Document Sample

Relationships among some concepts of multivariate negative dependence H. R. Nili Sani, M. Amini, 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 certain 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 negative superadditive dependence. Finally, we characterize 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 the 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. 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 certain 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 negative superadditive dependence. In Section 3, we prove analogous result of Ruschendorf (1981) for upper negative orthant dependence random variables. In fact, we characterize stochastic independence in the class of upper negative orthant dependence random variables. Corresponding author. E-mail address: m-amini@ferdowsi.um.ac.ir , nilisani@yahoo.com, bozorg@math.um.ac.ir, 1 MSC (2000): 60E15 1 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{ xm , y m }) and x y (max{ x1 , y1},, max{ xm , y m }) . Note that if f has continuous second partial derivatives, then supermodularity 2 f ( x) of f 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) . Definition 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 n n functions f i , i 1,2,...,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 ,, xn 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 ,, xn 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). 2 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 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 certain 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 are 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: 3 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 follows: 4 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 ), x3 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 shows 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, NUOD is equivalent to weak negative association for n 2 . ii). The condition of non-negativity in 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 5 2. Characterization of independence It is well known that for a normally distributed n-dimensional random variable, stochastical 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 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 are 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 ))du i , 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 presented in this paper show that the following implications holding among these concepts of dependence are strict for all n 3 : NUOD NOD SNOD NA NSD NUOD W NA LIND 6 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 Block, H.W. and Fang, Z.(1988). A multivariate extension of Hoeffding's lemma. Annals of Probability, 16, 1803-1820. Block, H.W. and Fang, Z.(1990).Setwise independence for some dependence structures.J. Multivariate Analysis,32, 103-119. 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. Christofiedes, T.C. and Vaggelatou, E. (2004). A connection between super modular ordering and positive/ negative association. Journal of Multivariate Analysis 88, 138-151. Joag-Dev, K. and Proschan, F. (1982). Negative association of random variables with applications. Ann. of Stat.11, 286-295. Joag-Dev, K. (1983). Independence via uncorrelatedness under certain dependence structures. Ann. Probab.Vol. 11, No.4, 1037-1041. Joe, H.(1997). Multivariate models and dependence concepts. Chapman and Hall, London. .. Hu, T. M u ller, A. and Scarsini, M. (2004). Some counterexamples in positive dependence. Journal of statistical Planning and Inference, 124. 153-158. Hu, T. and Yang, J. (2004) Further developments on sufficient conditions for negative dependence of random variables. Statistics & Probability Letters, 369- 381. Hu, T. (2000). Negatively superadditive dependence of random variables with applications. Chinese Journal of Applied Probability and Statistics, 16, 133-144. Hu, T. Ruan, L. and Xie, C. (2005). Dependence structures of multivariate Bernoulli random vectors. J.Multivariate analysis, 94(1), 172-195. .. M u ller, A. and Scarsini, M.(2000). Some remarks on the supermodular. J. Multivariate Analysis.73. 107-119. Lehmann,E.(1966). Some consepts of dependence. Ann.Math.Statist.,37,1137-1153. .. R u schendorf, L.(1981). Weak association of random variables. J. Multivariate Analysis, 11, 448-451. 7

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 4 |

posted: | 11/12/2011 |

language: | English |

pages: | 7 |

OTHER DOCS BY malj

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.