2. MULTIVARIATE DISPERSION ORDER UNDER THE NOTION OF DISTRIBUTION FUNCTIONS WITH THE SAME COPULA

Now, we characterize the multivariate dispersion order, [pr ]_Disp, for random variables with the same dependence structure in the sense that they have the same copula. A copula C is a cumulative distribution function with uniform margins on [0,1]. Furthermore, it is shown that if H is an n-dimensional distribution function with margins F₁,…,F_n, then there exists an n-copula C such that for all x in

, it holds that

Moreover, if F₁,…,F_n are continuous, then C is unique (for more details about copulas, see Nelsen [14]). It follows that if X = (X₁,…,X_n) and Y = (Y₁,…,Y_n) are two n-dimensional random variables, then they have the same copula if and only if

Within this setting, we can formulate the following theorem.

Theorem 2: Let X = (X₁,…,X_n) and Y = (Y₁,…,Y_n) be two n-dimensional random vectors such that they have the same copula, denoted as C. Then X [pr ]_Disp Y if and only if X_i [pr ]_Disp Y_i for all i = 1,…,n.

Proof: In light of Theorem 1, it is only necessary to prove that the component Φ_i of the function Φ has the following expression:

for i = 1,…,n. In other words, the component Φ_i of the function Φ, which maps the random vector X to Y, only depends on the ith marginal variable. Note that if (3) holds, the Jacobian matrix of Φ is a diagonal matrix in which the diagonal elements are the functions that map the univariate marginal distribution of X to the corresponding one of Y. Hence, the condition (1) in Theorem 1 is apparent.

The proof of (3) is by mathematical induction. For n = 1, it is trivial. Let us assume that it is true for i = 1,…,j − 1; then we need to show it for i = j. By induction hypothesis, it holds that

Furthermore, in light of the equality 2.9.1 in Nelsen [14], it is easy to show that

where C is the copula of the distribution function F. By assumption, F and G have the same copula; hence, using (4), it holds that

for all 0 ≤ p ≤ 1. Now, if we take u_i = F_Xi(x_i) in (5), we obtain that

Note that if we now consider

in (6), it is easy to verify that

Therefore, the proof is concluded. █

Corollary 1: Let (X₁…,X_n) be a multivariate random vector. Let us consider h_i a univariate strictly increasing expansion function for i = 1,…,n. Then it holds that

Proof: The corollary follows from the facts that any copula is invariant under monotone increasing transformations (see [14]) and the univariate dispersion order is invariant under strictly increasing expansion functions (see [17]). █

Note that this corollary provides many possible comparisons. In particular, if we take a real number a > 1, then it is easy to show that X [pr ]_Disp aX.

Example 1: Let

be two multivariate normal distributions. If ρ_ij^X = ρ_ij^Y for all i and j and σ_i^X < σ_i^Y, then X [prcue ]_Disp Y. It is well known that under the last assumption, X and Y have the same copula. Thus, the proof is apparent using Theorem 2.

3. DISPERSION PROPERTIES OF THE MULTIVARIATE STUDENT DISTRIBUTION

In this section, we apply some results obtained in the last section to the particular t-distribution family. For this purpose, we use the corresponding definition of the t-distribution from Bernardo and Smith [2, pp.139,140]. A continuous random vector X has a multivariate t-distribution or a multivariate Student distribution of dimension k, with parameters μ = (μ₁,…,μ_k), Σ, and n, where

, Σ is a symmetric positive-definite k × k matrix and n > 0 if its probability density function, denoted St_k(x|μ, Σ,n), is

where

Although not exactly equal to the inverse of the covariance matrix, the parameter Σ is often referred to as the precision matrix of the distribution or, equivalently, the inverse matrix of the dispersion matrix. In the general case, E [X] = μ and Var(X) = Σ⁻¹[n/(n − 2)].

Before introducing the results in this section, we need the definition of a univariate partial order strongly connected with the univariate dispersive ordering. We consider the tail ordering defined by Lawrence [12]. Let X and Y be two univariate random variables symmetric about zero; then we say that X is less in the tail order sense, denoted X [pr ]_r Y, if the ratio Q_Y(u)/Q_X(u) is nondecreasing (nonincreasing) for u ∈ (½,1) (u ∈ (0,½)). In the following theorem, we will use the definition of the tail ordering to order the univariate t-Student family.

Theorem 3: Let St₁(0,1,m) and St₁(0,1,m) be two univariate t-distributions. Then if n < m, it holds that St₁(0,1,m) [pr ]_Disp St₁(0,1,n).

Proof: To simplify, denote t_n as the univariate t-distribution with n degrees of freedom. Capéraà [3] showed that if n ≤ m, then t_m [pr ]_r t_n. In addition, Doksum [6] showed that for univariate absolutely continuous distributions with F(0) = G(0) = 0 such that f (0) ≥ g(0) > 0 and Q_Y(u)/Q_X(u) nondecreasing for all u ∈ (0,1), it holds that F [pr ]_Disp G.

Under the last discussion, we consider the random variable |t_n| with the density function given by

A straightforward computation shows that the distribution function of |t_n| is F_|tn|(x) = 2F_tn(x) − 1 for x ≥ 0. Hence, Q_|tn|(u) = Q_tn((u + 1)/2) for all u in the interval (0,1). Therefore, it is apparent, using the work of Capéraà [3], that if n ≤ m, then Q_|tn|(u)/Q_|tm|(u) is nondecreasing for all u ∈ (0,1). Since F_|tm|(0) = F_|tn|(0) = 0 and, of course f_|tm|(0) > f_|tn|(0), we obtain, using the result in [5] that |t_m| [pr ]_Disp |t_n|. It is easy to check, using properties of symmetry, that |t_m| [pr ]_Disp |t_n| implies that t_m [pr ]_Disp t_n. █

Note that although in the literature the degrees of freedom of a t-distribution are always associated with the dispersion and the lack of knowledge of the experiment, the previous result provides that the t-distribution family is ordered in a really strict dispersion order. To study in depth the implications of the univariate dispersion order, see Shaked and Shanthikumar [17].

We have ordered two univariate t-distributions pertaining to the degrees of freedom. If we consider the more general class when the precision is different, the following corollary holds.

Corollary 2: Let St₁(0,σ₁,m) and St₁(0,σ₂,n) be two univariate t-distributions that satisfy that for n ≤ m and σ₂ ≤ σ₁, St₁(0,σ₁,m) [pr ]_Disp St₁(0,σ₂,n).

Proof: The proof is apparent. █

Note that the precision matrix is related to the variance through the expression Var(St₁(0,σ₁,m)) = σ₁⁻¹(m/m − 2).

From this point forward, we will consider two multivariate t-distributions. We generalize the results obtained in Theorem 3 and Corollary 2 to the multivariate case.

Theorem 4: Let Y_n ∼ St_k(0, Σ,n) and Y_m ∼ St_k(0, Σ,m) be two multivariate t-distributions with the same precision matrix and different degrees of freedom. Then if n ≤ m, it holds that St_k(0, Σ,m) [pr ]_Disp St_k(0, Σ,n).

Although, of course, Theorem 4 is more general than Theorem 3, we first need the results of this one to prove the multivariate case.

Proof: Let Y_m = (Y_m,1,…,Y_m,k) and Y_n = (Y_n,1,…,Y_n,k) be the corresponding multivariate t-distributions. First, we need to prove that two multivariate t-distributions with the same precision matrix have the same copula. For this purpose, we define the random vector X such as

From Exercise 2.15 in Nelsen [14], it holds that the multivariate distributions X and Y_n have the same copula. We only have to prove that X =_st Y_m.

First, from the well-known result that the function Q_Ym,i(F_Yn,i)(·) for i = 1,…,k, maps the univariate random variable Y_n,i to Y_m,i, we know that all marginal distributions of X are generalized t-distributions. Hence, using Theorem 1 from Arellano-Valle and Bolfarine [1], we know that X is a multivariate t-distribution with parameters

. From the fact that X and Y_m have the same degrees of freedom, it is only necessary to show that Var(X) = Var(Y_m). Of course, it holds that Var(X_i) = Var(Y_m,i) for all i = 1,…,n. Hence, both matrices have the same diagonal elements.

Since X and Y_n have the same copula, it is easy to show, using Theorem 5.1.3 in Nelsen [14], that

where τ_X,Y is the population version of Kendall's tau for X and Y. Therefore, using Theorem 3.3 from Frahm, Junker, and Szimayer [8], it holds that they also have the same Pearson's coefficient, ρ, so

Using the fact that Y_n and Y_m have the same precision matrix, if we denote Var(Y_n) = (σ_ij,n) and Var(Y_m) = (σ_ij,m), it holds that

for all i,j = 1,…,k. Hence, it is apparent that cov(X_i,X_j) = cov(Y_m,i,Y_m,j) and, of course,

, which implies that Y_m =_st X.

We have already shown that Y_m and Y_n have the same copula. Hence, using first Corollary 2 and then Theorem 2, we have that Y_m [pr ]_Disp Y_n. █

At this point, we focus our attention on ordering two multivariate t-distributions with different precision matrices.

Theorem 5: Let Y₁ ∼ St_k(0, Σ₁,n) and Y₂ ∼ St_k(0, Σ₂,n) be two multivariate t-distributions with different precision matrices and the same degrees of freedom. Then the following conditions are equivalent:

1. Y₂ =_st k(Y₁), where k is one-to-one, linear, and expansion.
2. There is an orthogonal matrix Γ such that Σ₂⁻¹ ≥_L ΓΣ₁⁻¹Γ^t.
3. λ(Σ₂⁻¹) ≥ λ(Σ₁⁻¹) (where λ(·) is the vector of ordered eigenvalues and ≤ refers to the usual entrywise ordering).

Proof: It is analogous to Theorem 4 in Giovagnoli and Wynn [9]. █

Note that if both distributions have the same degrees of freedom, it is easy to find several linear transformations that map one t-distribution to the other one. Obviously, this not the case when they have different degrees of freedom; the possible transformations are clearly not linear.

It is necessary to take in account that Theorem 5 does not provide a characterization of the multivariate dispersion order. As we mentioned earlier, the [pr ]_Disp ordering is associated to a particular transformation given by the function in (2). Just looking at Example 4.1 in [7] for the multivariate Normal distributions, it is easy to show the linear expression of Φ when we are interested in comparing two multivariate t-distributions. This expression depends on the Cholesky decomposition of the matrices Σ₂⁻¹ and Σ₁⁻¹.

However, in the conditions of Theorem 5, it holds that St_k(0, Σ₁,n) [prcue ]_SD St_k(0, Σ₂,n). We emphasize that [prcue ]_SD is a weaker multivariate dispersion order than the [pr ]_Disp ordering. Hence, at least there exists an expansion function that maps one t-distribution to the other one. As we will show in Section 4, we only need to know about the [prcue ]_SD order in order to define a new measure of Bayesian influence.

Corollary 3 is needed to compare t-distributions when both degrees of freedom and precision matrices are different.

Corollary 3: Let Y₁ ∼ St_k(0, Σ₁,m) and Y₂ ∼ St_k(0, Σ₂,n) be two multivariate t-distributions with different precision matrices and degrees of freedom. Then if it holds that λ(Σ₂⁻¹) ≥ λ(Σ₁⁻¹) and n < m, then St_k(0, Σ₁,m) [prcue ]_SD St_k(0, Σ₂,n).

Proof: The result follows straightforward from the fact that

From the well-known result that a composition of two expansion functions is also an expansion function, it easily holds that St_k(0, Σ₁,m) [prcue ]_SD St_k(0, Σ₂,n). █

Note that Corollary 3 provides a sufficient condition for the [prcue ]_SD order. It is easy to show that under this condition, it also holds that

However, this last implication cannot be considered a sufficient condition. The reason is that the variance matrix is defined through both the precision matrix and degrees of freedom.

4. APPLICATION TO THE INFLUENTIAL OBSERVATIONS IN REGRESSION ANALYSIS

4.1. The Model

Johnson and Geisser [11] proposed a method of assessing the influence of specified subsets of the data when the goal is to predict future observations using predictive densities. For this purpose, they considered the following model: Y = Xβ + ε, where ε is an N × 1 random vector distributed as MN_n(0,θI) (N-dimensional multivariate normal (MN)) with mean vector 0 and covariance matrix θI, θ scalar, β is the p × 1 vector of regression coefficients, X is an N × p matrix of fixed “independent” variables, and Y is the N × 1 vector of responses on the “dependent” variable. Although they noted that a more general model is possible, they assumed the prior density for β and θ to be g(β,θ) ∝ θ⁻¹, which presumes that little prior information is available relative to that information inherent in the data. Assume the case when a particular subset of size k has been deleted; we denote this by (i), and the subset itself is indicated by i. Then the general linear model can then be expressed as

Thus, the predictive densities based on full and subset deleted datasets, when θ is unknown, are two multivariate t-distributions with parameters

where

and let S_(i), a_(i)², and s_(i)² be similarly defined.

4.2. The Problem

In this case, the problem of detecting influential observations is based on comparing two multivariate t-distributions. If we only study the comparison in terms of variability, it seems logical that if we delete a subset of data, then the obtained predictive density will be expected to be more dispersive than the one based on full data. In other words, it would be expected that f (·) [pr ]_SD f_(i)(·). This may be interpreted as the added variability, due to deletion of data subset i. However, it is not true that every subset of data with a fixed size k has the same influence. First, using Corollary 3, we checked that f (·) [pr ]_SD f_(i)(·) for all

subsets, k = 1,2,3. We do not consider it necessary to present these comparisons in this article. After these comparisons, and clearly inspired in Corollary 3, we can define a dispersion Bayesian influence in terms of variability (DBIV) measure to the i subset as

and we will order the subsets from least to most influential according to the magnitude of Q_i². Note that under the assumptions in Corollary 3, if we have the inequality λ(s_(i)²(I + H⁽ⁱ⁾)) ≥ λ(s²(I + H)), then it holds that f (·) [pr ]_SD f_(i)(·).

4.3. The Dataset and Conclusions

We consider data from the 1975 Florida Area Cumulus Experiment (FACE) previously discussed in great detail by Cook and Weisberg [4]. This experiment was conducted to determine the merits of using silver iodide to produce rainfall increases and to isolate factors contributing to treatment until additivity. There were 24 observations on 11 variables, including the response rainfall, an indicator variable determining whether clouds were seeded or not seeded, and 8 other variables, including interaction terms that were determined to be related to rainfall.

Initially, we will analyze the full dataset and then delete case 2 (observation 2) and reanalyze in its absence. For an initial analysis, a computer program was written to compute relevant statistics for all

subsets, k = 1,2,3 and N = 23,24 using the Maple 6 package.

A summary of these results for the full dataset, as well as for the full dataset with case 2 deleted, is given in Table 1. It is clear from Table 1 that observations 2, 18, 1, 15, 6, and 3 are most influential in the dispersion sense when k = 1. Moreover, case 2 is clearly an outlier and it significantly affects predictive inferences based in this dataset from the dispersion point of view. Also, it is interesting to note that when case 2 is deleted, the most influential observations in dispersion sense are 18, 1, 15, 5, 6, and 23. Although the order has been changed, we can observe that the differences among the influence measures are not significative. When k = 2 or 3, it is clear from Table 1 that case 2 will be included in most influential subsets. However, we choose to delete case 2 and perform a more careful analysis. Inspection of Table 1 reveals that observations 18 and 1 are not only the most individual influence observations but also the most influential pair. Also, we can observe that case 18 appears jointly with case 1 in the third most influential triple. Also, we note that the pair (1,18) appears in two influential triples. Consequently, the influence for case (1,18) must be taken into account in a dispersion sense.

Top Six Most Influential Subsets k = 1,2,3

Surprisingly, the most influential triple is not composed of the three most influential cases, but the third most influential triple coincides with cases that are most influential individually and in pairs.

With this analysis, we present a way to study the influence of specified subsets of the data in a dispersion sense complementary to the analysis of Johnson and Geisser [10]. This is not obviously an alternative method for studying the divergence between f (·) and f_(i)(·). The reason is why the study of the dispersion is, of course, location independent and the location component given by the means of the t-distributions is specially important to study the lack of fit of the model, as Johnson and Geisser [11] showed. However, this study provides an interesting point of view and we would like to emphasize that it was not necessary for the approximation of substituting the multivariate t-distribution for a scaled multivariate Normal density.