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
of f is equivalent to
 2 f ( x)
 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 f1 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
i 1
i 1
E ( f i ( X i ))   E ( f i ( X i )).
(c) Negatively upper orthant dependent (NUOD) if for all x1 ,, xn  R
n
1
P( X i  xi , i  1, , n)   P( X i  xi ).
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 , kA k X k and kB 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
 P[
n
(X i  x i )]  P [X i  x i , i  A ].P [X j  x j , j  A c ]
i 1
 P[
n
(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
p (1,1,1)  p (0, 0, 2)  p (0, 2, 0)  p (2, 0, 0) 
1
,
40
2
10
, 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
p (1, 0, 0)  p (0, 0,1)  p (0,1, 0) 
0  ai  1, 1  bi  2, i  1, 2,3, and i  j  k ,we have
3
P[
(X i  ai )] 
i 1
3
1
264
 P [X i  ai ].P [X j  a j , X k  ak ] 
,
40
40
37
1482
 P [X i  bi ].P [X j  b j , X k  b k ] 
,
40
1600
i 1
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
P[
(X i  bi )] 
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 ),
f ( x1 , x 2 )  
 1
15 2
x1  0.5, x 2  0.5
x1  0.5, x 2  0.5
and
1

( x3  15 ),
g ( x3 )  
1,
 15
x3  0.5
x3  0.5
we have
Cov (f (X 1 , X 2 ), g (X 3 )) 
7920 5940 1980
 4 
0
152
15
154
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 3  a3 ].P [X 1  a1 , X 2  a2 ] 
.
10
50
i 1
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
f1 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
P[
(X i  ai )] 
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
jT
are equivalent to stochastic independence
jT
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


n
0
i 1
 [P (
0
i

n
n
i
i
f i (X i )  u i )   P (f i (X i )  u i )] du i 0.

Where, f i ( X i )   I (ui , 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
i
i
E  f i ( X i )  Ef i ( X i ) implies independence of X 1 , X 2 ,
,X n .
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} ,
jT
jT
jT
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
jT
jT
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, 369381.
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