Given a bivariate random vector .X; Y /, we can find several notions of positive dependence based on conditional randomvariables such as .X jY > x/, .X jY 6 x/ or .X jY D x/, where by .X jA/, we denote any random variable whose distri-bution is the conditional distribution of X given an event A [1–3]. Some of these notions are of interest in the context ofreliability [4–11] and in some other contexts such as inequality theory [12] and risk theory [13]. Many of these notionsare defined by comparison of the conditional distributions with the marginal distribution under the theoretical assump-tion that X and Y are independent. These comparisons can be extended to compare general pairs of random vectors. Forexample, given two bivariate random vectors .X1; X2/ and .Y1; Y2/, and based on the PQD order notion, it is possible toprovide comparisons of the conditional random variables .X2jX1 > x/ and .Y2jY1 > x/, as well as for .X1jX2 > x/ and.Y1jY2 > x/. These comparisons are made in terms of the distribution functions and expected values of the conditionalrandom variables (see Section 9.A in [14]). As we will see later, some multivariate orders are based on comparisons ofprevious conditional random variables.
This topic is of interest in the context of reliability and risk theory where the detection of concordant behaviour (thecomponents tend to be all large together or small together) is especially important and, in particular, in comparing thedegree of concordance among two financial assets or two sets of components of a system. It is natural to consider vari-ability measures to compare random quantities and, in particular, these conditional random variables. For example, inrisk theory, the comparison can be made in terms of the variances in order to avoid situation of great uncertainty. Let usconsider the following real example in which we consider two bivariate random vectors of returns. In the first case, weconsider the returns of two Spanish energetic companies, Endesa (E) and Iberdrola (I), and let us denote by .RE; RI /the bivariate random vector of the corresponding returns. In the other case, we consider the returns of two Spanish bankcompanies, Santander (S) and BBVA (B), and let us denote the corresponding vector of returns by .RS; RB/. Data are ofpublic access and can be easily obtained from the Yahoo! Finance site. We have considered bivariate samples of size 400where the share value is measured from March 2009 until December 2010. If we denote by xt the share value at time t ,the rate of return at time t is defined by computing the rate .xt � xt�1/=xt�1. In Figure 1, we give a plot of the bivariate
aDpto. Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia,Spain
bDpto. Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Alicante, Apartado de Correos 99, 03080 Alicante, SpaincDpto. Estadística e Investigación Operativa, Facultad de Ciencias Empresariales, Universidad de Cádiz, C/ Duque de Nájera, 8, 11002 Cádiz,Spain
*Correspondence to: José M. Ruiz, Dpto. Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Murcia, Campus deEspinardo, 30100 Espinardo, Murcia, Spain.
Figure 1. Plots of sample values for .RE ; RI / and .RS ; RB/.
samples, which indicate, in both cases, a positive association between the bivariate returns. This is typical behaviour giventhat similar securities usually have some positive correlation with each other.
Additionally, we can ask ourselves how the values of one component affect the uncertainty or variability of the othercomponent; that is, we can ask about the behaviour, for example, of VarŒRE jRI > x� and VarŒRS jRB > x� (andVarŒRI jRE > x� and VarŒRB jRS > x�, respectively). This is of interest if we want to avoid high uncertainty or vari-ability. In Figure 2 , we provide empirical values for these quantities (we have computed the conditional variances up to apoint x where the sample size is not big enough to provide a good estimation of the variance). As we can see, the condi-tional variances for the energetic companies are smaller than the corresponding variances for the bank companies, whichshows that the uncertainty is smaller for the case of the energetic companies than that of the bank companies. Therefore, ifwe are interested in avoiding uncertainty, we should chose the energetic companies. It is worthy to mention that [15] and[16] used a similar graphical tool to detect empirically multivariate aging properties by plotting some classical multivariatedispersion measures associated with multivariate residual lifetimes.
This situation appears also in some parametric models. Let us consider the case of a bivariate Pareto distribution(denoted as P .˛1; ˛2; a/) with joint survival function
F .x1; x2/D
�x1
˛1Cx2
˛2� 1
��awhere xi > ˛i > 0 and a > 2.
It is not difficult to see that
VarŒXi jXj > xj �D˛2i˛2j
a
.a� 1/2.a� 2/x2j :
Let us consider now .X1; X2/ � P .˛1; ˛2; a/ and .Y1; Y2/ � P .˛1; ˛2; b/, clearly if a > b, then VarŒXi jXj > xj � 6VarŒYi jYj > xj � for i ¤ j 2 f1; 2g.
-0.08 -0.06 -0.04 -0.02 0.00 0.02
2e-
043e-
044e-
045e-
04
Returns
Con
ditio
nal v
aria
nces
Endesa | IberdrolaSantander | BBVA
-0.08 -0.06 -0.04 -0.02 0.00 0.02
0.00
015
0.00
025
0.00
035
0.00
045
Returns
Con
ditio
nal v
aria
nces
Iberdrola | EndesaBBVA | Santander
Figure 2. Estimation of conditional variances for .RE jRI > x/ and .RS jRB > x/.
The purpose of this paper was to study the comparison of random vectors in terms of comparisons of conditionalvariables, such as that in the previous examples. In particular, we consider the comparison in terms of dispersion andconcentration. As we will see in Section 2, the comparison of dispersion and concentration will be established in terms ofseveral criteria (stochastic orders) to compare random quantities, instead of measures such as the variance or the coeffi-cient of variation. Stochastic orders are considered to be more informative than the comparison of two single numbers (seePreface in [14]). The comparison of random quantities has been developed in the last 40 years, and for a detailed discussionon the subject, the reader can refer to [14]. In the univariate case, the comparison of concentration can be related to theconvex, star-shaped and superadditive and Lorenz orders. Some other univariate stochastic orders related to the previousones are the decreasing mean residual life (DMRL) and new better than used in expectation (NBUE) orders, as can be seenin [17] and [18]. In the multivariate case, a paper in the direction considered here is the one by Roy [10]. The comparisonof dispersion is related to the dispersive and right-spread orders, and some proposals have been made in the multivariatecase. In particular, the reader can refer to [19–24] for the introduction and discussion of several definitions.
The purpose of the present work was to propose and study new generalizations of the univariate dispersive and right-spread orders and the univariate DMRL and NBUE orders. The key point of this paper is an observation made in Section 2,where we show that the multivariate orders provided by Roy [10] and the multivariate hazard rate order provided by Hu,Khaledi and Shaked [25] can be defined in a unified way in terms of marginal conditional distributions. Definitions andresults that will be used along the paper are given in Section 2. Section 3 contains the definitions and the first propertiesof the new multivariate extensions for the dispersive, right-spread, DMRL and NBUE orders. Section 4 deals with therelationships between these new multivariate orders and some orders already known. Finally, in Section 5, we concludewith some examples and applications.
2. Preliminaries on stochastic orders
First, we recall some definitions of univariate stochastic orders that will be used along the paper. The reader can refer to[26] and [14] for definitions and properties.
Let X and Y be two non-negative and absolutely continuous random variables with distribution functions F and G,density functions f and g, hazard rate functions r � f=.1�F / and s � g=.1�G/, respectively. Denote by F � 1�F thesurvival function of a distribution function F and byG�1.p/D inffx W F.x/> pg the quantile function of the distributionfunction G. Let X and Y also have finite means, and let m.x/ D EŒX � xjX > x� and l.x/ D EŒY � xjY > x� be theirmean residual life functions and A.x/ D 1
x
R x0 r.t/dt and B.x/ D 1
x
R x0 s.t/dt be their hazard rate in average functions,
respectively.In the literature, there are several definitions of stochastic orders based on the quantile functions. In particular, we have
that
(a) X is said to be smaller than Y in the dispersive order (denoted by X 6disp Y ) if G�1F.x/ is a dispersive functionin the support of F ; that is, G�1F.y/�G�1F.x/> y � x where x 6 y in the support of F .
(b) X is said to be smaller than Y in the right-spread order (denoted by X 6rs Y ) if
Z 1F�1.p/
F .x/dx 6Z 1G�1.p/
G.x/dx for all p 2 .0; 1/:
(c) X is said to be smaller than Y in the convex order (denoted by X 6c Y ) if G�1F.x/ is a convex function in thesupport of F .
(d) X is said to be smaller than Y in the star-shaped order (denoted byX 6� Y ) ifG�1F.x/ is a star-shaped function inthe support of F . Recall that a function � W .0;C1/ 7! .0;C1/, such that �.0/ D 0, is said to be star-shaped if�.x/=x is increasing in x 2 .0;C1/.
(e) X is said to be smaller than Y in the superadditive order (denoted byX 6su Y ) ifG�1F.x/ is a superadditive func-tion in the support of F . Recall that a function � W R 7! R is said to be superadditive if �.x C y/ > �.x/C �.y/for any x; y 2R.
The dispersive, right-spread and convex orders are of interest in reliability, given that they can be interpreted in termsof some common reliability measures. Characterizations of the dispersive, right-spread and convex orders in terms of thehazard rate and mean residual life function are the following:
X 6disp Y , r.F �1.p//> s.G�1.p// for allp 2 .0; 1/
X 6rs Y ,m.F �1.p//6 l.G�1.p// for allp 2 .0; 1/
X 6c Y ,r.F �1.p//
s.G�1.p//is an increasing function inp 2 .0; 1/ :
(2.1)
Motivated by the characterization of the convex ordering, Kochar and Wiens [17] defined and studied the followingunivariate orders based on the mean residual life function:
(f) X is said to be smaller than Y in the DMRL order (denoted by X 6dmrl Y ) if mX .F�1.p//
mY .G�1.p//is decreasing in
p 2 .0; 1/.
(g) X is said to be smaller than Y in the NBUE order (denoted by X 6nbue Y ) if mX .F�1.p//
mY .G�1.p//6 E.X/
E.Y /for all p 2 .0; 1/.
Among the previous orders, we have the following relationships:
6c ) 6� ) 6su+ +6dmrl ) 6nbue
(2.2)
and
6disp)6rs : (2.3)
Finally, we consider two stochastic orders based on comparisons of the survival and hazard rate functions, which willbe used along the paper:
(h) X is said to be smaller than Y in the usual stochastic order (denoted by X 6st Y ) if F .x/6G.x/ for all x 2R.(i) X is said to be smaller than Y in the hazard rate order (denoted by X 6hr Y ) if r.t/> s.t/ for all t 2R.
All these orderings are transitive, reflexive and antisymmetric.Stochastic orders are related to the study of aging [27,28]. Next, we recall the relationship between some of the previous
orders and some aging notions (see [17, 29] for more details).
Theorem 2.1Let X and Y be non-negative random variables where Y is exponentially distributed with survival function given byG.x/D 1� e�x . Then, X is IFR [IFRA, NBU, DMRL, NBUE] if, and only if, X 6cŒ�;su;dmrl;nbue� Y.
Several attempts have been made to extend some of the previous orders to the multivariate case. For example, Shakedand Shanthikumar [30] and Fernández-Ponce and Suárez-Llorens [21] proposed multivariate dispersive orders based onthe standard construction. Roy [10] proposed definitions of the multivariate convex, star-shaped and superadditive orders(denoted by 6mc , 6m� and 6msu, respectively). Hu, Khaledi and Shaked [25] proposed a definition for the multivariate(weak) hazard rate order denoted by 6whr .
The multivariate convex, star-shaped, superadditive and weak hazard rate orders can be characterized in terms of theconditional distributions 0
@Xiˇˇ\j¤i
fXj > xj g
1A and
0@Yi
ˇˇ\j¤i
fYj > xj g
1A :
In particular, we have that X is smaller than Y in the multivariate convex [star-shaped, superadditive, weak hazard rate]order (X6mcŒm�;msu;whr� Y) if, for all .x1; : : : ; xn/ and each i D 1; : : : ; n,0
@Xiˇˇ\j¤i
fXj > xj g
1A6cŒ�;su;hr�
0@Yi
ˇˇ\j¤i
fYj > xj g
1A : (2.4)
These characterizations give us one of the main applications of this approach. In these characterizations, we compareunivariate marginal distributions, given some information for the rest of the components. In particular, if the componentsof the random vector denote the time to fail or to death of some units, systems or organisms, then we compare the univari-ate times to fail when we have the survival data information for the rest of the components. The previous definitions andcharacterizations suggest the definition of multivariate generalizations of some univariate orders, according to the previouscharacterization. In the next section, in particular, we introduce new notions when we consider the dispersive, right-spread,DMRL and NBUE orders.
3. New multivariate orderings based on conditional distributions
In this section, we introduce the new multivariate orders and study some of their properties. First, we consider thedispersive and right-spread orders, and later, we consider the DMRL and NBUE orders.
Given a random vector XD .X1; : : : ; Xn/ and .x1; : : : ; xn/ 2Rn, for i D 1; : : : ; n, let us denote by Fi .�jxi /, F �1i .�jxi /,F i .�jxi /, ri .�jxi /,Ai .�jxi / andmi .�jxi /, the distribution function, quantile function, survival function, hazard rate function,hazard rate in average function and mean residual life function, respectively, of the conditional random variable0
@Xiˇˇ\j¤i
fXj > xj g
1A ;
where xi D .x1; : : : ; xi�1; xiC1; : : : ; xn/ 2 Rn�1. Recall that r.x1; : : : ; xn/ D .r1.x1jx1/; : : : ; rn.xnjxn// andm.x1; : : : ; xn/ D .m1.x1jx1/; : : : ; mn.xnjxn// are known as the hazard gradient and the multivariate mean residual lifeof X, respectively [31, 32].
Analogously, for a random vector Y D .Y1; : : : ; Yn/, we denote by GIi .�jxi /, GI�1i .�jxi /, GIi .�jxi /, si .�jxi /, Bi .�jxi /and li .�jxi / the distribution function, quantile function, survival function, hazard rate function, hazard rate in averagefunction and mean residual life function, respectively, of the corresponding conditional random variable.
3.1. New multivariate dispersive and right-spread orderings and properties
In this section, we consider the generalization of the dispersive and right-spread orders according to (2.4).
Definition 3.2Let XD .X1; : : : ; Xn/ and YD .Y1; : : : ; Yn/ be two random vectors.
(i) X is said to be smaller than Y in the multivariate dispersive ordering (denoted by X 6mdisp Y) if, for all.x1; : : : ; xn/ 2Rn and each i D 1; : : : ; n,0
@Xiˇˇ\j¤i
fXj > xj g
1A6disp
0@Yi
ˇˇ\j¤i
fYj > xj g
1A :
(ii) X is said to be smaller than Y in the multivariate right-spread ordering (denoted by X 6mrs Y) if, for all.x1; : : : ; xn/ 2Rn and each i D 1; : : : ; n,0
@Xiˇˇ\j¤i
fXj > xj g
1A6rs
0@Yi
ˇˇ\j¤i
fYj > xj g
1A :
It is not difficult to see their transitivity, reflexivity and closure under marginalization and under conjunctions, as followsin the next result.
Proposition 3.3Let XD .X1; : : : ; Xn/ be a random vector, then X6mdispŒmrs� X.
(i) Let XD .X1; : : : ; Xn/, YD .Y1; : : : ; Yn/ and ZD .Z1; : : : ; Zn/ be three random vectors. If X6mdispŒmrs� Y andY6mdispŒmrs� Z, then X6mdispŒmrs� Z.
(ii) Let XD .X1; : : : ; Xn/ and YD .Y1; : : : ; Yn/ be two random vectors. Let I D .i1; : : : ; ir/� .1; : : : ; n/, 16 r 6 nand XI D .Xi1 ; : : : ; Xir / and YI D .Yi1 ; : : : ; Yir /. If X 6mdispŒmrs� Y, then XI 6mdispŒmrs� YI . That is, themdisp and mrs orders are closed under marginalization.
(iii) Let X1; : : : ;Xn be a set of independent random vectors where the dimension of Xi is ki , i D 1; : : : ; n.Let Y1; : : : ;Yn be a set of independent random vectors where the dimension of Yi is ki , i D 1; : : : ; n. IfXi 6mdispŒmrs� Yi for i D 1; : : : ; n, then (X1; : : : ;Xn/ 6mdispŒmrs� .Y1; : : : ;Yn/. That is, the mdisp and mrsorders are closed under conjunctions.
Next, we explore further the new multivariate orders. For the sake of simplicity, the comments will be given for bivariatedistributions, but similar comments hold for the n-dimensional case.
For a bivariate random vector .X1; X2/, we mean by positive dependence that X1 and X2 are likely to be large or tobe small together. We found an excellent exposition of all positive dependence concepts in [3]. Most of the dependence
concepts in the literature can be given in terms of the stochastic comparisons among conditional distributions and, inparticular, for the conditional distributions .X2jX1 > x/ and .X1jX2 > x/. For example, if
ŒX2 jX1 > x1�6st ŒX2 jX1 > x2�; for all x1 6 x2; (3.5)
then we say that .X1; X2/ is RTI.X2 j X1/ (right-tail increasing). The reason (3.5) is a positive dependence condition isthat for (3.5), X2 is more likely to take on larger values as X1 increases. The comparison of conditional distribution, insome of the orders considered previously, can be considered as a comparison of the degree of dependence of two vectors.Let us consider now two bivariate random vectors X D .X1; X2/ and Y D .Y1; Y2/ such that X 6mdisp Y or X 6mrs Y;then it is clear that .X2 jX1 > x/ is less variable or dispersed than .Y2 j Y1 > x/ for all x, and the same hold, replacingX1 by X2 and Y1 by Y2. This fact can be interpreted as a comparison of the degree of dependence in terms of variability.
From (2.1), we can easily provide characterizations of the multivariate convex order and the new multivariate dispersiveand right-spread orders, in terms of the hazard gradient and multivariate mean residual life.
Proposition 3.4Let XD .X1; : : : ; Xn/ and YD .Y1; : : : ; Yn/ be non-negative two random vectors.
(i) X6mc Y if, and only if, for all xi 2Rn�1C ,
ri .F�1i .pi jxi /jxi /
si .GI�1i .pi jxi /jxi /
is a non-decreasing function in pi , for all .p1; : : : ; pn/ 2 .0; 1/n and i D 1; : : : ; n.(ii) X6mdisp Y if, and only if, for all xi 2Rn�1,
ri .F�1i .pi jxi /jxi /> si .GI�1i .pi jxi /jxi /;
for all .p1; : : : ; pn/ 2 .0; 1/n and i D 1; : : : ; n.(iii) X6mrs Y if, and only if, for all xi 2Rn�1,
mi .F�1i .pi jxi /jxi /6 li .GI�1i .pi jxi /jxi /;
for all .p1; : : : ; pn/ 2 .0; 1/n and i D 1; : : : ; n.
Next, we provide a result for the preservation under transformations of the new multivariate dispersive and right-spreadorders.
Proposition 3.5Let X D .X1; : : : ; Xn/ and Y D .Y1; : : : ; Yn/ be two random vectors, and let �1; : : : ; �n W R! R be univariate strictlyincreasing convex or strictly decreasing concave functions. If X6mdispŒmrs� Y, then
By Theorem 3.B.10 in [14], it holds that0@�i .Xi /
ˇˇ\j¤i
fXj > ��1j .xj /g
1A6disp
0@�i .Yi /
ˇˇ\j¤i
fYj > ��1j .xj /g
1A : (3.8)
Clearly, condition (3.8) is equivalent to condition (3.6).The case dealing with the multivariate right-spread order is similar, using Theorem 3.C.4, to that in [14]. �
3.2. New multivariate decreasing mean residual life and new better than used in expectation orderings and properties
In this case, we consider multivariate extensions of the DMRL and NBUE orders.
Definition 3.6Let XD .X1; : : : ; Xn/ and YD .Y1; : : : ; Yn/ be two non-negative random vectors.
(i) X is said to be smaller than Y in the multivariate DMRL ordering (denoted by X 6mdmrl Y) if, for all.x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n,0
@Xiˇˇ\j¤i
fXj > xj g
1A6dmrl
0@Yi
ˇˇ\j¤i
fYj > xj g
1A :
(ii) X is said to be smaller than Y in the multivariate NBUE ordering (denoted by X 6mnbue Y) if, for all.x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n,0
@Xiˇˇ\j¤i
fXj > xj g
1A6nbue
0@Yi
ˇˇ\j¤i
fYj > xj g
1A :
It is not difficult to see their transitivity, reflexivity and closure under marginalization and under conjunctions, as followsin the next result.
Proposition 3.7 (i) Let XD .X1; : : : ; Xn/ be a non-negative random vector. Then X 6mdmrlŒmnbue� X.(ii) Let X D .X1; : : : ; Xn/, Y D .Y1; : : : ; Yn/ and Z D .Z1; : : : ; Zn/ be three non-negative random vectors. Then, if
X6mdmrlŒmnbue� Y and Y6mdmrlŒmnbue� Z, X6mdmrlŒmnbue� Z.(iii) Let X D .X1; : : : ; Xn/ and Y D .Y1; : : : ; Yn/ be two non-negative random vectors. Let I D .i1; : : : ; ir/ �
.1; : : : ; n/, 1 6 r 6 n and XI D .Xi1 ; : : : ; Xir / and YI D .Yi1 ; : : : ; Yir /. If X 6mdmrlŒmnbue� Y, thenX.1/ 6mdmrlŒmnbue� Y.1/. That is, the MDMRL and MNBUE orders are closed under marginalization.
(iv) Let X1; : : : ;Xn be a set of independent non-negative random vectors where the dimension of Xi is ki , i D 1; : : : ; n.Let Y1; : : : ;Yn be a set of independent random vectors where the dimension of Yi is ki , i D 1; : : : ; n. IfXi 6mdmrlŒmnbue� Yi for i D 1; : : : ; n, then .X1; : : : ;Xn/ 6mdmrlŒmnbue� .Y1; : : : ;Yn/. That is, the MDMRLand MNBUE orders are closed under conjunctions.
From (2.2), it holds the following relationships between the multivariate orders:
6mc ) 6m� ) 6msu+ +
6mdmrl ) 6mnbue :
As in the case of multivariate dispersive and right-spread orders, we explore further the meaning of these orders andprovide some potential applications.
Next, we provide a characterization of the new multivariate DMRL and NBUE orders based on the multivariate meanresidual life. The proof follows from definitions of the univariate DMRL and NBUE orders and it is omitted.
Proposition 3.8Let XD .X1; : : : ; Xn/ and YD .Y1; : : : ; Yn/ be two non-negative random vectors.
(i) X6mdmrl Y if, and only if, for all xi 2Rn�1C ,
mi .F�1i .pi jxi /jxi /
li .GI�1i .pi jxi /jxi /
is a non-increasing function in pi , for all .p1; : : : ; pn/ 2 .0; 1/n and i D 1; : : : ; n.(ii) X6mnbue Y if, and only if, for all xi 2Rn�1C ,
mi .F�1i .pi jxi /jxi /
li .GI�1i .pi jxi /jxi /
6 EŒXi jXj > xj ; j ¤ i �EŒYi jYj > xj ; j ¤ i �
;
for all .p1; : : : ; pn/ 2 .0; 1/n and i D 1; : : : ; n.
Roy [10] shows the relationships among some multivariate aging notions and the multivariate convex, star-shaped andsuperadditive orders. Given an n-dimensional random vector X, Roy [7] defines the following multivariate classes of lifedistributions:
(i) X is said to be MIFR [MDFR] if ri .xi jxi / is an increasing [decreasing] function in xi for all xi 2 Rn�1C and eachi D 1; : : : ; n.
(ii) X is said to be MIFRA [MDFRA] if Ai .xi jxi / is an increasing [decreasing] function in xi for all xi 2 Rn�1C andeach i D 1; : : : ; n.
(iii) X is said to be MNBU [MNWU] if, for all .x1; : : : ; xn/ 2RnC, yi > 0, i D 1; : : : ; n, it holds that
F .x1; : : : ; xi�1; xi C yi ; xiC1; : : : ; xn/F .x1; : : : ; xi�1; 0; xiC1; : : : ; xn/
(iv) X is said to be MDMRL [MIMRL] if mi .xi jxi / is a decreasing [increasing] function in xi for all xi 2 Rn�1C andeach i D 1; : : : ; n.
(v) X is said to be MNBUE [MNWUE] if
mi .xi jxi /6 Œ>�mi .0jxi /DEŒXi jXj > xj ; j ¤ i �
for all xi 2Rn�1C and each i D 1; : : : ; n.
Consider now a random vector Y with exponential multivariate distribution type Gumbel, that is, a random vector Ywith survival function given by the following expression [33]:
G.x1; : : : ; xn/D exp
24�X
i
�ixi �Xi>j
X�ijxixj � � � � � �1;2;:::;n
Yj¤i
xj
35 ; (3.9)
where .x1; : : : ; xn/ 2RnC.Roy [10] characterizes the classes MIFR, MIFRA and MNBU by using this multivariate extension of the exponential
distribution as follows.
Theorem 3.9Let X and Y be two non-negative random vectors where Y is a random vector type Gumbel with survival function givenby (3.9). Then
(i) X is MIFR if, and only if, X6mc Y.(ii) X is MIFRA if, and only if, X6m� Y.
(iii) X is MNBU if, and only if, X6msu Y.
We observe that these characterizations are analogous to those provided in Theorem 2.1. We give an analoguecharacterization based on the new multivariate orders DMRL and NBUE.
Theorem 3.10Let X D .X1; : : : ; Xn/ be a non-negative random vector with distribution function F.x/, and let Y be a Gumbel randomvector, that is, with survival function given by (3.9). Then
(i) X is MDMRL if, and only if, X6mdmrl Y.(ii) X is MNBUE if, and only if, X6mnbue Y.
Proof
(i) X6mdmrl Y holds if, and only if, for all .x1; : : : ; xn/ 2RnC and each i D 1; 2; : : : ; n,0@Xi
ˇˇ\j¤i
fXj > xj g
1A6dmrl
0@Yi
ˇˇ\j¤i
fYj > xj g
1A :
Besides,�Yi
ˇTj¤i fYj > xj g
�are exponential random variables. Thus, by the characterization of the univariate
class DMRL, we have that�Xi
ˇTj¤i fXj > xj g
�are DMRL random variables; that is, their mean residual life
functions mi .xi jxi / are increasing in xi for all xi 2 Rn�1C and each i D 1; : : : ; n, and this implies, by definition,that the random vector X is MDMRL.
(ii) X6mnbue Y if, and only if, for all .x1; : : : ; xn/ 2RnC and each i D 1; 2; : : : ; n,0@Xi
ˇˇ\j¤i
fXj > xj g
1A6nbue
0@Yi
ˇˇ\j¤i
fYj > xj g
1A :
The random variables
Yi
ˇˇ Tj¤i
fYj > xj g
!are exponential random variables. Thus, by the characterization of the
univariate class NBUE,
Xi
ˇˇ Tj¤i
fXj > xj g
!are random variables NBUE, and this implies, by definition, that the
random vector X is MNBUE.
�
4. Relationships among multivariate orders
In this section, we provide relationships among the new orders and previous stochastic orders.The multivariate dispersive order is related with the multivariate star-shaped order as is shown in the next proposition.
Proposition 4.11Let X D .X1; : : : ; Xn/ and Y D .Y1; : : : ; Yn/ be two non-negative random vectors, then X 6m� Y if, and onlyif, .logX1; : : : ; logXn/ 6mdisp .logY1; : : : ; logYn/. Equivalently, X 6mdisp Y if, and only if, .eX1 ; : : : ; eXn/ 6m�.eY1 ; : : : ; eYn/.
ProofIt holds that X6m� Y if, and only if,0
@Xiˇˇ\j¤i
fXj > xj g
1A6�
0@Yi
ˇˇ\j¤i
fYj > xj g
1A for all .x1; : : : ; xn/ 2R
nC and each i D 1; : : : ; n: (4.10)
From Theorem 4.B.1 in [14], it follows that (4.10) is equivalent to0@logXi
ˇˇ\j¤i
fXj > xj g
1A6disp
0@logYi
ˇˇ\j¤i
fYj > xj g
1A (4.11)
for all .x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n.If we consider x0j D log xj for j ¤ i , then (4.11) becomes0
for all .x01; : : : ; x0n/ and each i D 1; 2; : : : ; n.
The last inequality is equivalent to .logX1; : : : ; logXn/6mdisp .logY1; : : : ; logYn/. Thus, the assertion is proved. �Under the condition that the conditional random variables are ordered in the usual stochastic order, the multivariate
superadditive order implies the multivariate dispersive order as follows.
Proposition 4.12Let XD .X1; : : : ; Xn/ and YD .Y1; : : : ; Yn/ be two non-negative random vectors such that0
@Xiˇˇ\j¤i
fXj > xj g
1A6st
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
for all .x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n. If X6msu Y, then X6mdisp Y.
ProofBecause 0
@Xiˇˇ\j¤i
fXj > xj g
1A6st
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
and 0@Xi
ˇˇ\j¤i
fXj > xj g
1A6su
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
for all .x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n, Proposition 4.B.2 in [14] shows that0@Xi
ˇˇ\j¤i
fXj > xj g
1A6disp
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
or, equivalently, X6mdisp Y. �
Another weaker condition, under which X6msu Y implies X6mdisp Y, is given in the next proposition.
Proposition 4.13Let X D .X1; : : : ; Xn/ and Y D .Y1; : : : ; Yn/ be two non-negative random vectors. If X 6whr Y and X 6msu Y, thenX6mdisp Y.
ProofThe condition X6whr Y is equivalent to0
@Xiˇˇ\j¤i
fXj > xj g
1A6hr
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
for all .x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n. By Theorem 1.B.1 in [14], it holds that0@Xi
ˇˇ\j¤i
fXj > xj g
1A6st
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
for all .x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n. The stated comparison now follows from Proposition 4.12. �If we suppose that fXjX> xg6st fYjY> xg for all xD .x1; : : : ; xn/ 2RnC, we obtain the following corollary.
Corollary 4.14Let X D .X1; : : : ; Xn/ and Y D .Y1; : : : ; Yn/ be two non-negative random vectors. If fXjX > xg 6st fYjY > xg for allxD .x1; : : : ; xn/ 2RnC and X6msu Y, then X6mdisp Y.
ProofFirst, we observe that the condition fXjX > xg 6st fYjY > xg, for all x 2 RnC, implies, from the preservation under
marginalization of the multivariate stochastic order, the condition
fXi jX> xg6st fYi jY> xg;
for all xD .x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n, which is equivalent to0@Xi
ˇˇ\j¤i
fXj > xj g
1A6hr
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
for all xD .x1; : : : ; xn/ 2RnC and each i D 1; : : : ; n; that is, X6whr Y, and the result follows from Proposition 4.13. �In contrast of the aforementioned conditions, supposing the next limiting property, we have the following proposition.
Proposition 4.15Let X D .X1; : : : ; Xn/ and Y D .Y1; : : : ; Yn/ be two non-negative random vectors such that X 6msu Y. If it holds, foreach i D 1; : : : ; n and xi 2Rn�1C , that
limxi!0
G�1i .Fi .xi jxi /jxi /xi
> 1;
then X6mdisp Y.
ProofThe proof is straightforward from Theorem 4.B.3 in [14]. �
The next proposition lists some relations between some of the multivariate stochastic orders under aging multivariateconditions.
Proposition 4.16Let X D .X1; : : : ; Xn/ and Y D .Y1; : : : ; Yn/ be two non-negative random vectors. We assume, in (iv) and (v), that for
any .x1; : : : ; xn/ 2 RnC and for all i D 1; : : : ; n, the left extreme of the support of
Xi
ˇˇ Tj¤i
fXj > xj g
!is smaller than
the left extreme of the support of
Yi
ˇˇ Tj¤i
fYj > xj g
!.
(i) If X6whr Y and X or Y is MDFR, then X6mdisp Y.(ii) If X6mdisp Y and X or Y is MIFR, then X6whr Y.
(iii) If X is MNBU and Y is MNWU, then X6mdisp Y if, and only if, X6whr Y.(iv) If X6mrs Y and X or Y is MDMRL, then X6wmrl Y.(v) If X6wmrl Y and X or Y is MIMRL, then X6mrs Y, where X6wmrl Y if, and only if,
mi .xi jxi /6 li .xi jxi / for all xi 2Rn�1C and each i D 1; : : : ; n:
Proof
(i) Let us suppose that X is MDFR, which means that ri .xi jxi / is a decreasing function in xi for all xi 2 Rn�1C andeach i D 1; : : : ; n. Condition X6whr Y is equivalent to0
@Xiˇˇ\j¤i
fXj > xj g
1A6hr
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
By Theorem 3.B.20 in [14], we can state that0@Xi
ˇˇ\j¤i
fXj > xj g
1A6disp
0@Yi
ˇˇ\j¤i
fYj > xj g
1A ;
and by definition, this is equivalent to X6mdisp Y. The case when Y is MDFR is proved in a similar way.
(ii) Let us suppose that X is MIFR, which means that ri .xi jxi / is a decreasing function in xi for all xi 2 Rn�1C andeach i D 1; : : : ; n. Besides, X6mdisp Y, which is equivalent to0
@Xiˇˇ\j¤i
fXj > xj g
1A6disp
0@Yi
ˇˇ\j¤i
fYj > xj g
1A
By Theorem 3.B.20 in [14], we can state that0@Xi
ˇˇ\j¤i
fXj > xj g
1A6hr
0@Yi
ˇˇ\j¤i
fYj > xj g
1A ;
and by definition, this is equivalent to X6whr Y. The case when Y is MIFR is proved in a similar way.(iii) This statement can be proved in a similar way by using Theorem 3.B.20 c) in [14].(iv) Analogously, this statement can be proved in a similar way by using Theorem 3.C.5 in [14].(v) Analogously, this statement can be proved in a similar way by using Theorem 3.C.6 in [14].
�
5. Applications and examples
In this section, we provide several situations where these multivariate stochastic orders can be applied and examples ofparametric families of random vectors ordered according to some of the multivariate orders considered and that satisfysome of the multivariate aging notions considered in the paper. Recall that these multivariate aging notions can be char-acterized in terms of some multivariate stochastic orders and therefore can be seen as examples of comparisons in somemultivariate orders when one of the random vectors follows a multivariate Gumbel distribution. We want to point out alsothat we have considered only the case where the conditional random variables are of the type .X jY > x/, so we restrictourselves to situations where these conditional random variables are of interest. Anyway, it is possible to carry out a simi-lar study considering conditional random variables of the type .X jY 6 y/ or .X jY D x/ and some other situations wherethese type of conditioning could be of interest.
5.1. Comparisons of risks
In the context of actuarial theory, a non-negative random variable X represents the random amount that an insurance com-pany will pay to a policyholder, in case of claim. The comparison of risks is carried out through the comparison of somemeasures of interest. Two common measures are the value at risk and expected shortfall notions, which are evaluated at anypoint p 2 .0; 1/. The value at risk is simply the quantile function; that is, the value at risk is given by VaRŒX; p�� F �1.p/.The expected shortfall is the right-spread function; that is, ESŒX Ip��EŒ.X�F �1.p//C�D
R C1F�1.p/ F .x/dx. The disper-
sive and right-spread orders provide comparisons of risks based on the value at risk and the expected shortfall. Accordingto this, two risky situations A and B with random risks XA and XB , respectively, can be ranked in terms of the dispersiveorder or the right-spread order. If XA 6disp XB or XA 6rs XB , then we can say that B is more risky than A (see [13]for notation and discussion on this topic). In some situations, insurance companies do not have only one policy for somepolicyholders but two policies. For example, a policyholder can have a policy to insure the car and another policy to ensurethe house, with random claims X and Y . It is clear that X and Y should exhibit some kind of dependence, for example,positive dependence. So let us consider two risky situations A and B for policyholders in the situation described previ-ously, with random risks .XA; YA/ and .XB ; YB/. If we can compare these two random variables in terms of the mdisp ormrs orders, that is, if .XA; YA/ 6mdisp .XB ; YB/ or .XA; YA/ 6mrs .XB ; YB/, then from Proposition 3.5.iii, we not onlycompare the marginal risks in the dispersive order or right-spread order, but also provide a more detailed and illustrativecomparison of the risks, taking into account the dependence among risks. In particular, from the definition, we can providecomparisons of the type
which compares risks with the additional information that the level of claim of the other risk is above some certain thresh-old. Another example of this type is provided [34]. In this paper, the authors consider the following situation of insurancecompany indemnity claims. Each claim consists of an indemnity payment (the loss, X ) and an allocated loss adjustmentexpense (ALAE, Y ). Similar comments can be carried out for this case.
5.2. Comparisons of inequality and deprivation
Another application can be found in the context of inequality. First, we recall that the NBUE order implies the Lorenzorder. The Lorenz order is defined in terms of the Lorenz curve. The Lorenz curve appears in the context of income dis-tributions, where the incomes of the individuals of a population are considered (non-negative) random quantities. Given anon-negative random variable X with finite mean, the Lorenz curve is defined as
LX .p/D
R p0 F
�1.u/du
EŒX�forp 2 .0; 1/:
Now given two non-negative random variables X and Y , with finite means, we say that X is less than Y in the Lorenzorder, denoted by X �L Y if
LX .p/> LY .p/ for all p 2 .0; 1/:
As mentioned previously, the NBUE order implies the Lorenz order; that is, X 6nbue Y ) X 6L Y [17]. TheLorenz order and also the star, DMRL and NBUE orders are related to the comparison of concentration of two randomvariables. In particular, if X 6L Y , then the coefficient of variation of X is smaller than the corresponding coefficientfor Y . Also, it preserves the comparison of the Gini index. The Gini index, for a random variable X , is defined as twicethe area between the Lorenz curve and the ‘equalitarian line’ LX .p/ D p; that is, GIX D 1 � 2
R 10 LX .p/dp. In the
context of income distributions, the Lorenz curve and the Gini index are the most popular tools to analyse and compareincome inequality. However, they can be used only to compare random variables, but in some situations, the interestcan be focused on more than one random variable to describe the inequality in a population. Consider, for example, thefollowing situation. To study the inequality in a population, researchers can be interested not only in the household’sincome per year (X ) but also in the household’s properties (Y ). Clearly these two random variables are related. How can wecompare now the inequality among two different populations, taking into account these two random variables? Let us con-sider two countries A and B with random vectors .XA; YA/ and .XB ; YB/, respectively, as described previously. Thecomparison of the marginal Gini indexes is an incomplete study that does not take into account the dependence struc-ture of the two random variables. Another possibility is to compare not only the marginal Gini indexes but also theGini index of the incomes, given that the value of the properties is above any threshold x, that is, to compare GIXA.x/DGI.XAjYA>x/ withGIXB .x/DGI.XB jYB>x/ and a similar comparison for the Gini indexesGIYA.x/DGI.YAjXA>x/ withGIYB .x/D GI.YB jXB>x/. From previous discussion and previous results, we have that if .XA; YA/ 6mdmrl .XB ; YB/ or.XA; YA/6mnbue .XB ; YB/, then
GIXA.x/6GIXB .x/ andGIYA.x/6GIYB .x/ for all x > 0:
Recall that .XA; YA/ is MNBUE (MNWUE) if, and only if, .XA; YA/6mnbue .>mnbue/.XB ; YB/, where .XB ; YB/ hasa joint survival function given by (3.9). In this case, it is not difficult to see that GIXB .x/ D 0:5 D GIYB .x/. Therefore,GIXA.x/ and GIXB .x/ are bounded from above (below) by 0.5. It is very interesting to identify this situation becauseGini indexes greater than 0.5 suggest problems of inequality.
Let us see another example of this situation. Policymakers and researchers are often interested in assessing the changesin inequality and poverty over time. An example is the case studied in [35], which considers several parametric modelsfor the joint distribution of incomes for different years across the same population. In particular, they found that the bestbivariate approximation for incomes in Australia for 2001 and 2005, denoted by .AI2001; AI2005/, is the one provided by aGumbel copula with Singh–Maddala margins (see [35] for details of the estimated parameters). Following previ-ous discussion, we consider the conditional Gini indexes GIAI2001.x/ D GI.AI2001jAI2005>x/ and GIAI2005.x/ DGI.AI2005jAI2001>x/. Given that there is no closed expression for these values, we have considered the estimated val-ues for these quantities based on bivariate samples for this model of size 1000. The estimated values can be found inFigure 3, which shows that GIAI2001.x/ and GIAI2005.x/ are less than 0.5.
The comparison in the mdisp and mrs orders is related to the comparison of deprivation in the context of incomes andwealth. As can be seen in [36], an indicator of relative deprivation of an individual with income F �1.p/, when comparinghimself or herself to another individual with income F �1.q/, is given by
Figure 3. Plots of estimated conditional Gini values for .AI2001; AI2005/.
ı.p; q/D .F �1.q/�F �1.p//C;
and the expected relative deprivation of an individual at rank p is given by
ı.p/D
Z 1
0
ı.p; q/dq DEŒ.X �F �1.p//C�:
Therefore, the comparison in the dispersive order and the right-spread order are comparisons of relative depriva-tion among two populations, and previous comments are valid for the mdisp and mrs orders in the context of incomedistributions.
5.3. Examples
Finally, we consider several parametric examples where some of the previous notions can be studied.
Example 5.17Farlie–Gumbel–Morgenstern distributions
The survival function of the bivariate Farlie–Gumbel–Morgenstern distribution is given by
F .x1; x2/D F 1.x1/F 2.x2/Œ1C ˛F1.x1/F2.x2/�;
where F1 and F2 are the marginal distributions and �1 6 ˛ 6 1. Let us denote by FGM˛ a bivariate random vectorwith joint survival function provided by previous expression. For this model, Hu, Khaledi and Shaked [25] showed thatFGM˛ 6whr FGM˛0 for ˛ 6 ˛0. Also, Johnson and Kotz [31] showed that under certain conditions, this model is MDFR.If this is the case for FGM˛ or FGM˛0 , then from Proposition 4.16, we get that FGM˛ 6mdispŒmrs� FGM˛0 . Recall alsothat if FGM˛ is MIFR (MDFR), then FGM˛ 6 .>/mcY where Y follows a Gumbel type distribution (see (3.9)). Johnsonand Kotz [31] describe several cases of MIFR and MDFR distributions when the marginals are Weibull distributed.
Applications of this model can be found in the context of health care usage [37].
Example 5.18Conditionally specified models
Arnold, Castillo and Sarabia [38] provide several examples of bivariate random vectors where the conditional marginaldistributions are given explicitly. Let us consider one of these examples. Let X D .X1; X2/ and Y D .Y1; Y2/ be twobivariate random vectors, and suppose that for each x1; x2 > 0, the conditional random variables X1 given X2 > x2 andX2 given X1 > x1 are Weibull distributed with survival functions as follows
P.X1 > x1jX2 > x2/D expf�x�1X1 .˛C �Xx
�2X2 /g
and
P.X2 > x2jX1 > x1/D expf�x�2X2 .ˇC �Xx
�1X1 /g;
respectively. The Gumbel bivariate exponential distribution corresponds to the choice �1X D �2X D 1.
Analogously, suppose that for each x1; x2 > 0, the conditional random variables Y1 given Y2 > x2 and Y2 given Y1 > x1are Weibull distributed with survival functions as follows
P.Y1 > x1jY2 > x2/D expf�x�1Y1 .ıC �Yx
�2Y2 /g
and
P.Y2 > x2jY1 > x1/D expf�x�2Y2 .�C �Xx
�1Y1 /g;
respectively.For fixed x2 > 0, the hazard rate and quantile functions of the conditional distribution .X1jX2 > x2/ are given,
respectively, by
r.X1jX2>x2/.x1/D �1X.˛C �Xx
�2X2 /x
�1X�1
1 (5.13)
and
F �1.X1jX2>x2/.p/D
24� logf1� pg
˛C �Xx�2X2
351=�1X
Analogously, we can compute the hazard rate and the quantile functions for the rest of the aforementioned conditionaldistributions.
From (2.1), it is not difficult to see that all the following conditions, for all x1; x2 > 0, are sufficient to obtainX6mdisp Y:
(i) �1X > �1Y,(ii) �2X > �2Y,
(iii) .˛C �Xx�2X2 /
1=�1X > .ıC �Yx�2Y2 /
1=�1Y
(iv) .˛C �Xx�2X2 /
1=�1X > .ıC �Yx�2Y2 /
1=�1Y .
It is easy to see that the random vector X is MIFR in the sense of Roy if �1X > 1 and �2X > 1.Similar results can be given for some of the examples considered in [38].
Example 5.19Multivariate aging properties for exchangeable random vectors
Let X D .X; Y / be an exchangeable random vector of non-negative marginal random variables with distribution func-tion FX � FY � F and hazard rate function rX � rY � r , and let H.x; y/ D P.X > x; Y > y/ be its correspondingsurvival function. If K.u1; u2/ is the bivariate survival copula, it holds that H.x; y/DK.F .x/; F .y// [39].
Let .V1; V2/ be a random vector with uniform marginals and satisfying P.V1 > v1; V2 > v2/ D K.1� v1; 1� v2/ forall v1; v2 2 Œ0; 1�; that is, K is its survival copula.
For the sake of simplicity, let us denote by f .xjy/, F .xjy/ and r.xjy/, the density, survival and hazard rate functionsof fX jY > yg, respectively. Moreover, we consider the notation Ku1 �
@@u1K.
Theorem 5.20If the marginal random variable X is IFR and
.1�v1/Ku1 .1�v1;1�v2/
K.1�v1;1�v2/is an increasing function in v1 for all v2 2 Œ0; 1�, then
X is MIFR in the sense of Roy.
ProofNote that in case of exchangeability, X is MIFR in the sense of Roy if r.xjy/ is an increasing function in x for all y. Thesurvival function of fX jY > yg is given by
�Let us note that, fixed y 2R, it exists v2 2 Œ0; 1� such as F �1.v2/D y. From the increasingness of the quantile function,
it is easy to see that r.xjF �1.v2// is an increasing function in x for all v2 2 R if, and only if, r.F �1.v1/jF �1.v2// is anincreasing function in v1 for all v1 2 Œ0; 1�.
Taking into account these observations, we have that
r.F �1.v1/jF�1.v2//D r.F
�1.v1//.1� v1/Ku1.1� v1; 1� v2/
K.1� v1; 1� v2/:
Because X is IFR, then r.F �1.v1// is an increasing function in v1. On the other hand, by hypothesis,
.1� v1/Ku1.1� v1; 1� v2/
K.1� v1; 1� v2/
is an increasing function in v1 for all v2 2 Œ0; 1�. Putting these results together, we conclude that r.xjy/ is an increasingfunction in x for all y 2R, and the statement is proved. �
If we consider that K is the Clayton copula, then
.1� v1/Ku1.1� v1; 1� v2/
K.1� v1; 1� v2/D
.1� v1/�1=�
.1� v1/�1=� C .1� v2/�1=� � 1
is increasing in v1 for all v2 2 Œ0; 1�. In this way, if X and Y are IFR and if K is the Clayton copula, then XD .X; Y / is arandom vector MIFR in the sense of Roy.
Theorem 5.21If the marginal random variable X is IFRA and � logK.1�u1;1�u2/
� log.1�u1/is an increasing function in v1 for all v2 2 Œ0; 1�, then X
is MIFRA in the sense of Roy.
ProofIn this case, X is MIFRA in the sense of Roy if, and only if, fX jY > yg is IFRA for all y; that is, X is MIFRA if, and onlyif,
� logF .xjy/
x
is an increasing function in y. Taking into account that the survival function of fX jY > yg is given by (5.19), this isequivalent to say that
�.x/D� logK.F .x/; F .y/
x
is increasing in x for all y. Recall that �.x/ is increasing in x if, and only if,
�.F �1.u1//D� logK.1� u1; F .y//
F �1.u1/
is increasing in u1. Let u2 2 Œ0; 1� be such that F �1.u2/D y, then
�.F �1.u1//D� logK.1� u1; 1� u2/
F �1.u1/D� logK.1� u1; 1� u2/
� log.1� u1/
� log.1� u1/
F �1.u1/:
Note that the marginal random variable X is IFRA if, and only if, � logF .t/t
is increasing in t or, equivalently, if, and
only if, � log.1�u1/F�1.u1/
is increasing in u1. On the other hand, we suppose that � logK.1�u1;1�u2/� log.1�u1/
is increasing in v1. Putting
these results together, we can conclude that �.F �1.u1// is an increasing function in u1, and the proof is obtained. �
The new criteria, to compare multivariate distributions, considered in this paper suggest a way to compare multivariatedistributions in terms of conditional distributions of the type0
@Xiˇˇ\j¤i
fXj > xj g
1A :
This is in the spirit of previous works by, for example, Roy [10] and Hu, Khaledi and Shaked [25]. We have focused ourattention to the comparison of these random variables in terms of dispersion and concentration, and therefore, the mainfields of applications are reliability, risks and inequality. This approach can be used in other contexts, changing the typeof conditioning, for example, 0
@Xiˇˇ\j¤i
fXj � xj g
1A or
0@Xi
ˇˇ\j¤i
fXj D xj g
1A
or the comparison criteria, for example, in terms of variability, entropy and others.
Acknowledgements
We are thankful to an anonymous referee for his valuable comments, which have improved the presentation and contents of thispaper. Félix Belzunce, Julio Mulero and José M. Ruiz acknowledge support received from the Ministerio de Ciencia e Innovaciónunder grant MTM2009-08311 and the Fundación Séneca (CARM 08811/PI/08), and Alfonso Suárez-Llorens acknowledges supportreceived from the Ministerio de Ciencia e Innovación under grant MTM2009-08326.
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