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Relationships among some concepts of multivariate negative dependence M. Amini, H. R. Nili Sani, and A.Bozorgnia Department of Statistics, University of Birjand,Birjand,Iran. Department of Statistics, Ferdowsi University of, Mashhad,Mashhad,Iran. Abstract: In this paper, we provide counterexamples to show that some concepts of negative dependence are strictly stronger than others. In addition, we solve an open problem posed by Hu,et.al.(2005) referring to whether strong negative orthant dependence implies that negative superadditive dependence. Finally, we characterized independence in the class of negative upper orthant dependence random variables under some suitable moment conditions. Keywords: Negatively upper orthant dependent, Negative association, Characterization of independence, Negative superadditive dependence, Linear negative dependence, Strong negative orthant dependence.1 1. Introduction and Preliminaries Various results in probability and statistics have been derived under the assumption that some underlying random variables have the negative dependence property. A number of concepts of negative dependence have been introduced in recent years. Many implications among different dependence concepts are well known. The reader is referred to Joe(1997), Hu (2000), Hu and Yang (2004) and Hu et.al.(2004, 2005) for an extensive treatment of the topic. Furthermore, the characterization of stochastic independence via uncorrelatedness has been studied by many authors in some classes of negative or positive dependence. For example, Ruschendorf (1981) characterized the stochastic independence in the class of upper positive orthant dependence under some suitable moment conditions. Hu, (2000) proved that if X 1 , X 2 , X n are negative superadditive dependence and uncorrelated random variables then X 1 , X 2 , X n are stochastic independence. Block and Fang (1988, 1990) characterized the stochastic independence for some dependence structures also Joag-Dev (1983) characterized the stochastic independence in classes of negative association and strong negative orthant dependence random variables via uncorrelatedness. This paper is organized as follows: Section 1 recalls some well known concepts of negative dependence and presents some well known implications from them. In section 2, we provide some counterexamples and show that some concepts of negative dependence are strictly stronger than others. Moreover, we solve an open problem posed by Hu,et.al.(2005) referring to whether strong negative orthant dependence implies that negative superadditive dependence. In section 3, we prove analogous result of Ruschendorf (1981) for upper negative orthant dependence Corresponding author. E-mail address: m-amini@ferdowsi.um.ac.ir , nilisani@yahoo.com,, bozorg@math.um.ac.ir, 1 MSC(2000): 60E15 random variables. In fact, we characterize stochastic independence in the class of upper negative orthant dependence random variables. Definition 1: A function f : R m R is supermodular, if f ( x y ) f ( x y ) f ( x) f ( y ) for all x, y R m where x y (min{ x1 , y1 }, , min{ x m , y m }) and x y (max{ x1 , y1 }, , max{ x m , y m }) . Note that if f has continuous second partial derivatives, then supermodularity of f 2 f ( x) is equivalent to 0 for all 1 i j m and x R m (Muller and Scarsini, xi x j 2000). Let ( X 1 , X 2 , ... X n ), n 3 be a random vector defined on a probability space (, , P) Definitions 2: The random variables X 1 , X 2 , X n are : (a) (Joag-Dev and Proschan, 1982). Negatively associated (NA) if for every pair of disjoint nonempty subsets A1 , A2 of {1, ..., n} , Cov( f1 ( X i , i A1 ) , f 2 ( X i , i A2 )) 0, whenever f 1 and f 2 are coordinatewise nondecreasing functions and covariance exists. (b) Weakly negatively associated (WNA) if for all nonnegative and nondecreasing functions f i , i 1,2, n, . . n n E ( f i ( X i )) E ( f i ( X i )). i 1 i 1 (c) Negatively upper orthant dependent (NUOD) if for all x1 ,, x n R n P ( X i xi , i 1, , n) P ( X i xi ). 1 i 1 Negatively lower orthant dependent (NLOD) if for all x1 ,, x n R n P( X i xi , i 1, , n) P( X i xi ). (2) i 1 And negatively orthant dependent (NOD), if both (1) and (2) hold. (d) (Hu,2000). Negatively superadditive dependent (NSD) if E ( f ( X 1 , X 2 ,... X n )) E ( f (Y1 , Y2 ,...Yn )), (3) st where Y1 , Y2 , , Yn are independent variables with X i Yi for each i and f is a supermodular function such that the expectations in (3) exist. (e) Linearly negative dependent (LIND) if for any disjoint subsets A and B of {1,2,, n}and j 0, j 1, , n , kA k X k and kB k X k are NA. (f) (Joag-Dev, 1983). Strongly negative orthant dependent (SNOD) if for every set of indices A in {1,2,, n} and for all x R n , the following three conditions hold n P[ (X i x i )] P [X i x i , i A ].P [X j x j , j A c ] i 1 n P[ (X i x i )] P [X i x i , i A ].P [X j x j , j A c ] i 1 P [X i x i , i A , X j x j , j A c ] P [X i x i , i A ].P [X j x j , j A c ] The following implications are well known. i) If ( X 1 , X 2 , ... X n ) is NA then it is LIND, WNA and consequence NUOD. ii) If ( X 1 , X 2 , ... X n ) is NA then it is NSD.(Christofides and Vaggelatou, 2004). iii) If ( X 1 , X 2 , ... X n ) is NSD then it is NUOD.(Hu, 2000). iv) If ( X 1 , X 2 , ... X n ) is NA then it is SNOD and if ( X 1 , X 2 , ... X n ) is SNOD then it is NOD.( Joag-Dev, 1983) It is well known that some of negative dependence concepts do not imply others. Remark 1: i) Neither of the two dependence concepts NUOD and NLOD implies the other (Bozorgnia et.al, 1996) ii) Neither NUOD nor NLOD imply NA. (Joag-Dev and Proschan, 1982). iii) The NSD does not imply LIND and NA.(Hu, 2000). iv) The NSD does not imply SNOD. (Hu, et.al., 2005). We use the following Lemma that is important in the theory of negative dependence random variables. Lemma 1: (Bozorgnia et.al., 1996) Let X 1 , X 2 ,..., X n be NUOD random variables and let f1 , f 2 ,..., f n be a corresponding of monotone increasing, Borel functions which are continuous from the right, then f1 ( X 1 ), , f n ( X n ) are NUOD random variables. 2. Some counterexamples In this section, we present some counterexamples showing that some concepts of negative dependence are strictly stronger than others. Lemma 2: Neither of the two dependence concepts SNOD and LIND implies the other. Proof: i) ( LIND does not imply SNOD). Let ( X 1 , X 2 , X 3 ) have the following distribution. p (1,1,1) 0.05, p (1, 0, 0) p (0,1, 0) 0.225, p (0, 0,1) 0.22, p (0, 0, 0) 0.065, p (1,1, 0) 0.08, p (0,1,1) 0.06, p (1, 0,1) 0.075. It can be checked that the random variables X 1 , X 2 , X 3 are LIND and also NOD, since for all 0 ai 1, i 1, 2,3 . P (X 1 a1 , X 2 a2 , X 3 a3 ) 0.05 P (X a1 ).P (X a2 ).P (X a3 ) 0.07227, P (X 1 a1 , X 2 a2 , X 3 a3 ) 0.065 P (X a1 ).P (X a2 ).P (X a3 ) 0.1984. But the random variables X 1 , X 2 , X 3 are not SNOD, since for all 0 ai 1, i 1, 2,3 3 P[ (X i ai )] 0.05 P [ X 1 a1 ].P [ X 2 a2 , X 3 a3 ] 0.0473. i 1 ii) (SNOD does not imply LIND). Let (X 1 , X 2 , X 3 , X 4 ) have the joint distribution as given in Table 6 of Hu, et.al.(2005). Then the random variables X 1 , X 2 , X 3 , X 4 are SNOD but not LIND, since 9 8 P [X 1 1,Y 2 2] P [Y 1 1].P [Y 2 2] , 32 32 Where Y 1 X 1 , Y 2 X 2 X 3 X 4 . The next Lemma indicates that strong negative orthant dependence does not imply NSD which gives the answer to the question posed by Hu, et.al.(2005). Lemma 3: SNOD does not imply NSD. Proof: Let ( X 1 , X 2 , X 3 ) have the following distribution. 1 p (1,1,1) p (0, 0, 2) p (0, 2, 0) p (2, 0, 0) , 40 2 10 p (1, 0, 0) p (0, 0,1) p (0,1, 0) , p (1,1, 0) p (0,1,1) p (1, 0,1) . 40 40 It can be checked that ( X 1 , X 2 , X 3 ) is SNOD, since for all 0 ai 1, 1 bi 2, i 1, 2,3, and i j k ,we have 3 1 264 P[ (X i ai )] P [X i ai ].P [X j a j , X k ak ] , i 1 40 40 3 37 1482 P[ (X i bi )] P [X i bi ].P [X j b j , X k b k ] , i 1 40 1600 14 608 P [X i ai , X j b j , X k b k ] P [X i ai ].P [X j b j , X k b k ] , 40 1600 2 117 P [X i ai , X j a j , X k b k ] P [X i ai ].P [X j a j , X k b k ] , 40 1600 3 72 P [X i ai , X j a j , X k ak ] P [X i ai ].P [X j a j , X k ak ] , 40 1600 24 912 P [X i ai , X j b j , X k b k ] P [X i ai ].P [X j b j , X k b k ] , 40 1600 10 176 P [X i ai , X j a j , X k ak ] P [X i ai , X j a j ].P [X k ak ] , 40 1600 11 429 P [X i ai , X j a j , X k b k ] P [X i ai , X j a j ].P [X k b k ] , 40 1600 similarly it is easy to show that all conditions of Definition 2(f) are true. But ( X 1 , X 2 , X 3 ) is not NSD. Let f (x 1 , x 2 , x 3 ) max{x 1 x 2 x 3 1, 0} , this function is supermodular since it is a composition of an increasing convex real value function and an increasing supermodular function. For this function we get 56000 50494 Ef (X 1 , X 2 , X 3 ) Ef (Y 1 ,Y 2 ,Y 3 ) . 64000 64000 st Where Y 1 ,Y 2 ,Y 3 are independent random variables with X i Y i for all i 1, 2,3, . The following example shows that the converse implication NA LIND fails to hold. Example 2 : Let ( X 1 , X 2 , X 3 ) have the following distribution. 2 3 p (0, 0, 0) 0, p (0, 0,1) p (1, 0, 0) , p (1,1,1) , 15 15 2 p (0,1, 0) p (1,1, 0) p (0,1,1) p (1, 0,1) . 15 It is easy to show that X 1 , X 2 , X 3 are LIND. Now we define the two monotone functions f and g as following 1 1 ( x1 15 )( x 2 15 ), x1 0.5, x 2 0.5 f ( x1 , x 2 ) 1 x1 0.5, x 2 0.5 15 2 and 1 ( x3 15 ), x 3 0. 5 g ( x3 ) 1, x 3 0 .5 15 we have 7920 5940 1980 Cov (f (X 1 , X 2 ), g (X 3 )) 4 4 0 152 15 15 This show that X 1 , X 2 , X 3 are not NA. Example 3: (NOD implies neither NA nor LIND). Let ( X 1 , X 2 , X 3 ) have joint distribution as following: 2 p (0,0,0) p (1,0,1) 0, p (0,1, 0) p (0, 0,1) , 10 1 3 p (0,1,1) p (1,1, 0) p (1,1,1) , p (1, 0, 0) . 10 10 i. It is easy to see that X 1 , X 2 , X 3 are ND , 1 1 1 ii. If f (x 1 , x 2 ) I (x 1 , x 2 ), and g (x 3 ) I (x 3 ), then 2 2 2 1 8 E {f (X 1 , X 2 ).g (X 3 )} Ef (X 1 , X 2 ).Eg (X 3 ) . 10 100 Therefore the random variables X 1 , X 2 , X 3 are not NA. iii.The random variables X 1 , X 2 , X 3 are not LIND. Since if Y 1 X 1 X 2 and Y 2 X 3 , then 1 12 P [Y 1 1,Y 2 0] P [Y 1 1]P [Y 2 0] . 2 25 iv) The NOD does not imply SNOD, because for 0 ai 1, i 1, 2,3 , we have 3 1 4 P[ (X i ai )] P [X 3 a3 ].P [X 1 a1 , X 2 a2 ] . i 1 10 50 Remark 3: i) Lehmann (1966) proved that NUOD of X 1 and X 2 is equivalent to Cov( f1 ( X 1 ) , f 2 ( X 2 )) 0 , for all nonnegative and nondecreasing Borel functions f 1 and f 2 .Therefore for n 2 NUOD is equivalent to weak negative association. ii). The condition of non-negativity functions f i , i 1, 2,..., n in Definition b) is a necessary condition. To see this consider Example 3, 1 1 f 1 (x ) f 2 (x ) I (x ) and f 3 (x ) x . 2 4 Then 1 3 E {f 1 (X 1 ).f 2 (X 2 ).f 3 (X 3 )} Ef 1 (X 1 ).Ef 2 (X 2 ).Ef 3 (X 3 ) . 20 80 3. Characterization of independence It is well known that for a normally distributed n-dimensional random variables stochastically independence is equivalent to Cov( X ) I -the identity matrix.When n=2 this result is generalized to NUOD random variables in Lehmann(1966). Moreover, Joag-Dev and Proschan (1983) proved that If (X 1 , X 2 ,..., X n ) have N ( , ) -distribution, and then (X 1 , X 2 ,..., X n ) is NUOD if and only if ij 0, for all i j, i, j 1,2,...,n, where ( ij ) . In the following, we present two Theorems: Theorem 1 implies that WNA is equivalent to NUOD and in Theorem 2 shows that NUOD and E X j EX j are equivalent to stochastic independence jT jT of X 1 , X 2 ,..., X n . Theorem 1: The random variables X 1 , X 2 ,..., X n are WNA if and only if they NUOD. Proof: Let X 1 , X 2 ,..., X n be NUOD and f i , i 1,2,..., n be nonnegative and nondecreasing real value functions. Then by Lemma 1 f1 ( X 1 ), , f n ( X n ) are NUOD. The continuation of the proof is a simple generalization of Theorem 1 of Ruschendorf (1981) and the following equality, n n E ( f i (X i ) E (f i (X i )) i i n n n [P ( f i (X i ) u i ) P (f i (X i ) u i )] du i 0. 0 0 i 1 i i Where, f i ( X i ) I (u i , f i ( X i ))dui , 1 i n, I (u, x) 1 if x u and I (u, x) 0 0 elsewhere. This will complete the proof. Corollary: Let X 1 , X 2 , , X n be non-negative NUOD random variables, then n n E f i ( X i ) Ef i ( X i ) implies independence of X 1 , X 2 , ,X n . i i Now it is easy to prove the following Theorem. Theorem 2: Let X 1 , X 2 , , X n be NUOD random variables assuming that E X j exists for all T {1,2,, n} . If E X j EX j for all T {1,2,, n} , jT jT jT then X 1 , , X n are stochastically independent. Conclusions : The counterexamples have been presented in this paper show that the following implications holding among these concepts of dependence are strict for all n 3 : NUOD NOD S NOD NA NS D NUOD W NA LIND Moreover, we characterized the stochastic independence in the class of NUOD random variables under condition E X j EX j for all T {1,2,, n} . The jT jT characterization of stochastic independence in smaller class LIND , is still an open problem. References [1]. Bozorgnia, A., Patterson, R.F. and Taylor, R.L. (1996). Limit theorems for dependent random variables. World Congress Nonlinear Analysts, 92, Vol. I- IV(Tampa, FL, 1992), 1639-1650, de Gruyter, Berlin. [2]. Christofiedes, T.C. and Vaggelatou, E. (2004). A connection between supermodular ordering and positive/negative association. Journal of Multivariate Analysis 88, 138-151. [3]. Joag-Dev, K. and Proschan, F. (1982). Negative association of random variables with applications. Ann. of Stat.11, 286-295. [4]. Joag-Dev, K. (1983). Independence via uncorrelatedness under certain dependence structures. Ann. Probab.Vol. 11, No.4, 1037-1041. [5].Joe, H.(1997). Multivariate models and dependence concepts. Chapman and Hall, London. .. [6]. Hu, T. M u ller, A. and Scarsini, M. (2004). Some counterexamples in positive dependence. Journal of statistical Planning and Inference, 124. 153-158. [7]. Hu, T. and Yang, J. (2004) Further developments on sufficient conditions for negative dependence of random variables. Statistics & Probability Letters, 369-381. [8]. Hu, T. (2000). Negatively superadditive dependence of random variables with applications. Chinese Journal of Applied Probability and Statistics, 16, 133-144. [9]. Hu, T. Ruan, L. and Xie, C. (2005). Dependence structures of multivariate Bernoulli random vectors. J.Multivariate analysis, 94(1), 172-195. .. [10].M u ller, A. and Scarsini, M.(2000). Some remarks on the supermodular. J. Multivariate Analysis.73. 107-119. .. [11] R u schendorf, L. (1981). Weak association of random variables. J.Multivariate analysis,11,448-451.

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