A characterization of the multivariate excess wealth ordering

A characterization of the multivariate excess wealth ordering Author’s version Published in Insurance: Mathematics and Economics 49 (2011) 410417 doi:10.1016/j.insmatheco.2011.07.001 http://www.sciencedirect.com/science/article/pii/S0167668711000783 J.M. Fern´andez-Ponce1 Departamento de Estad´ıstica e I.O. C/ Tarfia s/n Universidad de Sevilla 41012-Sevilla, SPAIN F. Pellerey Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 I-10129 Torino, ITALY Rodr´ıguez-Gri˜nolo, M.R. Departamento Econom´ıa, M´etodos Cuantitativos e Historia Econ´omica Universidad Pablo de Olavide Ctra. de Utrera, Km 1 41013 Sevilla, SPAIN. Abstract In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate rightspread or excess wealth function, introduced by Fern´andez-Ponce et al. (1996), is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fern´andez-Ponce et al. (1998). The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are described. Keywords: Excess Wealth Function; Expansion Function; Multivariate Dispersion Ordering; Quantile; Upper-corrected Orthant. 1. Introduction Comparisons among univariate random variables in some stochastic sense have been extensively studied by many authors during the last thirty years. There exist many applications of these stochastic orderings, from economic theory to reliability and queueing theory (see Barlow and Proshan, 1975; Stoyan, 1983; Shaked and Shanthikumar, 1994). In particular, variability orders for univariate distributions have found a profound interest among researchers. Among these types of orders, the dispersion ordering has 1Corresponding author been well studied. Two interesting papers on the dispersion ordering (d.o.) are the ones by Shaked (1982) and Deshpand´e and Kochar (1983). This order was characterized through the number of crossings and the corresponding changes of sign for the distribution functions (see Hickey, 1986; Mu˜n´oz-P´erez, 1990; Saunders, 1984, and Bartoszewicz, 1995). Bartoszewicz (1995) characterized the d.o. using the TTT transforms and he found a relationship between the order based on the mean residual lives and the d.o. (see Bartoszewicz, 1997). Kochar (1996) studied the d.o. among order statistics from DFR distributions. The d.o. has been also considered to characterize lifetime Preprint submitted to Elsevier January 29, 2012 distributions satisfying particular aging properties; different characterizations for IFR and DFR random variables by means of the d.o can be seen, for example, in Pellerey and Shaked (1997). Other types of aging properties for lifetime distributions, weaker than IFR, have been proposed in Kochar and Wiens (1987). Thus, a characterization in dispersion sense for these distributions needed of a new comparison again based on dispersion but weaker than the classic d.o. For this reason, Fern´andez-Ponce et al. (1996) gave the concept of right-spread function, which characterized the aging notions defined in in Kochar and Wiens (1987). In a parallel direction and in an independent way, Shaked and Shanthikumar (1998) defined the same function which was named by them as the excess wealth function. The analysis of a new weak dispersion ordering, which was called excess wealth order, was developed in Shaked and Shanthikumar (1998) and Fern´andez- Ponce et al. (1998). Later several authors have characterized lifetime distributions using this partial order (see Kochar and Carri`ere, 1997 and Kochar, et al., 2001 among others). In this paper, a multivariate generalization of the excess wealth function is defined and studied, and a multivariate stochastic comparison is introduced. The relationships between this comparison and other orders based on dispersion, and in dependence properties, are investigated. Also, a new characterization of the CIS (Conditionally Increasing in Sequence) property, whose definition is recalled next, is provided. The paper is organized in the following way. In Section 2, we further study some properties of the multivariate quantiles which were introduced in Fern´andez-Ponce and Su´arez-Llor´ens (2003). In Section 3, we extend the concept of excess wealth function to the multivariate case. In Section 4, the excess wealth order for multivariate distributions is defined and studied. For example, it is shown that the multivariate excess-wealth order is weaker than the multivariate dispersion ordering. 2. Notation and preliminaries Some notation which will be used throughout the paper is given here. The usual stochastic ordering between univariate random variables is denoted by ≤ST , that is, X≤ST Y if P[X≤x]≥P[Y≤x] for all xin R. Fundamentally, random vectors will be dealt with which take on values in Rn. The space Rnis endowed with the usual componentwise partial order, which is defined as follows: let x= (x1,· · · , xn) and y= (y1,· · · , yn) be two vectors in Rn, then x≤yif xi≤yifor i= 1,· · · , n. Throughout the paper “increasing” means “non-decreasing” and “decreasing” means “non-increasing”. Particularly, a function ϕ:Rn−→ Rnis said to be an increasing function when ϕ(x)≤ϕ(y) for x≤y.The notation ∼st stands for equality in law. The vector of ones will be denoted by 1, i.e. 1= (1,· · · ,1), and the corresponding of zeros by 0. The multiple integral ∫AF(t1, . . . , tn)dt1. . . dtnwill be denoted as ∫AF(t)dt.The dimension of a random vector is clear from the context and unless otherwise stated it is assumed that it is n. Let Xbe a random vector in Rnwith distribution function (cdf) F(·).The multivariate u-quantile for X, also called regression representation, was introduced by O‘Brien (1975), Arjas and Lehtonen (1978) and R¨uschendorf (1981). The definition is as follows. Let un= (u1, . . . , un) be a vector in [0,1]n, the multivariate u-quantile for X, denoted by ∧ x(un),is defined as ∧ x1(u1) = F− X1(u1),∧ x2(u2) = F− X2|X1=∧ x1(u1)(u2), . . . , ∧ xn(un) = F− Xn|∩n−1 j=1 Xj=∧ xj(uj)(un) where F−(u) = inf{x:F(x)≥u}and ui= (u1, . . . , ui) for all i= 1, . . . , n. M¨uller and Scarsini (2001) used the multivariate uquantile to provide conditions for the stochastic comparison of random vectors with a common copula. Li et al. (1996) used this representation as a tool for the construction of multivariate distributions with given nonoverlapping multivariate marginals. Shaked and Shanthikumar (1998) also proposed the standard construction as an useful tool for the stochastic comparison of random vectors. R¨uschendorf (2004) used it for the stochastic comparison of risks with respect to supermodular ordering which is of particular interest in many applications. More recently, Fern´andez-Ponce and Su´arez-Llor´ens (2003) also defined several concepts for a multivariate random vector which will be used later on. The first concept is the multivariate x-rate vector, denoted by ⋆ x(x),and it is defined as ⋆ x1(x1) = P(X1≤x1),⋆ x2(x2) = P(X2≤ x2|X1=x1), ....., ⋆ xn(xn) = P(Xn≤xn|∩n−1 j=1 Xj=xj), where xi= (x1, . . . , xi) for all i= 1, . . . , n. The second concept is the right-upper orthant at a point z, denoted by C(z), and it is defined as C(z) = {x∈Rn:z≤x}.At the end, the upper-corrected orthant at a point zfor the random variable X, denoted as RX(z),is defined as RX(z) = {x∈Rn:x1≥F− X1(⋆ x1(z1)), . . . , xn≥F− Xn|∩n−1 j=1 Xj=xj(⋆ xn(zn))}. For notational purposes, to reduce the complexity in the proofs of the results, a slight modification of this notion is considered here, restricting the set only to the points that are in the support of the vector. By taking into account that the support of a random vector Xis defined as Supp(X) = {x∈Rn: P [X∈Bx(ε)] >0 for all ε > 0} where Bx(ε) is the centered ball at xwith radius ε, then the following definition can be given. 2 Definition 1. Let Xbe a random vector. Its upper corrected orthant at z∈Supp(X)is defined as RX(z) = {x∈Supp(X) : x1≥F− X1(⋆ x1(z1)), . . . , xn≥F− Xn|∩n−1 j=1 Xj=xj(⋆ xn(zn))}. It is easily shown that if Xis a random vector with independent components then RX(z) = C(z)∩Supp(X). Remark 1. Note that if t1≤t2then it could not be held that RX(t2)⊂RX(t1).For example, let Xbe a bivariate random vector with joint density function given by fX(t) =    2/3if t∈T1; 4/3if t∈T2; 0otherwise where T1is the triangle with vertices (0,0),(0,1) and (1,1),and T2is the triangle with vertices (0,0),(1,0) and (1,1). Let t1= (1 2,1 2). By straightforward computations, it is verified that RX(t1) = {x≥1 2, y ≥x+ 1 3}. Now, consider t2= (2 3,1 2). It holds that t1≤t2, and t2∈RX(t2). But, t2∈ RX(t1). Thus RX(t2)*RX(t1). From now on, assume that the following regularity conditions (RC) are verified by every cdf Fin the paper. 1. Fis a continuous function. 2. The vector b x(u) is differentiable at each component. 3. The conditional distribution of Xito X1, . . . , Xi−1 (Fi|1,...,i−1) is a continuous and strictly increasing function for i= 1, . . . , n. For convenience F1|0=F1. 4. F− Xi|X1,...,Xi−1(0) <∞for all i= 1, . . . , n. It is easy to verify that under the regularity conditions above there is a one to one correspondence between vectors un∈[0,1)nand the points x∈Supp(X). The next result is the main reason of interest in the notion of upper corrected orthant. Its proof directly follows from Proposition 2.1 in Fern´andez-Ponce and Su´arez-Llor´ens (2003). Proposition 1. Let Xbe a random vector. Then P[X∈RX(b x(un))] = n ∏ j=1 (1 −uj), for all un∈[0,1)n. Furthermore, let t= (t1, t2) be a point in Supp(X). Then it is easily shown that P(X∈RX(t)) = ∫+∞ t1∫+∞ F− 2|w1(F2|t1(t2)) f(w1, w2)dw2dw1 =∫+∞ t1fX1(w1)∫+∞ F− 2|w1(F2|t1(t2)) fX2|X1=w1(w2)dw2dw1. Consequently, it is obtained that P(X∈RX(t)) = P(X1> t1)P(X2> t2|X1=t1).(1) This equality can easily be generalized for the n−dimensional case by using induction arguments. This result means that the probability associated to the upper corrected orthant at the un−quantile does not depend on the distribution function. As it can be seen in Proposition 1, the upper corrected othant plays in the multivariate setting the same role as the u-quantile upper orthant for univariate distributions, being P[X∈RX(bx(u))] = P[X≥F− X(u)] = 1 −uin the univariate case. Thus, it is interesting to describe the main properties of upper corrected orthants. It is well-known that the univariate quantile function is an increasing function, i.e. u≤vif, and only if QX(u)≤ QX(v).This property is not verified by random vectors, in general. However if a type of dependence is held then this property can be verified. This type of dependence must be based on the growth of the corresponding conditional distributions. In this way, the following definition of conditionally increasing in sequence property can be found in Barlow and Proschan (1975). Also, two interesting results which will be used later are proved. Definition 2. The random variables X1, . . . , Xn are conditionally increasing [decreasing] in sequence (CIS)[CDS] if Xiis stochastically increasing [decreasing] in X1, . . . , Xi−1for i= 2, . . . , n. Theorem 1. X is a CIS random vector if and only if RX(x)⊂C(x)for all x∈Rn. Proof. First, we prove the sufficient condition. The proof will be by mathematical induction. The proposition is obviously true for n= 1.Assume that the proposition is true for n=m. We now need to show that it is true for n=m+ 1.If b x(um)≤tmthen, given that Xis CIS, bxm+1|b xm(um)(um+1)≤bxm+1|tm(um+1). However, we know that if tm+1 ∈RX[b x(um+1)] then tm+1 ≥bxm+1|tm(um+1).Therefore, tm+1 ∈C[b x(um+1)]. Hence we have completed the proof by the induction argument. The necessary condition is also proved by mathematical induction. The proposition is obviously true for n= 1. Assume that the proposition is true for n=m. It is sufficient to prove that, for all xj≤tj, j = 1, . . . , m FXm+1|∩m j=1 Xj=xj(xm+1)≥FXm+1|∩m j=1 Xj=tj(xm+1). (2) 3 If tm+1 ∈RX(x), then, for s= 1, . . . , m+1, it is held that ts≥F− Xs|∩s−1 j=1 Xj=tj[FXs|∩s−1 j=1 Xj=xj(xs)]. In particular, if tm+1 =F− Xm+1|∩m j=1 Xj=tj[FXm+1|∩m j=1 Xj=xj(xm)], given that RX(x)⊂C(x) for all x∈Rn, it holds that F− Xm+1|∩m j=1 Xj=tj[FXm+1|∩m j=1 Xj=xj(xm)] ≥xm+1.(3) Now, from the fact that FXm+1|∩m j=1 Xj=xj(x) is an increasing function in x, by solving the inequality (3), the inequality (2) is obtained. Theorem 2. X is a CDS random vector if and only if C(x)∩Supp(X)⊂RX(x)for all x∈Rn Proof. First, we prove the sufficient conditions. The proof will be obtained by mathematical induction. The proposition is obviously true for n= 1.Assume that the proposition is true for n=m. Now we need to show that it is true for n=m+ 1.If tm+1 ∈C(xm+1)∩Supp(X), then tm+1 ≥xm+1. Since Xis CDS, it follows that, for all xj≤tj, j = 1, . . . , m FXm+1|∩m j=1 Xj=tj(xm+1)≥FXm+1|∩m j=1 Xj=xj(xm+1). (4) On the other hand, given that FXm+1|∩m j=1 Xj=tj(x) is increasing in x, FXm+1|∩m j=1 Xj=tj(tm+1)≥FXm+1|∩m j=1 Xj=tj(xm+1).(5) From (4) and (5) it holds that FXm+1|∩m j=1 Xj=tj(tm+1)≥FXm+1|∩m j=1 Xj=xj(xm+1). Given that F− Xm+1|∩m j=1 Xj=tj(u) is an increasing function in u, then tm+1 ≥F− Xm+1|∩m j=1 Xj=tj[FXm+1|∩m j=1 Xj=xj(xm+1)]. Therefore, tm+1 ∈RX(x). The necessary condition is also proved by mathematical induction. The proposition is obviously true for n= 1. Assume that the proposition is true for n=m, then it is sufficient to prove that, for all xj≤tj, j = 1, . . . , m, it holds FXm+1|∩m j=1 Xj=tj(xm+1)≥FXm+1|∩m j=1 Xj=xj(xm+1). (6) If tm+1 ∈C(x)∩Supp(x)⊂RX(x), then ts≥F− Xs|∩s−1 j=1 Xj=tj[FXs|∩s−1 j=1 Xj=xj(xs)] and ts≥xs, for s= 1, . . . , m + 1. In particular, if tm+1 = xm+1, it holds that xm+1 ≥F− Xm+1|∩m j=1 Xj=tj[FXm+1|∩m j=1 Xj=xj(xm+1)], and, given that FXm+1|∩m j=1 Xj=tj(x) is an increasing function in x, the inequality (6) is obtained. Thus, Xis CDS. Now, the monotonicity of the b x(u) is characterized by the CIS or CDS property of the distribution function of X. Let ϕX: [0,1]n→Rnbe such that ϕX(u) = b x(u) and ϕ−1 X its corresponding inverse function. Theorem 3. X is a CIS random vector if and only if ϕX is increasing. Proof. For the sufficient condition, see Rubinstein et al. (1985). The necessary condition will be by mathematical induction. From Theorem 1, it is sufficient to prove that RX(x)⊂C(x). The statement is obviously true for n= 1. Assume that the proposition is true for n=m. Let tm+1 ∈RX(x), that is, ts≥F− Xs|∩s−1 j=1 Xj=tj[FXs|∩s−1 j=1 Xj=xj(xs)] for s= 1, . . . , m + 1.In particular, for s=m+ 1, it holds tm+1 ≥F− Xm+1|∩m j=1 Xj=tj[FXm+1|∩m j=1 Xj=xj(xm+1)] and, given that FXm+1|∩m j=1 Xj=tj(x) is increasing in x, it follows FXm+1|∩m j=1 Xj=tj(tm+1)≥FXm+1|∩m j=1 Xj=xj(xm+1). Now, let tj=bx(vj), xj=bx(uj) for j= 1, . . . , m and vm+1 =FXm+1|∩m j=1 Xj=tj(tm+1) and um+1 = FXm+1|∩m j=1 Xj=xj(xm+1). Then, by the hypothesis, bx(vm+1)≥bx(um+1), that is F− Xm+1|∩m j=1 Xj=bx(vj)[FXm+1|∩m j=1 Xj=bx(vj)(tm+1)] ≥F− Xm+1|∩m j=1 Xj=bx(uj)[FXm+1|∩m j=1 Xj=bx(uj)(xm+1)]. Therefore, tm+1 ≥xm+1 and the result is obtained. Note that this theorem shows that the CIQ property, which was given in Belzunce et al. (2008), is equivalent to the CIS property. Thus, as a consequence, all the multivariate dispersive orders considered in Belzunce et al. (2008), are equivalent for CIS vectors having a common copula. 4 Theorem 4. X is a CDS random vector if and only if ϕ−1 Xin increasing. Proof. Suppose that Xis a CDS random vector. The proof will be by mathematical induction. The proposition is obviously true for n= 1.Assume that the proposition is true for n=m. If Xis a CDS random vector, then F− Xm+1|∩m j=1 Xj=b x(uj)(um+1)≥F− Xm+1|∩m j=1 Xj=b x(vj)(um+1) (7) for all uj≤vj, j = 1, . . . , m. If b x(um+1)≤b x(vm+1), then by definition F− Xm+1|∩m j=1 Xj=bx(uj)(um+1)≤F− Xm+1|∩m j=1 Xj=bx(vj)(vm+1). (8) Finally, from (7) and (8) and given that F− Xm+1|∩m j=1 Xj=bx(vj)(u) is increasing in u, it follows that vm+1 ≥um+1 and the result is obtained. Conversely, suppose that b x(um+1)≤b x(vm+1) implies um+1 ≤vm+1. In the light of Theorem 2, it is sufficient to prove that C(x)⊂RX(x) for all x∈Rm. Let tm+1 ∈C(x), that is, ts≥xsfor s= 1, . . . , m + 1.(9) In particular, for s=m+ 1 and from the regularity conditions, the inequality (9) is equivalent to F− Xm+1|∩m j=1 Xj=tj[FXm+1|∩m j=1 Xj=tj(tm+1)] ≥ F− Xm+1|∩m j=1 Xj=xj[FXm+1|∩m j=1 Xj=xj(xm+1)]. Now, if for j= 1, . . . , m it is tj=bx(vj) and xj=bx(uj), and vm+1 =FXm+1|∩m j=1 Xj=tj(tm+1), um+1 =FXm+1|∩m j=1 Xj=xj(xm+1), then the above inequality can be rewritten as F− Xm+1|∩m j=1 Xj=bx(vj)(vm+1)≥F− Xm+1|∩m j=1 Xj=bx(uj)(um+1). (10) Moreover, by the hypothesis and the inequality (10), vm+1 ≥um+1 holds. Consequently, given that FXm+1|∩m j=1 Xj=tj(x) is increasing in x, it follows that tm+1 ≥F− Xm+1|∩m j=1 Xj=tj[FXm+1|∩m j=1 Xj=xj(xm+1)]. Therefore, tm+1 ∈RX(x) and the result is obtained. 3. The multivariate excess wealth function In this section, the multivariate excess wealth function is defined following the same steps given for definition in the univariate case in Fern´andez-Ponce et al. (1996). One purpose is to generalize the univariate version and to prove that it preserves some similar properties. Definition 3. Let Xbe a nonnegative random vector. Then the multiple expectation associated to X,when it exists, is defined as the real value µX=∫Supp(X) P[X∈RX(t)] dt. Remark 2. Let X= (X1, X2)be a bivariate random variable. Then under straightforward calculations is easily obtained by using (1) that ¯µX=νX−∫+∞ F− X1(0) F1(t)F− X2|X1=t(0)dt, where νX=∫∞ F− X1(0) F1(t)E[X2|X1=t]dt. Particularly, if Xrepresents a non-negative lifetime random variable then ¯µX≤νX. If Supp(X) = [0,+∞)2,then ¯µX=νXholds. Furthermore, if the components are independent random variables, then the multiple expectation coincides with the product of the marginal expectations. The value of νXin the bivariate case can be easily obtained if the first component has an exponential distribution. This fact is proved in the following proposition. Proposition 2. Let X= (X1, X2)be a couple of random lifetimes, and let X1have an exponential distribution.Then νX=E[X1]·E[X2]. Proof. Given that X1has an exponential distribution, it holds that X1=st Xe 1, where Xe 1is the lifetime having density function fXe 1(t) = FX1(t)/E[X1], t ≥0. It can also be noted that E[X2] = ∫+∞ 0E[X2|X1= t]dFX1(t), so that ∫+∞ 0 (E[X2|X1=t]−E[X2]) dFX1(t) = 0.(11) Now, let the function φ(t) defined as φ(t) = E[X2|X1= t]−E[X2]. By using (11), it holds that E[φ(X1)] = 0 and by taking into account that X1=st Xe 1, it follows 0 = E[φ(X1)] = E[φ(Xe 1)] =∫+∞ 0 (E[X2|X1=t]−E[X2]) dFXe 1(t) =∫+∞ 0 (E[X2|X1=t]−E[X2]) FX1(t) E[X1]dt. By using the previous inequality jointly with the fact that νX=∫+∞ 0E[X2|X1=t]FX1(t)dt and E[X1]>0, it is immediately obtained that νX=∫+∞ 0 E[X2|X1=t]FX1(t)dt =∫+∞ 0 E[X2]FX1(t)dt =E[X1]E[X2]. 5 Definition 4. Let Xbe a nonnegative random vector with finite multiple expectation. The multivariate excess wealth function associated to Xis defined as S+ X(u) = ∫RX[∧ x(u)] P[X∈RX(t)] dt,u∈[0,1)n.(12) Several interesting properties for the multivariate excess function can be shown as in the univariate case. Proposition 3. Let Xbe a nonnegative random vector with finite multiple expectation. Then i) S+ X(u)is a decreasing and a shift invariant function. ii) 0≤S+ X(u)≤S+ X(0) = µXfor all uin [0,1)n. iii) If the components of Xare independent, then S+ X(u) = n ∏ i=1 S+ Xi(ui). Proof. Trivial and therefore omitted. Particularly, it is interesting to obtain an expression for the bivariate excess wealth function easier to hand. Let X= (X1, X2) be a bi-dimensional random vector satisfying the RC conditions with density function fX(x1, x2).Obviously, it holds that b x(u1, u2) = (F− X1(u1), F− X2|X1=F− X1(u1)(u2)) and x∈RX(b x(u1,u2)) if x1≥F− X1(u1) and x2≥F− X2|X1=x1(u2). Consequently, by using (1) the bivariate excess wealth function can be expressed as S+ X(u1, u2) = ∫∞ F− X1(u1) ¯ FX1(t1)·S+ X2|X1=t1(u2)dt1.(13) Differentiating (12), it is easily shown that ∂S+ X(u1, u2) ∂u2∂u1 =P(X∈RX[∧ x(u)]) fX(b x(u)) (14) for all (u1, u2) in (0,1)2. 4. The multivariate excess wealth ordering In this section the multivariate excess wealth ordering is defined and some of its properties are proved. Definition 5. Let Xand Ybe two nonnegative random vectors with finite multiple expectations. Xis said to be smaller than Yin the sense of multivariate excess wealth order (denoted X≤ew Y) if S+ X(u)≤S+ Y(u)for all uin (0,1)n. Note that this ordering is the multivariate version of the excess wealth ordering which was studied in Shaked and Shanthikumar (1998) and Fern´andez-Ponce et al. (1998). Example 1. Let Xand Yhave a bivariate Pareto distribution with parameters θ1, θ2, a and θ′ 1, θ′ 2, a′, respectively. If 2< a′≤aand θi≤θ′ ifor i= 1,2, then X≤ew Y. In fact, if Xhas a joint Pareto density given by f(t1, t2) = a(a+ 1) θ1θ2(t1 θ1 +t2 θ2 −1)−(a+2) , with, a > 0, ti≥θi>0,easy calculations show that, for ti≥θi, FX2|X1=t1(t2) = t(a+1) 1 θ(a+1) 1(t1 θ1 +t2 θ2 −1)−(a+1) . The excess-wealth function of X2|X1=t1has the following expression: S+ X2|X1=t1(u2) = ∫+∞ F− X2|X1=t1(u2)¯ FX2|X1=t1(t2)dt2 =∫+∞ F− X2|X1=t1(u2) t(a+1) 1 θ(a+1) 1(t1 θ1+t2 θ2−1)−(a+1) dt1 =t1θ2 aθ1(1 −u2)a a+1 . Therefore, by using (4), the bivariate excess-wealth for the vector Xis given by SX(u1, u2) = ∫+∞ F− X1(u1) ¯ FX1(t1)S+ X2|X1=t1(u2)dt1 =∫+∞ F− X1(u1) θa 1t−a 1 t1θ2 aθ1 (1 −u2)a a+1 dt1 =θ2θ1(1 −u2)a a+1 (1 −u1)a−2 a a(a−2) , for a > 2. It is easily seen that the functions h(a) = (1 −u2)a a+1 (1 −u1)a−2 aand g(a) = 1 a(a−2) are decreasing in a. So, for 2< a′≤aand θi≤θ′ ii= 1,2, it follows that θ2θ1(1 −u2)a a+1 (1 −u1)a−2 a a(a−2) ≤ θ′ 2θ′ 1(1 −u2)a′ a′+1 (1 −u1)a′−2 a′ a′(a′−2) , 6 that is, X≤ew Y. It is easy to prove the following closure results. The notation ∼ew means that both X≤ew Yand Y≤ew Xhold. Theorem 5. i) Let X(1),...,X(m)be a set of independent random vectors and let Y(1), . . . , Y(m)be another set of independent random vectors. If X(i)≤ew Y(i)for i= 1, . . . , m, then (X(1), . . . , X(m))≤ew (Y(1), . . . , Y(m)). ii) Let {X(j):j= 1,2. . .}and {Y(j):j= 1,2, . . .}be two sequences of random vectors such that X(j)→st Xand Y(j)→st Yas j→ ∞,where →st denotes convergence in distribution. If X(j)≤ew Y(j)for every j= 1,2, . . ., then X≤ew Y. Proof. i) It is trivial by using Definition 5 and iii) of the Proposition 3. ii) It is easy to show that if X(j)→st X then b x(j)(u)→b x(u) and RX(j)[b x(j)(u)]→ RX[b x(u)] for all uin (0,1)n.Now by using the monotone convergence theorem and the fact that µX(j)<+∞for all j, it is easily shown that S+ X(j)(u)→S+ X(u).Thus the result is immediately obtained. Some properties and results of the excess wealth order are proved for the bivariate case. These properties can be easily generalized to any dimension. Theorem 6. Let X= (X1, X2)and Y= (Y1, Y2)be two random vectors. 1. If X∼st Y+cthen X∼ew Y. 2. If X∼ew Ythen f[b x(u)] = g[b y(u)] for all uin (0,1)2, where f(·)and g(·)are the corresponding density functions for Xand Y,respectively. Proof. (1) Trivial, since the multivariate excess wealth function is shift invariant, as it can be easily verified. (2) Suppose X∼ew Y,i.e., suppose it holds S+ X(u) = S+ Y(u) for all uin (0,1)2. Since Xand Ysatisfy the regularity conditions, it follows that the quantiles b x(u) and b y(u) are differentiable with respect to u= (u1, u2) at each component. Consequently, by using (14) and Proposition 1, it is immediately obtained the result. A multivariate dispersion ordering based on the standard construction was given by Fern´andez-Ponce and Su´arez- Llor´ens (2003). Given two n−dimensional random vectors Xand Y, we say that Xis smaller than Yin the multivariate dispersion order, denoted by X≤disp Y, if, and only if, ∥ˆx(v)−ˆx(u)∥2≤ ∥ ˆy(v)−ˆy(u)∥2, for all u,v∈[0,1]n, where ∥ · ∥2means the Euclidean distance. Fern´andez-Ponce and Su´arez-Llor´ens (2003) established that the multivariate dispersion ordering is characterized by an unique expansion function which maps each quantile for one variable in the corresponding quantile for the other one. Particularly, Arias-Nicol´as et al. (2005) studied the concept of multivariate dispersion order for multivariate distributions with the same dependence structure. Later on, Belzunce et al. (2008) studied a weaker multivariate dispersion ordering defined as Xis said to be less in the conditional dispersive order than Y, denoted by X≤c−disp Y, if, and only if, X1≤disp Y1, Xi| i−1 ∩ j=1 Xj= ˆxj(uj)≤disp Yi| i−1 ∩ j=1 Yj= ˆyj(uj). Now, it is proved that the multivariate conditional dispersion ordering implies the excess wealth ordering. Theorem 7. Let Xand Ybe two absolutely continuous random vectors. If X≤c−disp Ythen X≤ew Y. Proof. For simplicity, the bidimensional case is proved. This proof can easily be generalized to any dimension. By using the equality (1), the bivariate excess-wealth function of Xcan be expressed as S+ X(u) = ∫∞ F− X1(u1)∫∞ F− X2|X1=t1(u2) ¯ FX1(t1)¯ FX2|X1=t1(t2)dt1dt2. By taking w1=FX1(t1) and w2=FX2|X1=t1(t2), it follows that S+ X(u) = ∫1 u1∫1 u2 (1 −w1)(1 −w2) fX(ˆx(w)) dw1dw2 where fX(·) is the density function of X. Similarly, S+ Y(u) = ∫1 u1∫1 u2 (1 −w1)(1 −w2) fY(ˆy(w)) dw1dw2 where fY(·) is the density function of Y. It is clear that if X≤c−disp Y, then fX(ˆx(w)) ≥ fY(ˆy(w)) ∀w∈[0,1]2(see Belzunce et al., 2008). Therefore, the result is immediately obtained. An example where two random vectors are ordered in excess wealth, but not in multivariate dispersion sense, is given here. 7 Example 2. Suppose that X= (X1, X2)is a bivariate exponential random vector such that X1 and X2are independent, X1∼st Exp(2) and X2∼st Exp(1). Similarly, assume that Y= (Y1, Y2) is a bivariate exponential random vector such that Y1 and Y2are independent with Y1∼st Exp(1/3) and Y2∼st Exp(3). Consequently, it holds that X1<disp Y1and Y2<disp X2 which implies that X<c−disp Y. Furthermore, by using iii) of Proposition 3, it holds S+ X(u1, u2) = 1 2(1 −u1)(1 −u2) ≤(1 −u1)(1 −u2) = S+ Y(u1, u2) for all (u1, u2)in (0,1)2,i.e. X≤ew Y. The following result describes conditions under which the multivariate excess wealth order holds. Some previous definitions are necessary. Let Xand Ybe two univariate random variables with excess wealth function S+ X(u) and S+ Y(u),respectively. Definition 6. Let Xand Ybe two univariate random variables. Xis said to be smaller than Yin relative excess wealth sense (denoted X≤rew Y) if S+ X(v) S+ X(u)≤S+ Y(v) S+ Y(u)for all 0< u ≤v≤1. Note that, when the supports of Xand Yare [0,+∞), this ordering is equivalent to the DMRL ordering which was defined in Kochar and Wiens (1987) and further characterized in Fern´andez-Ponce, et al. (1996). Definition 7. Let Xand Ybe two univariate random variables. Xis said to be smaller than Yin wealth sense (denoted X≤WY) if X≤ST Yand X≤rew Y. Definition 8. Let X= (X1, X2)be a bidimensional random vector. Xis said to be conditional increasing in excess wealth (X∈CIEW )if X2|X1=t1≤ew X2|X1=t2for all t1≤t2. Recall that a distribution function F is said to be of powerlaw tail or Pareto type tail if, for x→+∞, 1−F(x)∼Kx−µ, where Kis a positive normalization constant or a slowly varying function. The exponent µ > 0 is called the tail exponent of the distribution. Proposition 4. Let X= (X1, X2)be a bidimensional random vector. If dF − X2|X1=t(u) dt is an increasing function in uand X2|X1=tis a power-law tail with exponent equals to one, for all t, then X∈CIEW . Proof. By using an expression for the right-spread function which can be seen in Fern´andez-Ponce et al. (1996), it is easy to obtain that S+ X(u) = ∫1 u (1 −ω)dF− X(ω). Now, integrating by parts and using the fact that the random variable X∼ST X2|X1=tis a power-law tail with exponent equals to one, it is held dS+ X2|X1=t(u) dt =∫1 u(dF− X2|X1=t(ω) dt −dF− X2|X1=t(u) dt )dω. and the result is immediately obtained. Theorem 8. Let X= (X1, X2)and Y= (Y1, Y2)be two bidimensional random vectors. If it is held that i) X1≤WY1, ii) X2|X1=t≤ew Y2|Y1=tfor all t∈R, iii) X∈CIEW or Y∈CIEW then X≤ew Y. Proof. Assume that Y∈CIEW. By using (ii), it is obtained that S+ X(u1, u2)≤∫∞ F− 1(u1) ¯ F1(t)S+ Y2|Y1=t(u2)dt. Now, consider the density functions ˜ f1(t) = ¯ F1(t) S+ X1(u1)I(t≥F− X1(u1)) and ˜g1(t) = ¯ G1(t) S+ Y1(u1)I(t≥F− Y1(u1)). The corresponding distribution functions are ˜ F1(x) = (1−S+ X1(F1(x)) S+ X1(u1))I(t≥F− X1(u1)) and ˜ G1(x) = (1−S+ Y1(G1(x)) S+ Y1(u1))I(t≥F− Y1(u1)). By taking into account (i), we obtain that ˜ F1≤ST ˜ G1. Thus, 8 ∫∞ 0 S+ X1(u1)S+ Y2|Y1=t(u2)d˜ F1(t)≤ ∫∞ 0 S+ Y1(u1)S+ Y2|Y1=t(u2)d˜ G1(t) = S+ Y(u1, u2) and the result is obtained. The statement can be shown in a similar manner if X∈CIEW is assumed. Example 3. Assume that X1and X2are independent random variables with exponential distributions whose means are equal to λ1(i.e. Xi∼Exp(λ1)for i= 1,2). Now, let Y1and Y2be two random variables such that Y1∼Exp(λ2)and Y2|Y1=t∼Exp(λ2(t+ 1)) λ1≤λ2. It is easy to verify that the random vectors X= (X1, X2) and Y= (Y1, Y2)verify the conditions of Theorem 8. Consequently, X≤ew Y. The following result describes some necessary conditions for the excess wealth order. An immediate application, dealing with conditions for comparisons in dependence sense of random vectors, will be given next. For the statement of this result, we recall the definition of the IGFR property. Let Xbe a nonnegative random variable with density function fX, survival function ¯ FXand failure rate function r(x) = fX(x)/¯ FX(x). The generalized failure rate is defined as R(X) = xr(x). If the function R(x) in increasing in x, then it is said that Xhas an increasing generalized failure rate (IGFR) distribution (the DGFR property can be defined analogously). Clearly, if Xhas increasing failure rate then it is also IGFR. However, the reversed need not hold, given that there exist many distributions having decreasing failure rate which are IGFR (See Belzunce et al. , 1998, and Lariviere, 2006). Also, recall that if Xis a random vector then its centered random vector is e X=X−E(X). Theorem 9. Let X= (X1, X2)and Y= (Y1, Y2)be two bidimensional random vectors and e Xand e Ytheir corresponding centered random vectors. If it is held that i) X1∼ST Y1, ii) X1has a IGFR distribution, iii) Supp(e X)⊆Supp(e Y), iv) X≤ew Y, then cov(X1, X2)≤cov(Y1, Y2). Proof. For an easier notation, F1(·) is used to denote FX1(·) and FY1(·). Assume first that E(X2) = E(Y2). By taking into account that S+ X(u, 0) = ∫+∞ F− 1(u) F1(t)(E(X2|X1=t)−F− X2|X1=t(0))dt, and a similar formula is held for the random vector Y, by using (iv) it can be obtained the following inequality ∫+∞ F− 1(u) F1(t) (E(Y2|Y1=t)−E(X2|X1=t)) dt− ∫+∞ F− 1(u) F1(t)(F− Y2|Y1=t(0) −F− X2|X1=t(0))dt ≥0. By using (iii) is immediately obtained that F− Y2|Y1=t(0) −F− X2|X1=t(0) ≥0 for all t∈R. Consequently, ∫+∞ F− 1(u) F1(t) (E(Y2|Y1=t)−E(X2|X1=t)) dt ≥0,∀u. Let dW(t) = F1(t)f1(t) F1(t)(E(Y2|Y1=t)−E(X2|X1=t)) dt and h(t) = t·f1(t) F1(t). By assumption (ii), h(t) is increasing, and, by inequality above, ∫+∞ sdW(t)≥0 for all s=F− 1(u)≥0. Now, using Lemma 7.1 (a) in Barlow and Proschan (1975), it is obtained that ∫+∞ 0h(t)dW(t) = ∫+∞ 0F1(t)f1(t) F1(t)t(E(Y2|Y1=t)−E(X2|X1=t)) dt = ∫+∞ 0f1(t) (E(tY2|Y1=t)−E(tX2|X1=t)) dt ≥0. Thus, E(Y1Y2)≥E(X1X2) and by (i) and (ii) the result holds. Now, assume that E(X2)=E(Y2).Note that X≤ew Yif, and only if e X≤ew e Y.Thus, in a similar manner, it is obtained that E(X1f X2)≤E(Y1f Y2), and the result is immediately obtained. The following corollary points out that, under appropriate assumption on aging for the first component, the excess wealth order can be considered as positive dependence order, since it implies the comparison between the covariances for vectors in the same Frechet class. Corollary 1. Let X= (X1, X2)and Y= (Y1, Y2)be two bidimensional random vectors having the same marginal distributions. If X1has a IGF R distribution, then X≤ew Y⇒cov(X1, X2)≤cov(Y1, Y2). Acknowledgements The authors thank the anonymous referees and Prof. Alfonso Su´arez-Llor´ens for their valuable suggestions that helped to improve our work. 9