Applied Mathematics and Computation 146 (2003) 601–610 www.elsevier.com/locate/amc
On stochastic inequalities and dependence orderings Broderick O. Oluyede Department of Mathematics and Computer Science, Georgia Southern University, P.O. Box 8093, Statesboro, Georgia 30460-8093, USA
Abstract Inequalities, local dependence orderings of reliability or survival functions of random variables are introduced and discussed. The notions of hazard dependence, stochastic orderings and exchangeability are studied and various important inequalities obtained. Examples and applications including inequalities and association for families of distribution functions satisfying the notions of stochastic and hazard dependence are presented. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Positive dependence; Families of distributions; Stochastic ordering
1. Introduction Dependent random variables exhibit the property that values of one variable, large or small tend to be accompanied or associated systematically with large or small values of other variable. We present results on the dependence structure for a general class of families of distributions. It is natural to investigate whether distributions are more locally dependent according to some dependence ordering as the parameter increases or decreases. There are several notions of dependence in the literature, notable are those given by Lehmann [5], Karlin [3], Esary and Proschan [1], Shaked and Shanthikumar [8] to mention just a few. See references therein. In this note, some basic concepts of stochastic and hazard dependence of random variables are defined and compared. We present results on local positive E-mail address:
[email protected] (B.O. Oluyede). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00606-9
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dependence as well as hazard dependence ordering. It is shown that these orderings have the generalized monotone invariance property. Examples and applications of families of distributions satisfying the properties of dependence are given. In Section 2, utility notions, basic definitions and stochastic dependence are given and compared. In Section 3, we present results on stochastic inequalities and exchangeability. In Section 4, examples and applications of families of distribution function satisfying the notions of dependence are presented.
2. Utility notions and dependence In this section we give definitions and some utility notions of dependence. We establish relationship between the notion right corner set increasing (RCSI) and hazard dependence. Definition 2.1. Absolutely continuous random variables X and Y having a joint density function f ðx; yÞ and reliability function F ðx; yÞ are hazard positive (negative) dependent, HPD (HND), if and only if Z 1 Z f ðx; yÞ f ðu; yÞ du 1 f ðx; vÞ dv Pð6Þ ; ð2:1Þ F ðx; yÞ F ðx; yÞ y F ðx; yÞ x where f ðx; yÞ=F ðx; yÞ is the bivariate hazard function, and Z 1 f ðu; yÞ du o ¼ f log F ðx; yÞg; oy F ðx; yÞ x Z 1 f ðx; vÞ dv o ¼ f log F ðx; yÞg ox F ðx; yÞ y are conditional hazard functions. Also, Z 1 Z f ðx; yÞ f ðu; yÞ du 1 f ðx; vÞ dv o f log F ðx; yÞg: ¼ oxoy F ðx; yÞ F ðx; yÞ y F ðx; yÞ x
ð2:2Þ
Similar definitions can be given for discrete random variables X and Y . The random variables X and Y are independent if and only if Z 1 Z 1 f ðx; yÞ f ðu; yÞ du f ðx; vÞ dv ¼ : ð2:3Þ F ðx; yÞ F ðx; yÞ F ðx; yÞ x y Definition 2.2. A random vector (X ; Y ) or its distribution function F ðx; yÞ is said to be totally positive of order 2, TP2 if and only if F ða1 ; b1 Þ F ða2 ; b2 Þ P F ða1 ; b2 Þ F ða2 ; b1 Þ for all a1 < a2 and b1 < b2 .
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Theorem 2.1. Let (X ; Y ) be absolutely continuous random vector with distribution function F ðx; yÞ. Then P ðX > a2 ; b1 < Y 6 b2 Þ P ða1 < X 6 a2 ; Y > b2 Þ 6 P ðX > a2 ; Y > b2 Þ P ða1 < X 6 a2 ; b1 < Y 6 b2 Þ for all a1 < a2 and b1 < b2 if and only if F ðx; yÞ is totally positive of order 2, TP2 . Proof P ðX > a2 ; b1 < Y 6 b2 Þ P ða1 < X 6 a2 ; Y > b2 Þ 6 P ðX > a2 ; Y > b2 Þ P ða1 < X 6 a2 ; b1 < Y 6 b2 Þ for all a1 < a2 and b1 < b2 is seen to be equivalent to F ða2 ; b2 ÞfF ða1 ; b1 Þ F ða1 ; b2 Þ F ða2 ; b1 Þ þ F ða2 ; b2 Þg P fF ða2 ; b1 Þ F ða2 ; b2 ÞgfF ða1 ; b2 Þ F ða2 ; b2 Þg for all a1 < a2 and b1 < b2 ; () F ða1 ; b1 Þ F ða2 ; b2 Þ P F ða1 ; b2 Þ F ða2 ; b1 Þ for all a1 < a2 and b1 < b2 . Corollary 2.1. Let (X ; Y ) be absolutely continuous. If F ðx; yÞ is TP2 in x and y (1 < x; y < 1), then (X ; Y ) is HPD. Definition 2.3 [2]. The random vector (X ; Y ) or its distribution function, is said to be RCSI if P ðX > x; Y > yjX > x0 ; Y > y 0 Þ is increasing in x0 and in y 0 for all x and y. Theorem 2.2. Let (X ; Y ) be an absolutely continuous random vector. If (X ; Y ) is RCSI, then (X ; Y ) is HPD. Proof. P fX > x; Y > yjX > x0 ; Y > y 0 g is increasing in x0 and in y 0 for all x and y ()
P ðX > x; Y > y; X > x0 ; Y > y 0 Þ P ðX > x; Y > y; X > x00 ; Y > y 00 Þ 6 P ðX > x0 ; Y > y 0 Þ P ðX > x00 ; Y > y 00 Þ
for all x0 < x00 and y 0 < y 00 . ()
P ðX > x0 ; Y > y 00 Þ P ðX > x00 ; Y > y 0 Þ 6 0 0 P ðX > x ; Y > y Þ P ðX > x00 ; Y > y 00 Þ
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for all x0 < x00 and y 0 < y 00 . () P ðX > x00 ; Y > y 0 Þ P ðX > x00 ; Y > y 00 Þ þ P ðX > x00 ; y 0 < Y 6 y 00 Þ þ P ðx0 < X 6 x00 ; y 0 < Y 6 y 00 Þ þ P ðx0 < X 6 x00 ; Y > y 00 Þ P P ðx0 < X 6 x00 ; Y > y 00 Þ þ P ðX > x00 ; Y > y 00 Þ P ðX > x00 ; Y > y 00 Þ þ P ðX > x00 ; y 0 < Y 6 y 00 Þ : This implies P ðX > x00 ; Y > y 0 ÞP ðx0 < X 6 x00 ; y 0 < Y 6 y 00 Þ P P ðX > x00 ; y 0 < Y 6 y 00 ÞP ðx0 < X 6 x00 ; Y > y 0 Þ; which implies P ðX > x0 ; Y > y 0 ÞP ðx0 < X 6 x00 ; y 0 < Y 6 y 00 Þ P P ðX > x0 ; y 0 < Y 6 y 00 ÞP ðx0 < X 6 x00 ; Y > y 0 Þ: Let x0 ¼ x, x00 ¼ x þ Dx, y 0 ¼ y, y 00 ¼ y þ Dy, where Dx > 0, Dy > 0. Dividing the last inequality by DxDy and letting Dx ! 0, Dy ! 0, it follows that F ðx; yÞ f ðx; yÞ P
Z
1
f ðu; yÞ du
x
Z
1
f ðx; vÞ dv:
y
3. Stochastic dependence and exchangeability In this section, we present important results on stochastic inequalities and exchangeable random variables. Theorem 3.1. Let (X ; Y ) and (U ; V ) be independent pairs of random vectors with distribution functions H and F respectively. If there exists a monotone function g : R ! R such that P ðY 6 gðX ÞÞ P P ðX 6 gðY ÞÞ then P ðY þ V 6 gðX þ U ÞÞ P P ðX þ U 6 gðY þ V ÞÞ:
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605
Proof P ðY þ V 6 gðX þ U ÞÞ ¼
Z Z
P ðY 6 gðX þ uÞÞ dF ðu; vÞ
Z Z
P ðX 6 gðY þ uÞ vÞ dF ðu; vÞ
P
¼ P ðX þ V 6 gðY þ uÞÞ Z Z ¼ P ðV 6 gðy þ uÞ xÞ dH ðx; yÞ Z Z P P ðU 6 gðy þ V Þ xÞ dH ðx; yÞ ¼ P ðX þ U 6 gðY þ V ÞÞ:
Theorem 3.2. Let (Xi ; Yi ) and (Ui ; Vi ) be independent pairs of random vectors i ¼ 1; 2. Suppose (Ui ; Vi ) are exchangeable random vectors and the functions gi : R2 ! R are monotone and are such that P ðY1 6 g1 ðX1 ; X2 Þ; Y2 6 g2 ðX1 ; X2 ÞÞ P P ðX1 6 g1 ðY1 ; Y2 Þ; X2 6 g2 ðY1 ; Y2 ÞÞ; then P ðY1 þ V1 6 g1 ðX1 þ U1 ; X2 þ U2 Þ; Y2 6 g2 ðX1 þ U1 ; X2 þ U2 ÞÞ P P ðU1 þ X1 6 g1 ðY1 þ V1 ; Y2 þ V2 Þ; U2 þ X2 6 g2 ðY1 þ V1 ; Y2 þ V2 ÞÞ: Proof. Let gi , i ¼ 1; 2 be nondecreasing and continuous and let Fi and Hi be the distribution functions of (Ui ; Vi ) and (Xi ; Yi ), i ¼ 1; 2 respectively. Then P ðY1 þ V1 6 g1 ðX1 þ U1 ; X2 þ U2 Þ; Y2 þ V2 6 g2 ðX1 þ U1 ; X2 þ U2 ÞÞ Z Z Z Z ¼ P ðY1 þ v1 6 g1 ðX1 þ u1 ; X2 þ u2 Þ; Y2 þ v2 6 g2 ðX1 þ u1 ; X2 þ u2 ÞÞ ¼
Z
Z
2 Y
dFi ðui ; vi Þ
i¼1
P ðY1 6 g1 ðX1 þ u1 ; X2 þ u2 Þ v1 ; Y2
6 g2 ðX1 þ u1 ; X2 þ u2 Þ v2 Þ Z P
Z
2 Y
dFi ðui ; vi Þ
i¼1
P ðX1 6 g1 ðY1 þ u1 ; Y2 þ u2 Þ v1 ; X2
6 g2 ðY1 þ u1 ; Y2 þ u2 Þ v2 Þ
2 Y i¼1
dFi ðui ; vi Þ
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¼ P ðX1 þ V1 6 g1 ðY1 þ U1 ; Y2 þ U2 Þ; X2 þ V2 6 g2 ðY1 þ U1 ; Y2 þ U2 ÞÞ Z Z ¼ P ðV1 6 g1 ðy1 þ U1 ; y2 þ U2 Þ x1 ; V2 6 g2 ðy1 þ U1 ; y2 þ U2 Þ x2 Þ Z P
Z
2 Y
dHi ðxi ; yi Þ
i¼1
P ðU1 6 g1 ðy1 þ V1 ; y2 þ V2 Þ x1 ; U2
6 g2 ðy1 þ V1 ; y2 þ V2 Þ x2 Þ
2 Y
dHi ðxi ; yi Þ
i¼1
¼ P ðU1 þ X1 6 g1 ðY1 þ V1 ; Y2 þ V2 Þ; U2 þ X2 6 g2 ðY1 þ V1 ; Y2 þ V2 ÞÞ:
Theorem 3.3. Let X ¼ ðX1 ; X2 Þ be a random vector with distribution function FX which is PF2 [3]. Let Y ¼ ðY1 ; Y2 Þ be a random vector with distribution function FY which is TP2 and suppose X and Y are independent. Then Z ¼ X þ Y has a distribution function H which is HPD. Proof. The joint distribution of Z ¼ ðZ1 ; Z2 Þ is given by Z Y 2 HZ ðz1 ; z2 Þ ¼ FXi ðzi yi Þ dFY ðyÞ: i¼1
By Q2 assumption FXi is PF2 , and FXi ðzi yi Þ if TP2 in (zi ; yi ), i ¼ 1; 2, so that i¼1 FXi ðzi yi Þ dFY ðyÞ is TP2 in (z1 ; z2 ; y1 ; y2 ). Hence by the basic composition formula [3] H is TP2 . Consequently, H is HPD. Theorem 3.4. Let fFh ðxÞg and fGh ðyÞg be two families of distribution functions each with monotone increasing likelihood ratio. Then for any mixing distribution K the distribution Z Fh ðxÞGh ðyÞ dKðhÞ; H ðx; yÞ ¼ X
where X is a Borel set in Rn and K is a probability measure on X, is HPD. Proof. By assumption, both Fh ðxÞ and Gh ðyÞ have monotone increasing likelihood ratio, so that for h, w 2 X, h 6¼ w, h < w fFh ðxÞFw ðx0 Þ Fw ðxÞFh ðx0 ÞgfGh ðyÞGw ðy 0 Þ Gh ðy 0 ÞGw ðyÞg P 0 for x < x0 ; y < y 0 () H ðx; yÞH ðx0 ; y 0 Þ P H ðx; y 0 ÞH ðx0 ; yÞ for x < x0 and y < y 0 . Clearly, H is TP2 , so that H is HPD.
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Proposition 3.1. Let X ¼ ðX1 ; X2 Þ be a random vector with independent components, each Xi , i ¼ 1; 2 has a PF2 distribution FX . Let U be a random variable independent of X ¼ ðX1 ; X2 Þ with distribution function FU . Suppose Z ¼ ðX1 þ U ; X2 þ U Þ, then the joint distribution of Z ¼ ðZ1 ; Z2 Þ is LPD. Proof. The joint distribution of (z1 ; z2 ) is given by Z Y 2 FXi ðzi uÞ dFU ðuÞ: HZ ðz1 ; z2 Þ ¼ i¼1
The result follows from Theorem 3.3.
Corollary 3.1. Let X ¼ ðX1 ; X2 Þ have independent components with densities fXi , i ¼ 1; 2. Let Y0 > 0 be a random variable. If either fXi ðu=vÞ is TP2 in 1 < u < 1 and v > 0 or fXi ðuvÞ is TP2 in 1 < u < 1 and v > 0 then the random vectors Z ¼ ðX1 Y0 ; X2 Y0 Þ and Z ¼ ðX1 =Y0 ; X2 =Y0 Þ are HPD. Let X1 , X2 be exchangeable random variables and let Y1 , Y2 be iid random variables such that Xi , Yi have common marginal distributions. Then P ½X1 2 B; X2 2 B P P ½Y1 2 B; Y2 2 B holds for all Borel measurable sets B R. In fact " # " # n n \ \ P ðXi 2 BÞ P P ðYi 2 BÞ i¼1
ð3:1Þ
ð3:2Þ
i¼1
for all Borel measurable sets B R. A common question of interest is whether the inequalities (3.1) and (3.2) also holds when the Yi Õs are less positively dependent than the Xi Õs in some fashion. This leads to partial ordering of types of positive dependence for random variables (including exchangeable random variables). Rinott and Pollack [7] obtained the following results. Let X1 , X2 ; . . . ; Xn be exchangeable normal random variables with mean l, variance r2 and correlation coefficient q2 . Let Y1 , Y2 ; . . . ; Yn be exchangeable normal random variable with mean l, variance r2 and correlation coefficient q1 . If 0 6 q1 < q2 , then E
n Y i¼1
/ðXi Þ P E
n Y
/ðYi Þ
ð3:3Þ
i¼1
holds for all Borel measurable functions / : R ! ½0; 1Þ such that the expectation exists. It therefore follows that corrð/ðX1 Þ; /ðX2 ÞÞ P corrð/ðY1 Þ; /ðY2 ÞÞ holds.
ð3:4Þ
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Tchen [9] defines a bivariate distribution G to be more positively quadrant dependent (PQD) than a bivariate distribution function F having the same marginals if Gðx; yÞ P F ðx; yÞ
ð3:5Þ
for all ðx; yÞ 2 R2 (denoted by F PQD G). Kimeldorf and Sampson [4] gave the following definition of a TP2 positive dependence ordering. Let I1 , I2 be real intervals and F and G be bivariate distributions with the same pair of marginals. The distribution G is more TP2 than F written F T G if for all intervals I1 < I2 , J1 < J2 , F ðI1 ; J1 ÞF ðI2 ; J2 ÞGðI1 ; J2 ÞGðI2 ; J1 Þ 6 GðI1 ; J1 ÞGðI2 ; J2 ÞF ðI1 ; J2 ÞF ðI2 ; J1 Þ; ð3:6Þ where F ðIi ; Jj Þ represents the probability assigned by F to the rectangle Ii Jj . Definition 3.1. Let F and G be two absolutely continuous bivariate distribution functions with the same marginal distributions. We say G is more LPD than F if o2 ln Gðx; yÞ o2 ln F ðx; yÞ P oxoy oxoy
ð3:7Þ
for all ðx; yÞ 2 R2 written F LPD G. Proposition 3.2. Let F and G be absolutely continuous distribution functions with the same marginal distributions. If Gðx; yÞ P F ðx; yÞ for all ðx; yÞ 2 R2 , then F LPD G. Proof. Since Gðx; yÞ P F ðx; yÞ for all ðx; yÞ 2 R2 , ln Gðx; yÞ P ln F ðx; yÞ for all ðx; yÞ 2 R2 . The result follows.
Definition 3.2. A positive dependence ordering has the generalized monotone invariance property if for independent pairs (Xi ; Yi ), (Ui ; Vi ), i ¼ 1; 2 of random variables and monotone functions f and g ðUi ; Vi Þ ðXi ; Yi Þ ) ðf ðU1 ; U2 Þ; gðV1 ; V2 ÞÞ ðf ðX1 ; X2 Þ; gðY1 ; Y2 ÞÞ:
ð3:8Þ
Lemma 3.1. Suppose ðU ; V Þ PQD ðX ; Y Þ; (Z1 ; Z2 ), (X ; Y ) and (U ; V ) are independent pairs of random variables; and f : R2 ! R and g : R2 ! R are increasing. Then ðf ðU ; Z1 Þ; gðV ; Z2 ÞÞ PQD ðf ðX ; Z1 Þ; gðY ; Z2 ÞÞ:
B.O. Oluyede / Appl. Math. Comput. 146 (2003) 601–610
Proof. Kimeldorf and Sampson [4].
609
Theorem 3.5. The local positive dependence ordering has the generalized monotone invariance property. Proof. Let f and g be fixed increasing functions. Define f 0 ðs; tÞ ¼ f ðt; sÞ and g0 ðs; tÞ ¼ gðs; tÞ. By Lemma 3.1, ðf ðU1 ; U2 Þ; gðV1 ; V2 ÞÞ PQD ðf ðU1 ; X2 Þ; gðV1 ; X2 ÞÞ ¼ ðf 0 ðX2 ; U1 Þ; g0 ðX2 ; V1 ÞÞ PQD ðf ðX1 ; X2 Þ; gðY1 ; Y2 ÞÞ: Apply Proposition 3.2 to obtain ðf ðU1 ; U2 Þ; gðV1 ; V2 ÞÞ HPD ðf ðX1 ; X2 Þ; gðY1 ; Y2 ÞÞ:
Proposition 3.3. For all increasing functions f ðX ; Y Þ T ðU ; V Þ ) ðf ðX Þ; Y Þ T ðf ðU Þ; V Þ ) ðf ðX Þ; Y Þ HPD ðf ðU Þ; V Þ:
4. Examples and applications In this section, we present some examples satisfying the concepts of local dependence, hazard dependence and stochastic orderings discussed in earlier sections. Theorem 4.1. Let (X ; Y ) be a random vector having a bivariate extreme value distribution, then (X ; Y ) is HPD. Proof. Marshall and Olkin [6] have shown that (X ; Y ) has a minimum extreme value distribution if and only if Z 1 ln F ðz1 ; z2 Þ ¼ ln P ðZ1 > z1 ; Z2 > z2 Þ ¼ max½az1 ; ð1 aÞz2 dlðaÞ; 0
where l is a probability measure and X ¼ g1 ðZ1 Þ and Y ¼ g2 ðZÞ are nondecreasing functions. We note that F is TP2 , so that o2 ln F ðz1 ; z2 Þ P 0: oz1 oz2 But (X ; Y ) is TP2 ) ðX ; Y Þ is RCSI ) ðX ; Y Þ is HPD.
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Example 4.1. For the class of 2 2 tables with given marginals F HPD G () F T G () F PQD G: Example 4.2. Let Ui , i ¼ 1; 2, and W be lifelengths of components of a reliability system. If there are two series or parallel systems each consisting of two components (or subsystems) one of which is common to both systems, then (X ¼ minðU1 ; W Þ; Y ¼ minðU2 ; W Þ) is the lifelengths of the two systems. Suppose U1 , U2 and W are independent and let the distribution of W be denoted by K. Then Z 1 H ðx; yÞ ¼ P ðminðU1 ; wÞ > xÞP ðminðU2 ; wÞ > yÞ dKðwÞ: ð4:1Þ 1
Note that H ðx; yÞ is the survival function of (X ; Y ), and is TP2 in x and y (x P 0; y P 0) since P ðminðUi ; wÞ > xÞ is TP2 in x and w, i ¼ 1; 2. P ðminðUi ; wÞ > xÞ ¼ H i ðxÞdðx; wÞ; where dðx; wÞ ¼
1 0
i ¼ 1; 2;
if w > x; if w 6 x;
d is TP2 . By Corollary 2.1 it follows that (X ; Y ) is HPD.
References [1] J.D. Esary, F. Proschan, Relationship among some concepts of bivariate dependence, Ann. Math. Statist. 43 (1972) 651–655. [2] R. Harris, A multivariate definition for increasing hazard rate distribution functions, Ann. Math. Statist. 41 (1970) 713–717. [3] S. Karlin, in: Total Positivity, Vol. I, Stanford University Press, Stanford, 1968. [4] G. Kimeldorf, A.R. Sampson, Monotone dependence, Ann. Statist. 6 (1987) 895–903. [5] E.L. Lehmann, Some concepts of dependence, Ann. Math. Statist. 37 (1966) 1137–1153. [6] A.W. Marshall, I. Olkin, Domains of attraction of multivariate extreme value distributions, Ann. Probab. 11 (1983) 168–177. [7] Y. Rinott, M. Pollack, A stochastic order induced by a concept of positive dependence and monotonicity of asymptotic test sizes, Ann. Statist. 8 (1980) 190–198. [8] M. Shaked, J.G. Shanthikumar, Stochastic Order and Their Applications, Academic Press, New York, 1994. [9] C. Tchen, Inequalities for distributions with given marginals, Ann. Probab. 8 (1980) 814–827.