Weak of Random Variables

Relationships among some concepts of multivariatenegativedependence

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 certainconcepts 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 dependenceimplies negativesuperadditive dependence.Finally,wecharacterizeindependence 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]

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 are negative superadditive dependence and uncorrelated random variables then 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 Huet.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.

Definition 1:A function is supermodular, if

for all

where,

and .

Note that if has continuous second partial derivatives, then supermodularity of is equivalent to for alland(Muller and Scarsini, 2000).Let be a random vector defined on a probability space .

Definition 2: The random variables are :

(a) (Joag-Dev and Proschan, 1982). Negatively associated (NA) if for every pair of disjoint nonempty subsets of ,

Whenever andare coordinatewise nondecreasingfunctions and covariance exists.

(b) Weakly negativelyassociated(WNA)if for all nonnegative and nondecreasing functions

Negatively lower orthant dependent (NLOD) if for all

And negatively orthant dependent (NOD) if both (1) and (2) hold.

(d) (Hu,2000). Negatively superadditive dependent (NSD) if

where areindependent variableswith for each i and is a supermodular function such that the expectations in (3) exist.

(e) Linearly negative dependent(LIND) if for any disjoint subsets and of and , and are NA.

(f)(Joag-Dev, 1983).Strongly negative orthant dependent (SNOD) if for every set of indices A in and for all , the following three conditions hold

The following implications are well known.

i) If is NA then it isLIND, WNA and consequence NUOD.

ii) If is NA then it isNSD.(Christofides and Vaggelatou, 2004).

iii) If is NSD then it isNUOD.(Hu, 2000).

iv)If is NA then it is SNOD and if isSNOD 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 be NUOD random variables and let be a corresponding of monotone increasing, Borel functions which are continuous from the right, then 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 LINDimplies the other.

Proof:i) ( LIND does not imply SNOD).Let have the followingdistribution.

It can be checked that the random variables areLIND and arealso NOD, since for all.

But the random variables are notSNOD,since for all

ii) (SNOD does not imply LIND). Let have the joint distribution as given in Table 6 of Hu et.al.(2005). Then the random variables are SNOD but not LIND, since

where

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 have the following distribution:

It can be checked that is SNOD, since for all

and ,we have

Similarly, it is easy to show that all conditions of Definition 2(f) are true.But is not NSD. Let, 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

Where,are independent random variables with for all.

The following example shows that the converse implication fails to hold.

Example 2: Let have the followingdistribution:

It is easy to show that are LIND. Now we define the two monotone functions and as follows:

and

we have

This shows that are not NA.

Example 3: (NODimplies neitherNAnorLIND). Let have joint distribution as following:

It is easy to see that are ND ,
Ifthen

Therefore, the random variables are not NA.

The random variables are not LIND. Since if and , then

iv)The NOD does not imply SNOD because for , we have

Remark 3:i) Lehmann (1966) proved that NUOD of and is equivalent tofor all nonnegative and nondecreasing Borel functions and.Therefore,NUOD is equivalent to weak negative associationfor.

ii).The condition ofnon-negativity infunctions in Definitionb) is a necessary condition.To see this,consider Example 3,

Then

Characterization of independence

It is well known that for a normally distributed n-dimensional random variable,stochastical independence is equivalent to -the identity matrix.When n=2, this result is generalized to NUOD random variables inLehmann(1966). Moreover, Joag-Dev and Proschan (1983)proved thatIf have-distribution, and then is NUOD if and only if for all where. In the following, we present two Theorems: Theorem 1 implies that WNA is equivalent to NUOD and Theorem 2 shows that NUOD and are equivalent to stochastic independence of.

Theorem 1:Therandom variables are WNA if and only if they are NUOD.

Proof:Letbe NUOD and be nonnegative and nondecreasing real value functions. Then,by Lemma 1 are NUOD.The continuation of the proof is a simple generalization of Theorem 1 of Ruschendorf (1981)andthe following equality,

Where, if and elsewhere. Thiswill complete the proof.

Corollary: Letbe non-negativeNUODrandom variables, then impliesindependence of .

Now, it is easy to prove the following Theorem.

Theorem2: Let be NUOD random variables assuming that exists for all.If for all , thenare stochasticallyindependent.

Conclusions:

The counterexamples presented in this paper show that the following implications holding among these concepts of dependence are strict for all:

Moreover, we characterized the stochastic independence in the class of NUOD random variables under condition for all. The characterization of stochastic independence in smaller class, LIND is still an open problem.

References

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[1]MSC(2000): 60E15