ε i f by ?ε i f (x) = f (x + εei ) ? f (x), 1 ≤ i ≤ n where ei is the i-th unit vector. De?nition 4.1 Let f : IRn → IR. 1) f is ?-monotone if for every subset I = {i1 , . . . , ik } ? {1, . . . , n} and ε1 , . . . , εk > 0 holds
εk 1 ?ε i1 . . . ?ik f (x) ≥ 0

(4.1)

for all x.

(4.2)

2) f is supermodular if for all 1 ≤ i < j ≤ n, ε, δ > 0 and all x
δ ?ε i ?j f (x) ≥ 0.

(4.3)

3) f is directionally convex if (4.3) holds for all i ≤ j . Denote by F ? , F sm , F dcx the set of all ?-monotone resp. supermodular resp. directionally convex functions then F ? ? F sm and F dcx ? F sm . Remark 4.2 The class of supermodular and directionally convex functions were investigated in early papers of Lorentz (1953) and Ky Fan and Lorentz (1954) in the context of functional inequalities (see also the extensive chapter in Marshall and Olkin (1979)). In Cambanis, Simons and Stout (1976) and Tchen (1980) supermodular functions are called quasimonotone. ?-monotone functions were introduced in Ru (1980). Twice di?erentiable functions f are supermodular (directionally convex) if ?2 f (x) ≥ 0 ?xi ?xj for all x and i < j (resp. for i ≤ j ). (4.4)

Ordering of Risks Di?erentiable functions f are ?-monotone if for all i1 < i2 < · · · < ik , 1≤k≤n ?k f (x) ≥ 0. ?xi1 . . . ?xik De?nition 4.3 For P, Q ∈ M 1 (IRn ) de?ne a) P ≤uo Q ‘upper orthant ordering’ if P ([x, ∞]) ≤ Q([x, ∞]), ? x ∈ IRn

17

(4.5)

(4.6)

b) ≤sm , ≤dcx denote the supermodular ordering resp. directionally convex ordering generated by F sm resp. F dcx . There are corresponding positive/negative dependence notions. De?nition 4.4 1) P is positive (negative) upper orthant dependent ?P ∈ P U OD (resp. P ∈ N U OD) if
n n

Pi ≤uo P
i=1

resp. P ≤uo
i=1

Pi ≥
n i=1 Efi (Xi )

(4.7) for all nondecreasing

2) P is weakly associated if E fi ≥ 0.

n i=1 fi (Xi )

Remark 4.5 A similar notion also exists for lower orthants and is denoted by ≤lo resp. PLOD and NLOD. This notion was introduced in Lehmann (1966). The following equivalence holds: P ∈ PUOD if and only if P is weakly associated. (4.8)

(For n = 2 due to Lehmann (1966), for n ≥ 2 to Ru (1981c).) In fact more generally it was shown in Bergmann (1978) that: P ≤uo Q if and only if for X ? P, Y ? Q
n n

E
i=1

fi (Xi ) ≤ E
i=1

fi (Yi ),

(4.9)

for fi nondecreasing, fi ≥ 0. The maximal generator of the upper orthant order is the set of ?-monotone functions. Theorem 4.6 (?-monotone functions, Ru (1980a)) If P, Q ∈ M 1 (IRn ), then: P ≤uo Q if and only if f dP ≤ f dQ for all f ∈ F ? (4.10)

which are integrable w.r.t. P and Q, i.e. ≤uo is equivalent to ≤F ? .

18

L. R¨ uschendorf

A similar result of course also holds for the lower orthant order and can be combined to characterize the ‘concordance ordering’; P ≤con Q if P ≤uo Q and P ≤lo Q. A random vector X is called WA (weakly associated) if Cov(f (XJ ), g (XI )) ≥ 0 for any disjoint subsets J, I of {1, . . . , n} and monotonically nondecreasing functions f, g of these components. Concerning the supermodular ordering the analogous result is the following. Theorem 4.7 (Supermodular functions) Let P, Q ∈ M 1 (IRn ), a) n = 2 Cambanis, Simons and Stout (1976). For P, Q ∈ M (P1 , P2 ) holds: P ≤uo Q ?? P ≤sm Q b) n ≥ 2 ‘The Lorentz Theorem’, Tchen (1980), Ru (1983). For P ∈ M (P1 , . . . , Pn ) holds P ≤sm P+ , (4.12) (4.11)

where P+ is the measure corresponding to the upper Fr? echet-bound (the comonotonic measure). c) Christo?des and Vaggelatou (2003). If X is a weakly associated random vector, then X has positive supermodular dependence , i.e.
n i=1

P Xi ≤sm P X .

(4.13)

Remark 4.8 a) The interesting result in (4.13) can be stated in the form: Positive dependence implies increasing of the risk. This is of essential interest in risk theory. b) (4.12) was proved in Tchen (1980) by discrete approximation and reduction to the Lorentz (1953) inequalities. In Ru (1979, 1983) the problem of generalized Fr? echet-bounds was identi?ed with a rearrangement problem for functions and then reduced to the Lorentz inequality. c) For P and Q with identical (n-1)-dim. marginal distributions one obtains P ≤uo Q ? P ≤sm Q (Tchen (1980), Ru (1980a, Theorem 3b)) d) There are some useful composition rules which allow to use ≤sm for several models of interest (see M¨ uller and Stoyan (2002) for results and references). e) From Tchens proof of b) it is clear that P? ≤sm P if the lower Fr? echet-bound P? , is a df (cf. also M¨ uller and Stoyan (2002, p. 120)). (4.14)

Ordering of Risks

19

f ) Since for any ? convex the function ψ (x) = ?(x1 + · · · + xn ) is supermodular one obtains as consequence of (4.12) the statement of (3.9) that
n n

Xi ≤cx
i=1 i=1

Fi?1 (U )

(4.15)

i.e. the sums are maximal in convex order for the comonotone case, in other words the comonotone positively dependent portfolio is the riskiest possible. g) Comparison of P ∈ M (P1 , . . . , Pn ) and Q ∈ M (Q1 , . . . , Qn ) w.r.t. the supermodular ordering ≤sm is only possible if the marginals are identical P ≤sm Q implies Pi = Qi , 1 ≤ i ≤ n. (4.16)

So in comparison to ≤F ? =≤uo the ≤sm ordering is restricted to one marginal class while ≤uo allows comparisons between P and Q if the marginals increase stochastically: P ≤uo Q implies Pi ≤st Qi , 1 ≤ i ≤ n. (4.17)

The comparison by F ? is however for a smaller class of functions F ? ? F sm . On the other hand criteria for ≤sm are not as simple as those for ≤uo . The directionally convex order ≤dcx is a ‘typical’ risk order. It allows comparisons in cases where the marginals increase convexly: P ≤dcx Q implies Pi ≤cx Qi , 1 ≤ i ≤ n. (4.18)

From the copula representation (2.7) of distributions with given marginals the following is immediate: If P ∈ M (P1 , . . . , Pn ), Q ∈ M (Q1 , . . . , Qn ) have the same copula C and Pi ≤st Qi , 1 ≤ i ≤ n, then P ≤st Q. (4.19)

(≤st is the multivariate stochastic order w.r.t. increasing functions, see Ru (1981b, Proposition 7). The situation is more complicated if the marginals increase in convex order. Here the analog of (4.19) is wrong, see M¨ uller and Scarsini (2001). The reason is that negative dependence can destroy this conclusion, as the following simple example of that paper shows. Example 4.9 Consider n = 2 and rv ’s X=(W, -W), Y = (W, ?EW ) for some integrable random variable W . Then Yi ≤cx Xi , i = 1, 2, but X1 +X2 = W ?W = 0 while Y1 + Y2 = W ? EW i.e. X1 + X2 ≤cx Y1 + Y2 . However in the positive direction M¨ uller and Scarsini (2001) proved the interesting result, that under a strong positive dependence assumption the analog of (4.19) is true. Let F+ denote as usual the upper Fr? echet-bound of F (F1 , . . . , Fn ). Theorem 4.10 (Directionally convex ordering) Let Fi , Gi be one dimensional df ’s, 1 ≤ i ≤ n.

20 a) ’The Ky-Fan-Lorentz Theorem’ Ru (1983). If Fi ≤cx Gi , 1 ≤ i ≤ n then F+ ≤dcx G+ .

L. R¨ uschendorf

(4.20)

b) M¨ uller and Scarsini (2001). If F ∈ F (F1 , . . . , Fn ) and G ∈ F (G1 , . . . , Gn ) have the same conditionally increasing (CI ) copula C and if Fi ≤cx Gi , 1 ≤ i ≤ n, then F ≤dcx G. (4.21)

Remark 4.11 M¨ uller and Scarsini (2001) give a proof of a) using mean preserving spread (see Theorem 3.12.13 of their paper) while the proof in Ru (1983) is based on the Ky Fan and Lorentz Theorem. The second main ingredient of the proof of b) is the a.s. standard construction of random vectors in (2.4): ? (V ) = (h (V ), . . . , h (V , . . . , V )), where the functions h are monotonX = τF 1 1 n 1 n i ically nondecreasing for CI distribution functions. (4.21) is not valid any more under the weaker dependence assumption of association or of conditional increasing in sequence CIS (see M¨ uller and Scasini (2001)). The following weakening of the WA-notion was introduced in Ru (2003): X is smaller than Y in the weakly conditional in sequence order – X ≤W CS Y – if for all t, 1 ≤ i ≤ n ? 1 and f monotonically nondecreasing Cov(1(Xi > t), f (X(i+1) )) ≤ Cov(1(Yi > t), f (Y(i+1) )) (4.22)

where X(i+1) = (Xi+1 , . . . , Xn ). X is called weakly associated in sequence (WAS) if X ? ≤W CS X, where X ? is the corresponding version of X with independent components; equivalently for all t P X(i+1) |Xi >t ≥st P X(i+1) . The following result extends and uni?es Theorem 4.7 and 4.10. Theorem 4.12 (WCS-Theorem, Ru (2003)) Let X, Y be random vectors with marginals Pi , Qi . a) If Pi = Qi , 1 ≤ i ≤ n and X ≤W CS Y then X ≤sm Y b) If Pi ≤cx Qi , 1 ≤ i ≤ n and X ≤W CS Y then X ≤dcx Y . The ordering ≤W CS combines an increase in positive dependence with a convex increase of the marginals. Some examples for this ordering are given in Ru (2003). In particular one obtains as corollary: Corollary 4.13 If F ∈ F (F1 , . . . , Fn ) and Fi ≤cx Gi , 1 ≤ i ≤ n, then a) F ≤dcx G+ (4.24) (4.23)

Ordering of Risks b) If X ? F , then
n n

21

Xi ≤cx
i=1 i=1

1 G? i (U ),

(4.25)

where U is uniformly distributed on (0, 1). Finally we state some new criteria for the ≤sm and ≤dcx ordering in functional dependence models which are related to B¨ auerle (1997, Theorem 3.1) and B¨ auerle and M¨ uller (1998). For the proofs see Ru (2003). Let (Ui ) be independent rv ’s and (Vi ), V any random variables independent of (Ui ). Further let Xi = gi (Ui , Vi ), Yi = gi (Ui , V ), Zi = gi (Ui , Vi ), Wi = gi (Ui , V ) (4.26)

where Vi ? V and gi (u, ·), gi (u, ·) are monotonically nondecreasing. Let X = (X1 , . . . , Xn ), Y = (Y1 , . . . , Yn ), Z , W denote the corresponding vectors and let ≤ccx denote the componentwise convex order. X , Y , Z , W describe functional models where the dependence is obtained in functional form from some inner and outer factors Ui resp. Vi . These type of models are of particular relevance in various applications in insurance and in economics. Theorem 4.14 For the X , Y , Z , W speci?ed as in (4.26) holds: a) B¨ auerle (1997). X ≤sm Y , Z ≤sm W b) If for all v , gi (Ui , v ) ≤cx gi (Ui , v ) then Z ≤ccx X , W ≤ccx Y and Z ≤dcx Y . c) If gi (Ui , v ) ≤cx gi (Ui , v ) then X ≤ccx Z , Y ≤ccx W , and X ≤dcx W . For the proofs of b) and c) see Ru (2003). Remark 4.15 The random vectors Z , Y and X , W which are compared w.r.t. ≤dcx in Theorem 4.14 do not have the same dependence structure (copula), which was a basic assumption for the proof of the ≤dcx ordering result in Theorem 4.10 b). Also the X and W vectors are not necessarily positive dependent. Since we do not postulate any independence for the (Vi ), we can describe any multivariate df F by a random vector of the form as for X . Thus this comparison result applies to many models. Similarly as in B¨ auerle (1997) one could add in Theorem 4.14 a further random in?uence component W and consider models of the form Xi = gi (Ui , Vi , W ), Yi = gi (Ui , Vi , W ) etc.

References
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L. R¨ uschendorf 4. Bergmann, R. (1978). Some classes of semi-ordering relations for random vectors and their use for comparing covariances, Mathematische Nachrichten, 82, 103–114. 5. Bergmann, R. (1991). Stochastic orders and their application to a uni?ed approach to various concepts of dependence and association, In Stochastic Orders and Decision Under Risk (Ed., K. Mosler and M. Scarsini), pp. 48–73, Volume 19 of IMS Lecture Notes - Monograph Series, Hayward, California (Institute of Mathematical Statistics). 6. Cambanis, S., Simons, G. and Stout, W. (1976). Inequalities for Ek (X, Y ) when the marginals are ?xed, Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete, 36, 285–294. 7. Chong, K.-M. (1974). Some extensions of a theorem of Hardy, Littlewood and Polya and their applications, Canadian Journal of Mathematics, 26, 1321–1340. 8. Christo?des, C., Vaggelatou, E. (2004). A connection between supermodular ordering and positive/negative association, Journal of Multivariate Analysis, 88, 138–151. 9. Cohen, A. and Sackrowitz, B. H. (1995). On stochastic ordering of random vectors, Journal of Applied Probability, 32, 960–965.

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23

18. Embrechts, P., H¨ oing, A. and Juri, A. (2003). Using copulae to bound the value-at-risk for functions of dependent risks, Finance & Stochastics, 7, 145–167. 19. Fan, K. and Lorentz, G. G.(1954). An integral inequality, American Mathematical Monthly, 61, 626–631. 20. Frank, M. J., Nelsen, R. B. and Schweizer, B. (1987). Best-possible bounds for the distribution for a sum – a problem of Kolmogorov. Probability Theory and Relates Fields, 74, 199–211. 21. Franken, P. and Kirstein, B. M. (1977). Zur Vergleichbarkeit zuf¨ alliger Prozesse, Mathematische Nachrichten, 78, 197–205. 22. Franken, P. and Stoyan, D. (1975). Einige Bemerkungen u ¨ber monotone und vergleichbare markowsche Prozesse, Mathematische Nachrichten, 66, 201–209. 23. Fr? echet, M. (1951). Sur le tableaux de corr? elation dont les marges sont donn? ees, Annales de l’Universit? e de Lyon, Section A, 14, 53–77. 24. Hoe?ding, W. (1940). Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts und des Instituts f¨ ur Angewandte Mathematik der Universit¨ at Berlin, 5, 179–233. 25. Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications, Annals of Statistics, 11, 286–295. 26. Joe, H. (1997). Multivariate Models and Dependence Concepts, Volume 73 of Monographs on Statistics and Applied Probability, Chapman & Hall, London. 27. Kaas, R., Dhaene, J., Vyncke, D., Goovaerts, M. J. and Denuit, M. (2001) A simple geometric proof that comonotonic risks have the convex largest sum, In Proceedings of the ?fth International Congres on Insurance, Mathematics and Economics, State College. 28. Kalmykov G. I. (1962). On the partial ordering of one-dimensional Markov processes on the real line, Theory of Probability and its Applications, 7, 456–459. 29. Kamae, T. and Krengel, U. (1978). Stochastic partial ordering, Annals of Probability, 6, 1044–1049. 30. Kamae, T. and Krengel, U., O’Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces, Annals of Probability, 5, 899–912. 31. Kellerer, H. G. (1984). Duality theorems for marginal problems, Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete, 67, 399–432. 32. Lai, T. L. and Robbins, M. (1976). Maximally dependent random variables, Proc. Nat. Acad. Sci. USA, 73, 286–288.

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