2. PROPERTIES OF UPPER ORTHANT DISPERSIVE ORDERING

In this section we establish an interesting property of the upper orthant dispersive ordering that if two n-dimensional random vectors X and Y have the same dependence structure in the sense that they have the same copula, then dispersive ordering among the marginal distributions implies upper orthant dispersive ordering and vice versa. The notion of copula has been introduced by Sklar [23], and studied by, among others, Kimeldrof and Sampson [13] under the name of uniform representation and by Deheuvels [5] under the name of dependence function. A copula C is a cumulative distribution function with uniform margins on [0,1]. Given a copula C, if one defines

then F is a multivariate distribution function with margins as F₁,F₂,…,F_n. For any multivariate distribution function F with margins as F₁,F₂,…,F_n, there exists a copula C such that (2.1) holds. If F is continuous, then C is unique and can be constructed as follows:

It follows that if X and Y are two n-dimensional random vectors with margins as (F₁,F₂,…,F_n) and (G₁,G₂,…,G_n), respectively, and if they have the same copula, then

For i = 1,…,n, let us denote by H_i,u^X the cumulative distribution function (cdf) of the conditional distribution (X_i|∩_j≠i{X_j > F_j⁻¹(u_j)}) and by H_i,u^Y that of (Y_i|∩_j≠i{Y_j > G_j⁻¹(u_j)}). To prove the next theorem, we first prove the following lemma, which may be of independent interest.

Lemma 2.1: If two n-dimensional random vectors X and Y have the same copula, then, for i = 1,…n,

Proof: Proving (2.4) is equivalent to proving that for i = 1,…n and u ∈ [0,1]ⁿ⁻¹,

which is true because of (2.3), since X and Y have the same copula. █

Theorem 2.1: Let X and Y be two n-dimensional random vectors with the same copula. Then

if and only if X_i ≤_disp Y_i, i = 1,…,n.

Proof: By definition,

It follows from (2.4) that if X and Y have the same copula, then for i = 1,…n and u ∈ [0,1]ⁿ⁻¹,

for i = 1,…n and for every β ∈ [0,1].

It is easy to see that the right-hand side of (2.6) is increasing in β if and only if G_i⁻¹ F_i(x) − x is increasing in x (i.e., if and only if X_i ≤_disp Y_i, i = 1,…,n). This proves the desired result. █

Recently, Müller and Scarsini [16] have investigated some other multivariate stochastic orders for which results parallel to Theorem 2.1 hold for those orders. The following interesting property of the upper orthant dispersive ordering immediately follows from Theorem 2.1.

Corollary 2.1: Let Y = (a₁ X₁ + b₁,a₂ X₂ + b₂,…,a_n X_n + b_n). Then for

.

Proof: Since X and Y have the same copula, the required result follows immediately from Theorem 2.1. █

Example 2.1 (Multivariate Normal Distributions): Let X follow the p-variate multivariate Normal distribution with mean vector μ and dispersion matrix Σ = ((σ_ij)), with σ_ij = ρ_ijσ_iσ_j, ρ_ii = 1, σ_i > 0, and i,j = 1,…,p. Let Y follow the p-variate multivariate Normal distribution with mean vector μ′ and dispersion matrix Σ′ = ((σ_ij′)), with σ_ij′ = ρ_ijσ_i′σ_j′, σ_i′ > 0, and i,j = 1,…,p. It is known that X and Y have the same copula. It follows from Theorem 2.1 that

if and only if σ_i ≤ σ_i′ for i = 1,…,p. This result in conjunction with Example 1.1 leads us to the following result for comparing two bivariate Normal distributions.

Let X and Y follow bivariate Normal distributions with dispersion matrices

respectively. If 0 < σ_i ≤ σ_i′ for i = 1,2 and |ρ′| ≤ |ρ| < 1, then

.

It will be interesting to find necessary and sufficient conditions under which two multivariate normal random vectors will be ordered according to upper orthant dispersive ordering in the general case.

It follows immediately that if two random vectors are ordered according to upper orthant dispersive ordering, then so are their corresponding subsets. In particular, their marginal distributions will be then ordered according to univariate dispersive ordering.

Theorem 2.2: Let X and Y be two n-dimensional random vectors such that

. Then

where I = {i₁,i₂,…,i_k} ⊂ {1,2,…,n}, X_I = (X_i1,…,X_ik), Y_I = (Y_i1,…,Y_ik), and k = 1,…,n.

The proof of the next result is also immediate.

Theorem 2.3: Let X₁,…,X_m be a set of independent random vectors for which the dimension of X_i is k_i, i = 1,…,m. Let Y₁,…,Y_m be another set of independent random variables for which the dimension of Y_i is k_i, i = 1,…,m. Then

Remark 2.1: A consequence of (2.7) is that if X₁,…,X_n is a collection of independent univariate random variables and Y₁,…,Y_n is another set of independent random variables, then X_i ≤_disp Y_i, i = 1,…,n implies

.

In general, there does not seem to be any direct connection between upper orthant dispersive ordering and the multivariate dispersive ordering as introduced by Fernandez-Ponce and Suarez-Llorens [7]. According to their definition, X ≤_disp Y may not imply that X_i ≤_disp Y_i for i = 1,…,n. Also, the multivariate dispersive ordering as defined by them may not be preserved under permutations of the variables. On the other hand, the upper orthant dispersive ordering is invariant under the same permutation of the two vectors and their marginals are also ordered according to univariate dispersive ordering. Obviously, if

, then tr Σ_X ≤ tr Σ_Y, where Σ_X and Σ_Y denote the dispersion matrices of X and Y, respectively.

3. THE CASE OF NONNEGATIVE RANDOM VARIABLES

In this section, we will restrict our attention to the case in which the random vectors under consideration are nonnegative or, more generally, they have a finite common left end point of their supports. We will see that certain results hold in this case that may not hold in the general case. The following assumption will be made at some places in this article.

Assumption A: The random variables {X_i|∩_j≠i{X_j > F_j⁻¹(u_j)}} and {Y_i|∩_j≠i{Y_j > G_j⁻¹(u_j)}} have a finite common left endpoint of their supports for all u and for i = 1,…,n.

In the univariate case, for nonnegative random variables, there is an intimate connection between hazard rate ordering and dispersive ordering and which is made more explicit in the following result of Bagai and Kochar [1]. We use this theorem to prove some of the results of this section.

Theorem 3.1: Let X and Y be two univariate random variables with distribution functions F and G, respectively, such that F(0) = G(0) = 0. Then the following hold:

For a bivariate random vector (S,T), we say that T is right tail increasing in S if P[T > t|S > s] is increasing in s for all t, and we denote this relationship by RTI(T |S). If S and T are continuous lifetimes, then T is right tail increasing in S if and only if r(s|T > t) ≤ r(s|T > 0) = r_S(s) for all s > 0 and for each fixed t. The RTI property is weaker than the RCSI (right corner set increasing) property, but stronger than PQD (positive quadrant dependence). In the next theorem, we study the effect of positive dependence on upper orthant dispersive ordering for nonnegative random vectors.

Theorem 3.2: Let X = (X₁,X₂) be a bivariate random vector such that the left end point of the support of {X_i|X_j > F_j⁻¹(u)} is finite and independent of u ∈ [0,1] for i,j = 1,2. Let X^I = (X₁^I,X₂^I) be a random vector of independent random variables such that

.

(b) If

is IFR for i = 1,2, then X_i is RTI in X_j, i ≠ j, i,j = 1,2.

Proof:

(b)

implies for all u ≥ 0,

Theorem 3.3: Let X and Y be two n-dimensional random vectors satisfying Assumption A and such that

. Then

implies that

where C_i,j^X (C_i,j^Y) denotes the copula of (X_i,X_j) ((Y_i,Y_j)).

Proof:

from which it follows that

and which, in turn, implies

under Assumption A since dispersive ordering implies stochastic ordering when the random variables have a finite common left end point of their supports. If we denote by F_i,j (G_i,j) the joint survival function of (X_i,X_j) ((Y_i,Y_j)), then (3.6) can be written as

where C(u,v) = 1 − u − v + C(u.v). █

If (3.4) holds and the margins of (X_i,X_j) and (Y_i,Y_j) are equal, then we say that (Y_i,Y_j) is more PQD than (X_i,X_j) (cf. [10, p.36]). Note that C_i,j^X(u_i,u_j) ≡ u_iu_j in the case X_i and X_j are independent and C_i,j^X(u_i,u_j) ≥ u_iu_j for all u_i,u_j ∈ [0,1] in the case X_i and X_j are PQD. Thus, according to Theorem 3.3, if Assumption A holds and if X and Y have the same margins and

, then the Y_i's are more dependent than the X_i's according to PQD ordering. We obtain the following result as a special case.

Corollary 3.1: Let X = (X₁,X₂) be a bivariate random vector such that the left end point of the support of {X_i|X_j > F_j⁻¹(u)} is finite and independent of u ∈ [0,1], for i ≠ j and i,j = 1,2. Let X^I = (X₁^I,X₂^I) be a random vector of independent random variables such that

. Then

Contrast this result with Theorem 3.2(b), which is a stronger one since RTI implies PQD. However, here no assumption on the monotonicity of the hazard rates is made in the second case.

Remark 3.1: Assumption A is very crucial for Theorem 3.3 and Corollary 3.1 to hold. As a counterexample, let Y₁ and Y₂ be two independent U(0,1) random variables. Let X₁ = X₂ be uniformly distributed over (0,1) also. Note that X₁ and X₂ are strongly positively dependent, as they satisfy the Frechet upper bound. Let us compare (X₁,X₂) with (Y₁,Y₂) according to upper orthant dispersive ordering. The relevant conditional distributions to compare are

The left-hand conditional distribution is U(u,1) and the right-hand conditional distribution is U(0,1). Hence,

, but C_1,2^X ≥ C_1,2^Y, contradictory to (3.4). The reason for this contradiction is that unless Assumption A is satisfied, dispersive ordering may not imply stochastic ordering.

Theorem 3.4: Let X and Y be two n-dimensional random vectors satisfying Assumption A and such that

. Then

implies

for all increasing convex functions h₁ and h₂ for which the above covariances exist.

Proof: Without loss of generality, let i = 1 and j = 2. The survival functions of (h₁(X₁),h₂(X₂)), h₁(X₁), and h₂(X₂) are, respectively, H(x₁,x₂) = F(h₁⁻¹(x₁), h₂⁻¹(x₂)), H₁(x₁) = F₁(h₁⁻¹(x₁)), and H₂(x₂) = F₂(h₂⁻¹(x₂)). Similarly, the survival functions of (h₁(Y₁),h₂(Y₂)), h₁(Y₁), and h₂(Y₂) are, respectively, K(x₁,x₂) = G(h₁⁻¹(x₁),h₂⁻¹(x₂)), K₁(x₁) = G₁(h₁⁻¹(x₁)), and K₂(x₂) = G₂(h₂⁻¹(x₂)). Covariance between h₁(X₁) and h₂(X₂), if it exists, can be expressed as

where h₁⁻¹(x₁) = F₁⁻¹(u) and h₂⁻¹(x₂) = F₂⁻¹(v). The assumption

implies that X_i ≤_disp Y_i, from which it follows that f_i(F_i⁻¹(u)) ≥ g_i(G_i⁻¹(u)) and F_i⁻¹(u) ≤ G_i⁻¹(u), i = 1,2, under Assumption A. Now h_i′(x) is increasing in x since h(x) is convex. Combining these facts, the required result follows from (3.8). █

Theorem 3.5: Let X and Y be two n-dimensional random vectors satisfying the Assumption A and let φ₁,…,φ_n be increasing convex functions on

. Then

Proof: Note that the cdf of φ_j(X_j) is F_φj(x) = F_j(φ_j⁻¹(x)), with its inverse as F_φj⁻¹(u) = φ_j(F_j⁻¹(u)). We have to prove that for i = 1,…,n, u_j ∈ (0,1), j = 1,…,n, j ≠ i,

that is,

which is equivalent to

Using Assumption A, for i = 1,…,n, u_j ∈ (0,1), j = 1,…,n, j ≠ i,

implies

Now the required result follows from Theorem 2.2 of Rojo and He [18] since φ_i's are increasing convex functions. █

Hu et al. [9] gave the following definition of multivariate (weak) hazard rate ordering.

Definition 3.1: Let X and Y be n-dimensional random vectors with hazard gradients r_X and r_Y, respectively. We say that X is smaller than Y according to weak hazard rate ordering (written as X ≤_whr Y) if

for

; that is, if

In the next theorem we establish results analogous to Theorem 3.1 between upper orthant dispersive ordering and multivariate weak hazard rate ordering under Assumption A.

Theorem 3.6: Let X and Y be two n-dimensional random vectors satisfying Assumption A.

Proof:

(b) From the assumption

, it follows that

Remark 3.2: If r_X⁽ⁱ⁾(x₁,…,x_i,…,x_n) increases in x_i for i = 1,…,n, then we say that the random vector X has a multivariate increasing hazard rate distribution (cf. Johnson and Kotz [11]). The condition r_X⁽ⁱ⁾(x₁,…,x_i,…,x_n) increasing in x_j, j = 1,…,n, j ≠ i, i = 1,…,n describes a condition of positive dependence that is equivalent to saying that the random vector X has RCSI; that is,

increases in x_i′, i = 1,…,n.

We will now study some preservation properties of the upper orthant dispersive order under random compositions. Such results are often referred to as preservations under “random mapping” (see Shaked and Wong [22]), or preservations of “stochastic convexity” (see Shaked and Shanthikumar [20, Chap.6] and Denuit, Lefèvre, and Utev [6], and references therein).

Let

be a family of n-dimensional survival functions, where

is a subset of the real line. Let X(θ) denote a random vector with survival function F_θ. For any random variable Θ with support in

and with distribution function H, let us denote by X(Θ) a random vector with survival function G given by

Theorem 3.7: Consider a family of n-dimensional survival functions

as above. Let Θ₁ and Θ₂ be two random variables with supports in

and distribution functions H₁ and H₂, respectively. Let Y₁ and Y₂ be two random vectors such that Y_i =_st X(Θ_i), i = 1,2; that is, suppose that the survival function of Y_i is given by

If

then

Proof: Hu et al. [9] proved that assumptions (a) and (b) imply that Y₁ ≤_whr Y₂. Now, we show that for i = 1,…,n, r_Y1⁽ⁱ⁾(x₁,…,x_n) is decreasing in x_j, j = 1,…,n, then the required result will follow from Theorem 3.6(a). Assumption (a) is equivalent to F_θ(x) being TP₂ (recall from Karlin [12] that a function

is said to be totally positive of order 2 (TP₂) if f (x₁,y₁) f (x₂,y₂) ≥ f (x₁,y₂) f (x₂,y₁), for x₂ ≥ x₁ and y₂ ≥ y₁) in (θ,x_j), j = 1,…,n. r_X(θ)⁽ⁱ⁾(x₁,…,x_n) decreasing in x_j, j = 1,…,n, j ≠ i is equivalent to F_θ(x₁,…,x_n) being TP₂ in (x_i,x_j), i,j = 1,…,n, j ≠ i. Using these observations, it follows that G₁(x) is TP₂ in (x_i,x_j), i,j = 1,…,n (cf. Karlin [12]), which is equivalent to r_Y1⁽ⁱ⁾(x₁,…,x_n) decreasing in x_j, j = 1,…,n, j ≠ i. It is worth noting that r_X(θ)⁽ⁱ⁾(x₁,…,x_n) decreasing in x_i is equivalent to the fact that {X_i|∩_j≠i{X_j > x_j}} is a DFR random variable, i = 1,…,n. Now, G₁(x) can be written as

that is, (Y₁,…,Y_n) is a sort of mixture of DFR random variables; therefore, −∂ log G_j(x)/∂x_i is decreasing in x_i, which is equivalent to r_Y1⁽ⁱ⁾(x₁,…,x_n) decreasing in x_i. This completes the proof. █

4. EXAMPLES AND APPLICATIONS

Example 4.1 (Multivariate Pareto Distributions): For a > 0, let X_a = (X_a,1,…,X_a,n) have the survival function F_a given by

see, for example, Kotz, Balakrishnan, and Johnson [14, p.600]. The corresponding density function is given by

Hu et al. [9] showed that X_a1 ≤_whr X_a2 whenever a₁ ≥ a₂. On the other hand,

, is decreasing in x_j, j = 1,…,n. Then from Theorem 3.6(a) it follows that

whenever a₁ ≥ a₂.

Example 4.2 (Bivariate Farlie–Gumbel–Morgenstern Distributions): For α ∈ (−1, 1), let X_α = (X_α,1,X_α,2) have the survival function F_α given by

and Y_α = (Y_α,1,Y_α,2) have the survival function G_α given by

where F₁, F₂, G₁, and G₂ are arbitrary univariate survival functions (which happen to be the marginal survival functions of X_α,1, X_α,2, Y_α,1, and Y_α,2, respectively, independently of α). Assume that X_α,i ≤_disp Y_α,i, i = 1,2. It is easy to see that C^X_α(u,v) = C^Y_α(u,v). Then, from Theorem 2.1, it follows that

.

Example 4.3 (Multivariate Gumbel Exponential Distributions): For positive parameters λ = {λ_I : I ⊆ {1,2,…,n},I ≠ [empty ]}, let X_λ = (X₁,X₂,…,X_n) have the survival function F_λ given by

see Kotz et al. [14, p.406]. For another set of positive parameters λ* = {λ_I* : I ⊆ {1,2,…,n},I ≠ [empty ]}, let Y_λ* = (Y₁,Y₂,…,Y_n) have the survival function G_λ*. Let X_i =_st Y_i, i = 1,…,n; that is, λ_i = λ_i*. We show that if λ ≥ λ*, then for i = 1,…,n,

Since X_i =_st Y_i, i = 1,…,n, (4.1) is equivalent to

Let x_i = (x₁,…,x_i−1,x_i+1,…,n) . The survival function of (X_i|∩_j≠i{X_j > x_j}), denoted by F_i(x_i;x_i), is

Similarly, the survival function of {Y_i|Y_j > x_j, j ≠ i}, denoted by G_i(x_i;x_i), is

Now, the ratio

is increasing in x_i; that is,

On the other hand, the random variable {X_i|X_j > x_j, j ≠ i} has an exponential distribution that is DFR. Combining this observation with (4.6), it follows from Theorem 3.1(a) that (4.2) holds. Now, applying Theorem 3.3 to this example, we get that C_i,j^λ is decreasing in λ, where C_i,j^λ denotes the copula of (X_i,X_j)

Application 4.1 (Order Statistics): Let X₁,…,X_n (Y₁,…,Y_n) be a random sample from a univariate distribution with strictly increasing distribution function F (G). Bartoszewicz [3] has shown that F ≤_disp G implies X_i:n ≤_disp Y_i:n, i = 1,…,n, where X_i:n (Y_i:n) is the ith-order statistic of the X sample (Y sample). We will strengthen this result to prove that F ≤_disp G implies

We first show that in the case of random samples from continuous distributions, the copulas of order statistics are independent of the parent distributions. Note that

. Since the function G⁻¹F is strictly increasing, it follows from Theorem 2.4.3 of Nelsen [17] that C(X_i:n,X_j:n) = C(Y_i:n,Y_j:n), for i,j = 1,…,n. It now immediately follows from Theorem 2.1 that F ≤_disp G implies (4.7).

Since the order statistics from a random sample are positively associated (cf. Boland, Hollander, Joag-Dev, and Kochar [4]) and since (X_1:n,…,X_n:n) and (Y_1:n,…,Y_n:n) have the same copula, the conditions of Theorem 3.4 are satisfied. Hence, for i,j ∈ {1,…,n},

for all increasing convex functions h₁ and h₂ for which the above covariances exist. This result was originally proved by Bartoszewicz [2] using a different method. A similar result can be established for record values.

Application 4.2 (Record Values): Let X₁,…,X_n,… (Y₁,…,Y_n,…) be a sequence of random variables from a univariate distribution F (G). It is known that F ≤_disp G implies R_m^X ≤_disp R_m^Y, where R_m^X (R_m^Y) is the mth record value of the X sequence (Y sequence). We first show that in the case of random sequences from continuous distributions, the copulas of record values are independent of the parent distributions. Then it will immediately follow from Theorem 2.1 that F ≤_disp G implies

Let M(x) = −log F(x); then the distribution function of R_m^X can be expressed as F_Rm^X(x) = G_m(M(x)), where G_m(x) is the distribution function of a Gamma random variable with scale parameter one and shape parameter m. Similarly, let N(x) = −log G(x); then the distribution function of R_m^Y is F_Rm^Y(x) = G_m(N(x)). Now, both M(X₁),…,M(X_n),… and N(Y₁),…,N(Y_n),… are sequences of independent and identically distributed (i.i.d.) exponential random variables with mean one. Using this observation, it follows that

where R_m* is the mth record value of a sequence of i.i.d. exponential random variables with mean one.

This proves the desired result.