On Concordance Measures for Discrete Data and Dependence Properties of Poisson Model

Abstract

We study Kendall's tau and Spearman's rho concordance measures for discrete variables. We mainly provide their best bounds using positive dependence properties. These bounds are difficult to write down explicitly in general. Here, we give the explicit formula of the best bounds in a particular Fréchet space in order to understand the behavior of the ranges of these measures. Also, based on the empirical copula which is viewed as a discrete distribution, we propose a new estimator of the copula function. Finally, we give useful dependence properties of the bivariate Poisson distribution and show the relationship between parameters of the Poisson distribution and both tau and rho.

1. Introduction

The best known dependence property is “lack of dependence,” or what is known as stochastic independence. In many applications, independence between two random variables is assumed; this can be a strong assumption in the undertaken analysis. Taking into account the dependence structure between the variables leads to appropriate modeling approaches and correct conclusions. To study stochastic dependence, concordance concept and positive dependence are well used tools. This is because many dependence properties can be described by means of the joint distribution of the variables and these measures and properties are often margins free. In this paper we study two concordance measures, Kendall’s tau Kruskal [1] and Spearman’s rho Lehmann [2]. These measures have several properties known as Rényi's axioms; for more details see Rényi [3]. Among these axioms, we focus on the range of the association measure.

Many researches have been concerned with the study of tau and rho in the case of continuous variables. Schweizer and Wolff [4], in one seminal paper, show that the study of concordance measures for continuous random variables can be characterized as the study of copulas [5]. However, for noncontinuous variables, this interrelationship generally does not hold. There are few papers concerning the discrete version of Kendall's tau and Spearman's rho. Conti [6] gives definitions of two approaches of indifference and links them to concordance and discordance properties of the data. Tajar et al. [7] propose a copula-type representation for random couples with binary margins. They show that appropriate measures of association for binary random variables do not depend on the marginal distribution of the variables under study. Mesfioui and Tajar [8] and Denuit and Lambert [9] have shown independently that the range of tau and rho in the discrete case is not the unit interval as in the continuous case. Nešlehovà [10] considers an alternative transformation of an arbitrary random variable to a uniform distribution variable in order to study the rank measures for noncontinuous random variables.

In this paper, we focus on the range of the concordance measures. Aside from identifying the best bounds of tau and rho in the case of discrete random variables, we present some dependence properties of the bivariate Poisson model and discuss their relationship with the concordance measures tau and rho. The paper is organized as follows. The next section provides a method of constructing the ranges of tau and rho for discrete data. Section 3 develops explicit expressions for the best bounds of tau and rho in the discrete Fréchet space with the same marginal. Section 4 provides a new estimator of the copulas based on the so-called empirical copulas. Section 5 discusses some dependence properties of the bivariate Poisson model.

2. Defintions and Properties

Following Hoeffding [11], Kruskal [1], and Lehmann [2], Schweizer and Wolff [4] express Kendall's tau and Spearman's rho for continuous random vector (𝑋,𝑌) in terms of the joint distribution 𝐻(𝑥,𝑦) of (𝑋,𝑌) and the margins 𝐹(𝑥) for 𝑋 and 𝐺(𝑦) for 𝑌. A general representation for each of 𝜏 and 𝜌 has been first proposed by Kowalczyk and Niewiadomska-Bugaj [12]; namely

𝜏=𝐸_𝐻[𝐻(𝑋,𝑌)]+𝐸_𝐻[𝐻(𝑋⁻,𝑌⁻)]+𝐸_𝐻[𝐻(𝑋⁻,𝑌)]+𝐸_𝐻[𝐻(𝑋,𝑌⁻)]−1,𝜌=3{𝐸_Π[𝐻(𝑋⁻,𝑌)]+𝐸_Π[𝐻(𝑋,𝑌⁻)]+𝐸_Π[𝐻(𝑋⁻,𝑌⁻)]+𝐸_Π[𝐻(𝑋,𝑌)]−1}, (2.1)(2.2) where 𝐻(𝑥⁻,𝑦⁻)=𝑃[𝑋<𝑥,𝑌<𝑦], 𝐻(𝑥,𝑦⁻)=𝑃[𝑋≤𝑥,𝑌<𝑦], 𝐻(𝑥⁻,𝑦)=𝑃[𝑋<𝑥,𝑌≤𝑦], and Π(𝑥,𝑦)=𝐹(𝑥)𝐺(𝑦).

Several results in this paper are based on the monotonicity property of Kendall's 𝜏 and Spearman's 𝜌. This property has first been proposed for continuous variables by Yanagimoto and Okamoto [13] (see also [14]). Tchen [15] obtained similar monotonicity property for 𝜏 and 𝜌 when the supports of the joint distributions consist in a finite number of atoms. Mesfioui and Tajar [8] extend various dependence relationships between Kendall's 𝜏 and Sperman's 𝜌 in Capéraà and Genest [16] and Nelsen [5], to the discrete case. One key result of their paper is the generalization to any kind of random variables for continuous and/or discrete variables.

For the remainder of the paper, we recall the property of concordance orderings, defined as follows.

Let (𝑋₁,𝑌₁) and (𝑋₂,𝑌₂) be random vectors with identical marginals and respective cdf's 𝐻₁ and 𝐻₂. The random couple (𝑋₂,𝑌₂) is said to be more concordant than (𝑋₁,𝑌₁), denoted by (𝑋₁,𝑌₁)≺_𝑐(𝑋₂,𝑌₂), if 𝐻₁(𝑥₁,𝑥₂)≤𝐻₂(𝑥₁,𝑥₂) holds for all 𝑥₁,𝑥₂∈ℝ.

In the following proposition, we propose a flexible method to establish the monotonicity property given in Mesfioui and Tajar [8] for purely discrets random vectors. The proof is direct and easy to understand and extends the result to the general random vectors.

Proposition 2.1. Let (𝑋₁,𝑌₁) and (𝑋₂,𝑌₂) be two random couples with respective distribution function 𝐻₁ and 𝐻₂ in Γ(𝐹,𝐺), the Fréchet space of all distribution functions with fixed marginals 𝐹 and 𝐺. Then, (𝑋₁,𝑌₁)⪯_𝑐(𝑋₂,𝑌₂)⇒𝜏_𝐻1≤𝜏_𝐻2,(𝑋₁,𝑌₁)⪯_𝑐(𝑋₂,𝑌₂)⇒𝜌_𝐻1≤𝜌_𝐻2. (2.3)(2.4)

Proof. Using Fubini's theorem, we note that 𝐸_𝐻1[𝐻₂(𝑋,𝑌)]=𝐸_𝐻2[𝐻₁(𝑋⁻,𝑌⁻𝐸)],_𝐻1[𝐻₂(𝑋⁻,𝑌)]=𝐸_𝐻2[𝐻₁(𝑋,𝑌⁻𝐸)],_𝐻1[𝐻₂(𝑋,𝑌⁻)]=𝐸_𝐻2[𝐻₁(𝑋⁻𝐸,𝑌)],_𝐻1[𝐻₂(𝑋⁻,𝑌⁻)]=𝐸_𝐻2[𝐻₁(𝑋,𝑌)], (2.5) where 𝐻_𝑖 denotes the survival functions associated to 𝐻_𝑖, 𝑖=1,2.
Now without loss of generality if we assume that 𝐻₁≤𝐻₂, which is equivalent to 𝐻₁≤𝐻₂, we then get 𝐸_𝐻1[𝐻₁(𝑋,𝑌)]≤𝐸_𝐻1[𝐻₂(𝑋,𝑌)]=𝐸_𝐻2[𝐻₁(𝑋⁻,𝑌⁻)]≤𝐸_𝐻2[𝐻₂(𝑋⁻,𝑌⁻)]=𝐸_𝐻2[𝐻₂(𝑋,𝑌)]. (2.6) Similarly, we obtain 𝐸_𝐻1[𝐻₁(𝑋⁻,𝑌)]≤𝐸_𝐻2[𝐻₂(𝑋⁻𝐸,𝑌)],_𝐻1[𝐻₁(𝑋,𝑌⁻)]≤𝐸_𝐻2[𝐻₂(𝑋,𝑌⁻𝐸)],_𝐻1[𝐻₁(𝑋⁻,𝑌⁻)]≤𝐸_𝐻2[𝐻₂(𝑋⁻,𝑌⁻)]. (2.7) Combining the later inequalities with (2.1), we then obtain (2.3). It is easy seen that (2.4) is immediate from (2.2).

For any bivariate distribution function 𝐻 with univariate marginals 𝐹 and 𝐺, one has

max[0,𝐹(𝑥)+𝐺(𝑦)−1]≤𝐻(𝑥,𝑦)≤min[𝐹(𝑥),𝐺(𝑦)]. (2.8) The extreme distributions 𝐻_min(𝑥,𝑦)=max[0,𝐹(𝑥)+𝐺(𝑦)−1] and 𝐻_max(𝑥,𝑦)=min[𝐹(𝑥),𝐺(𝑦)] are often refereed as Fréchet bounds (see [17]). These bounds play a central role to construct optimal ranges of 𝜏 and 𝜌 as stated in the following corollary.

Corollary 2.2. Let (𝑋,𝑌) be a random couple with distribution function 𝐻 in Γ(𝐹,𝐺). Then, 𝜏_min≤𝜏_𝐻≤𝜏_max,𝜌_min≤𝜏_𝐻≤𝜌_max, (2.9) where 𝜏_min, 𝜌_min and 𝜏_max, 𝜌_max denote the values of Kendall's 𝜏 and Spearman's 𝜌 corresponding to the Fréchet lower and upper bounds in Γ(𝐹,𝐺), respectively.

As stated earlier, the main objective in this paper is to examine the bounds of 𝜏 and 𝜌 in the Fréchet space Γ(𝐹,𝐺) when 𝐹 and 𝐺 are discrete. To do that, let (𝑋,𝑌) be a discrete random couple with cdf 𝐻∈Γ(𝐹,𝐺). Since Kendall's 𝜏 and Spearman's 𝜌 are scale invariants, they remain unchanged under strictly increasing transformations of the marginal distributions. We can then suppose, without any loss of generality, that 𝑋 and 𝑌 are valued in ℤ, the set of all integers. Therefore, we can see from (2.1) and (2.2) that 𝜏 and 𝜌 can be written as

𝜏=^∞∞_{𝑖=−∞𝑗=−∞}𝑇_𝑖𝑗𝑆_𝑖𝑗−1,𝜌=3^∞∞_{𝑖=−∞𝑗=−∞}𝑇_𝑖𝑗[𝐹(𝑖)−𝐹(𝑖−1)][𝐺(𝑗)−𝐺(𝑗−1)]−3, (2.10)(2.11) where

𝑇_𝑖𝑗𝑆=𝐻(𝑖,𝑗)+𝐻(𝑖−1,𝑗−1)+𝐻(𝑖,𝑗−1)+𝐻(𝑖−1,𝑗),_𝑖𝑗=𝐻(𝑖,𝑗)−𝐻(𝑖,𝑗−1)−𝐻(𝑖−1,𝑗)+𝐻(𝑖−1,𝑗−1). (2.12)(2.13)

In order to obtain the best bounds 𝜏_min, 𝜌_min and 𝜏_max, 𝜌_max, the minimum and maximum values corresponding to lower and upper bound of 𝜏 and 𝜌, respectively, we replace 𝐻 in (2.10) and (2.11) by the Fréchet bounds 𝐻_min(𝑖,𝑗)=max[𝐹(𝑖)+𝐺(𝑗)−1,0] and 𝐻_max(𝑖,𝑗)=min[𝐹(𝑖),𝐺(𝑗)], respectively.

For discrete data, the ranges of 𝜏 and 𝜌 are different from the usual unit interval [−1,1]. This is a violation of the monotone dependence properties of concordance measures, as stated in Nelsen [5]. To correct this problem, we propose the following corrections:

𝜏_𝑐=𝜏𝜏_max−𝜏if𝜏≥0,𝜏_min𝜌if𝜏<0,_𝑐=𝜌𝜌_max−𝜌if𝜌≥0,𝜌_minif𝜌<0. (2.14) The main importance of these corrections is that they allow to interpret the levels of the new measures, 𝜏_𝑐 and 𝜌_𝑐, as percentages. Illustrations of these transformations are proposed in Section 5 with the bivariate Poisson distribution.

3. Explicit Bounds of Discrete 𝜏 and 𝜌 in Γ(𝐹,𝐹)

The aim of this section is to study the effect of the marginal distributions on the range of 𝜏 and 𝜌 for discrete data. Note that it is difficult to obtain explicit expressions of the extreme values of 𝜏 and 𝜌 in Γ(𝐹,𝐺) for noncontinuous distribution 𝐹 and 𝐺. This problem is very complicated and requires several assumptions on 𝐹 and 𝐺. In order to analyze the behavior of these bounds, we consider the particular space Γ(𝐹,𝐹), where 𝐹 is a discrete distribution function. To this end, consider the integer function defined by

𝜙(𝑖)=min{𝑗∈ℤ∶𝐹(𝑖)+𝐹(𝑗)>1},𝑖∈ℤ. (3.1) This function plays an important role to explicit lower bounds of 𝜏 and 𝜌 in the space Γ(𝐹,𝐹). The next proposition presents explicit optimal bounds of Spearman's 𝜌.

Proposition 3.1. The best bounds for 𝜌 in the space Γ(𝐹,𝐹) are given by 𝜌_max=3𝐸[1−𝐹²(𝑋)−𝐹²𝜌(𝑋−1)],_min=3𝐸[𝜓(𝑋)+𝜓(𝑋−1)−1], (3.2)(3.3) where 𝜓(𝑖)=^∞_{𝑗=𝜙(𝑖)}[𝐹(𝑖)+𝐹(𝑗)−1][𝐹(𝑗+1)−𝐹(𝑗−1)]. (3.4)

Proof. Let 𝐻(𝑖,𝑗)=min[𝐹(𝑖),𝐹(𝑗)]. From (2.12), we observe that 𝑇_𝑖𝑗=[𝐹(𝑖)+3𝐹(𝑖−1)]𝕀_[𝑖=𝑗]+2[𝐹(𝑖)+𝐹(𝑖−1)]𝕀_[𝑖<𝑗]+2[𝐹(𝑗)+𝐹(𝑗−1)]𝕀_[𝑖>𝑗], (3.5) and writing 𝐹(𝑖)−𝐹(𝑖−1)=𝑝_𝑖, we get from (2.11) that 𝜌_max=3^∞_𝑖=−∞[𝐹(𝑖)+3𝐹(𝑖−1)][𝐹(𝑖)−𝐹(𝑖−1)]𝑝_𝑖+6^∞_𝑖=−∞[𝐹(𝑖)+𝐹(𝑖−1)]𝑝_𝑖^∞_𝑗=𝑖+1[𝐹(𝑗)−𝐹(𝑗−1)]+6^∞_𝑖=−∞𝑝_𝑖^𝑖−1_𝑗=−∞[𝐹(𝑗)−𝐹(𝑗−1)][𝐹(𝑗)+𝐹(𝑗−1)]−3, (3.6) which may be simplified as 𝜌_max=3𝐸{[𝐹(𝑋)+3𝐹(𝑋−1)][𝐹(𝑋)−𝐹(𝑋−1)]}+6𝐸{[𝐹(𝑋)+𝐹(𝑋−1)][1−𝐹(𝑋)]}+6𝐸[𝐹²(𝑋−1)]−3. (3.7) The result then follows from the fact that 𝐸[𝐹(𝑋)+𝐹(𝑋−1)]=1. Now, choose 𝐻(𝑖,𝑗)=sup[𝐹(𝑖)+𝐹(𝑗)−1,0] and put 𝐻⁺(𝑖,𝑗)=𝐹(𝑖)+𝐹(𝑗)−1. From (2.11), we see that 𝜌_min=3^∞∞_{𝑖=−∞𝑗=𝜙(𝑖)}𝐻⁺(𝑖,𝑗)𝑝_𝑖𝑝_𝑗+3^∞∞_{𝑖=−∞𝑗=𝜙(𝑖)}𝐻⁺(𝑖,𝑗)𝑝_𝑖+1𝑝_𝑗+1+3^∞∞_{𝑖=−∞𝑗=𝜙(𝑖)}𝐻⁺(𝑖,𝑗)𝑝_𝑖𝑝_𝑗+1+3^∞∞_{𝑖=−∞𝑗=𝜙(𝑖)}𝐻⁺(𝑖,𝑗)𝑝_𝑖+1𝑝_𝑗−3. (3.8) It follows that 𝜌_min=3^∞_𝑖=−∞(𝑝_𝑖+𝑝_𝑖+1)^∞_{𝑗=𝜙(𝑖)}[𝐹(𝑖)+𝐹(𝑗)−1](𝑝_𝑗+𝑝_𝑗+1)−3, (3.9) which may be rewritten as 𝜌_min=3^∞_𝑖=−∞[𝜓(𝑖)+𝜓(𝑖−1)−1]𝑝_𝑖, (3.10) where 𝜓(𝑖)=^∞_{𝑗=𝜙(𝑖)}[𝐹(𝑖)+𝐹(𝑗)−1][𝐹(𝑗+1)−𝐹(𝑗−1)]. (3.11) The result is therefore obtained from (3.11) and (3.10).

Using (2.10) with 𝐻(𝑖,𝑗)=min[𝐹(𝑖),𝐹(𝑗)], we notice that the upper bound of Kendall's 𝜏 in the space Γ(𝐹,𝐹) can be expressed as

𝜏_max=2𝐸[𝐹(𝑋−1)]. (3.12) Note that the sharp upper bound given in Denuit and Lambert [9] coincides with (3.12) in Γ(𝐹,𝐹). However, the behavior of Kendall's tau lower bound in terms of the distribution 𝐹 is not evident. The following proposition gives an explicit form of this bound in Γ(𝐹,𝐹).

Proposition 3.2. The best lower bounds of 𝜏 in Γ(𝐹,𝐹) is 𝜏_min=2𝐸[𝐹[𝜙(𝑋−1)]]−2^∞_𝑘=−∞𝜉(𝑘)−2, (3.13) where 𝜉(𝑘)=[𝐹(𝑘−1)+𝐹[𝜙(𝑘−1)]−1][𝐹(𝑘)+𝐹[𝜙(𝑘−1)−1]−1]𝕀_{[𝜙(𝑘)<𝜙(𝑘−1)]}. (3.14)

Proof. From (2.12) and (2.13), we observe that 𝑆_𝑖𝑗𝑇_𝑖𝑗=𝐻²(𝑖,𝑗)+𝐻²(𝑖−1,𝑗−1)−𝐻²(𝑖−1,𝑗)−𝐻²(𝑖,𝑗−1)+2𝐻(𝑖,𝑗)𝐻(𝑖−1,𝑗−1)−2𝐻(𝑖−1,𝑗)𝐻(𝑖,𝑗−1)_{∞∞𝑖=−∞𝑗=−∞}[𝐻²(𝑖,𝑗)+𝐻²(𝑖−1,𝑗−1)−𝐻²(𝑖−1,𝑗)−𝐻²(𝑖,𝑗−1)]=1. (3.15) Consider now 𝐻(𝑖,𝑗)=sup[𝐹(𝑖)+𝐹(𝑗)−1,0] and write 𝐻⁺(𝑖,𝑗)=𝐹(𝑖)+𝐹(𝑗)−1. From (2.10), we get 𝜏_min=2^∞∞_{𝑖=−∞𝑗=𝜙(𝑖−1)+1}𝐻⁺(𝑖,𝑗)𝐻⁺(𝑖−1,𝑗−1)−2^∞∞_{𝑖=−∞𝑗=max[𝜙(𝑖−1),𝜙(𝑖)+1]}𝐻⁺(𝑖−1,𝑗)𝐻⁺(𝑖,𝑗−1). (3.16) Using the fact that 𝐻⁺(𝑖,𝑗)𝐻⁺(𝑖−1,𝑗−1)−𝐻⁺(𝑖−1,𝑗)𝐻⁺(𝑖,𝑗−1)=−𝑝_𝑖𝑝_𝑗, (3.17) we have 𝜏_min=−2^∞_𝑖=−∞𝑝_𝑖[^∞_{𝑗=𝜙(𝑖−1)+1}𝑝_𝑗]𝕀_{[𝜙(𝑖)=𝜙(𝑖−1)]}−2^∞_𝑖=−∞𝑝_𝑖[^∞_{𝑗=𝜙(𝑖−1)+1}𝑝_𝑗]𝕀_{[𝜙(𝑖)<𝜙(𝑖−1)]}−2^∞_𝑖=−∞[𝐹(𝑖−1)+𝐹(𝜙(𝑖−1))−1][𝐹(𝑖)+𝐹(𝜙(𝑖−1)−1)−1]𝕀_{[𝜙(𝑖)<𝜙(𝑖−1)]}, (3.18) which is equivalent to 𝜏_min=−2^∞_𝑖=−∞𝑝_𝑖[1−𝐹[𝜙(𝑖−1)]]−2^∞_𝑖=−∞𝑟(𝑖) (3.19) with 𝑟(𝑖)=[𝐹(𝑖−1)+𝐹(𝜙(𝑖−1))−1][𝐹(𝑖)+𝐹(𝜙(𝑖−1)−1)−1]𝕀_{[𝜙(𝑖)<𝜙(𝑖−1)]}, (3.20) which completes the proof.

Remark 3.3. Let 𝐹_𝑛,𝑝 be a binomial distribution with parameters 𝑛 and 𝑝, and denote the extreme values of 𝜏 and 𝜌 in Γ(𝐹_𝑛,𝑝,𝐹_𝑛,𝑝) by 𝜏_max(𝑛,𝑝) and 𝜌_max(𝑛,𝑝). One can show the following symmetry properties, namely: 𝜏_max(𝑛,𝑝)=𝜏_max(𝑛,1−𝑝),𝜌_max(𝑛,𝑝)=𝜌_max(𝑛,1−𝑝). (3.21) Indeed, since 𝐹_𝑛,𝑝(𝑘)=𝐹_{𝑛,1−𝑝}(𝑛−𝑘), then from (3.7), we have 𝜏_max(𝑛,𝑝)=2^𝑛_𝑘=0𝐹_𝑛,𝑝(𝑘−1)[𝐹_𝑛,𝑝(𝑘)−𝐹_𝑛,𝑝(𝑘−1)]=2^𝑛_𝑘=0𝐹_{𝑛,1−𝑝}(𝑘−1)[𝐹_{𝑛,1−𝑝}(𝑘)−𝐹_{𝑛,1−𝑝}(𝑘−1)]=𝜏_max(𝑛,1−𝑝). (3.22) Similar arguments provide 𝜌_max(𝑛,𝑝)=𝜌_max(𝑛,1−𝑝).

In this section, we examine the symmetry of the ranges of 𝜏 and 𝜌 associated to discretef data. In continuous case, it is well known that the ranges of these parameters are symmetric, that is, 𝜏_max=−𝜏_min and 𝜌_max=−𝜌_min. This conclusion is of course invalid for noncontinuous data. In order to clarify this question, we consider again the space Γ(𝐹,𝐹) with discrete distribution 𝐹. We present below a situation which ensures that 𝜌_max=−𝜌_min and 𝜏_max=−𝜏_min. As consequence of Propositions 3.1 and 3.2 and (3.12), one can establish the following results.

Corollary 3.4. In space, Γ(𝐹,𝐹), if 𝐸[𝐹²(𝑋)]=𝐸[𝜓(𝑋)] and 𝐸[𝐹²(𝑋−1)]=𝐸[𝜓(𝑋−1)], then 𝜌_max=−𝜌_min.

Corollary 3.5. In space, Γ(𝐹,𝐹), if 𝜙(𝑖)=𝜙(𝑖−1), 𝑖∈ℤ, and 𝐸[𝐹(𝑋)]=𝐸[𝐹(𝜙(𝑋))], then 𝜏_max=−𝜏_min.

4. Empirical Copulas Viewed as a Discrete Distribution

It is well recognized that copula provides a flexible approach to model the joint behavior of random variables. In fact, this method allows to represent a bivariate distribution as function of its univariate marginals through a linking function called a copula. Specifically, if 𝐻 is a distribution function of a bivariate random vector (𝑋,𝑌) with continuous marginals, then Sklar [18] ensures that there exists a unique copula 𝐶∶[0,1]²→[0,1] such that for all (𝑥,𝑦)∈ℝ²,

𝐻(𝑥,𝑦)=𝐶[𝐹(𝑥),𝐺(𝑦)]. (4.1) Hence, 𝐶 is a bivariate distribution function with uniform marginals on [0,1] that captures all the information about the dependence among the components of (𝑋,𝑌). For a comprehensive introduction to a copula, the reader is referred to monographs by Nelsen [5].

Suppose that the random sample (𝑋₁,𝑌₁),…,(𝑋_𝑛,𝑌_𝑛) is given from some pair (𝑋,𝑌) of continuous variable with copula 𝐶(𝑢,𝑣). To estimate the copula 𝐶, Deheuvels [19] proposes the so-called empirical copula defined by

𝐶_𝑛1(𝑢,𝑣)=𝑛^𝑛_𝑖=1𝕀(𝐹_𝑛(𝑋_𝑖)≤𝑢,𝐺_𝑛(𝑌_𝑖)≤𝑣), (4.2) where 𝐹_𝑛 and 𝐺_𝑛 are the empirical distribution functions of 𝑋 and 𝑌 based on the sample 𝑋₁,…,𝑋_𝑛 and 𝑌₁,…,𝑌_𝑛 given by

𝐹_𝑛1(𝑡)=𝑛^𝑛_𝑖=1𝕀(𝑋_𝑖≤𝑡),𝐺_𝑛1(𝑡)=𝑛^𝑛_𝑖=1𝕀(𝑌_𝑖≤𝑡). (4.3) Let 𝑅_𝑖 be the rank of 𝑋_𝑖 among the sample 𝑋₁,…,𝑋_𝑛 and 𝑇_𝑖 stands the rank of 𝑌_𝑖 among the sample 𝑌₁,…,𝑌_𝑛. Observe that 𝐶_𝑛 is a function of ranks (𝑅₁,𝑇₁),…,(𝑅_𝑛,𝑇_𝑛), because 𝐹_𝑛(𝑋_𝑖)=𝑅_𝑖/𝑛 and 𝐺_𝑛(𝑌_𝑖)=𝑇_𝑖/𝑛, 𝑖=1,…,𝑛, namely,

𝐶_𝑛1(𝑢,𝑣)=𝑛^𝑛_𝑖=1𝑅𝕀(^𝑖𝑛𝑇≤𝑢,^𝑖𝑛≤𝑣). (4.4) From this representation, one can consider 𝐶_𝑛(𝑢,𝑣) as a discrete bivariate distribution with uniform marginals taking values in the set {1/𝑛,2/𝑛,…,1}. Observe that

𝐶_𝑛(𝑢,𝑣)=𝐶_𝑛(𝑖𝑛,𝑗𝑛𝑖)for(𝑢,𝑣)∈[𝑛,𝑖+1𝑛𝑗]×[𝑛,𝑗+1𝑛]. (4.5) Now, one can observe that the 𝐶_𝑛 is not copula. Indeed, 𝐶_𝑛(𝑢,1)=⌊𝑛𝑢⌋/𝑛≠𝑢, where ⌊𝑛𝑢⌋ denotes the integer part of 𝑛𝑢.

Our goal in this section is to transform the empirical copula in order to obtain a new estimator 𝐶^∗_𝑛 which is a copula. To this end, let (𝑍_𝑛,𝑊_𝑛) be a discrete random vector with distribution function 𝐶_𝑛 which is defined in (4.2). The idea is to transform the uniform discrete random variables 𝑍_𝑛 and 𝑊_𝑛 into a continuous variables 𝑍^∗_𝑛 and 𝑊^∗_𝑛 by defining

𝑍^∗_𝑛=𝑍_𝑛−𝑈_𝑛,𝑊^∗_𝑛=𝑊_𝑛−𝑉_𝑛, (4.6) where 𝑈_𝑛 and 𝑉_𝑛 are independents and uniformly distributed in [0,1/𝑛]. We also suppose that the random vectors 𝑍_𝑛 and 𝑈_𝑛 (resp, 𝑊_𝑛 and 𝑉_𝑛) are independents. The next result shows that the distribution function of the continuous version (𝑍^∗_𝑛,𝑊^∗_𝑛) is a copula.

Proposition 4.1. The distribution function 𝐶^∗_𝑛 of the random vector (𝑍^∗_𝑛,𝑊^∗_𝑛) is a copula which may be expressed in terms of the empirical copula as follows: 𝐶^∗_𝑛(𝑢,𝑣)=[1−𝑛𝑢+⌊𝑛𝑢⌋][1−𝑛𝑣+⌊𝑛𝑣⌋]𝐶_𝑛(⌊𝑛𝑢⌋𝑛,⌊𝑛𝑣⌋𝑛)+[𝑛𝑢−⌊𝑛𝑢⌋][1−𝑛𝑣+⌊𝑛𝑣⌋]𝐶_𝑛(⌊𝑛𝑢⌋+1𝑛,⌊𝑛𝑣⌋𝑛)+[1−𝑛𝑢+⌊𝑛𝑢⌋][𝑛𝑣−⌊𝑛𝑣⌋]𝐶_𝑛(⌊𝑛𝑢⌋𝑛,⌊𝑛𝑣⌋+1𝑛)+[𝑛𝑢−⌊𝑛𝑢⌋][𝑛𝑣−⌊𝑛𝑣⌋]𝐶_𝑛(⌊𝑛𝑢⌋+1𝑛,⌊𝑛𝑣⌋+1𝑛),𝑢,𝑣∈[0,1], (4.7) where ⌊𝑥⌋ is the integer part of 𝑥.

Proof. For any 𝑢∈[𝑖/𝑛,(𝑖+1)/𝑛], 𝑖=0,…,𝑛−1, one sees from the definition of 𝑍^∗_𝑛 that 𝑃(𝑍^∗_𝑛1≤𝑢)=𝑛^𝑛_𝑘=1𝑃(𝑈_𝑛≥𝑘𝑛−𝑢), (4.8) and by using the fact that 𝑃(𝑈_𝑛≥𝑘𝑛−𝑢)=𝕀_{(𝑘≤𝑖)}+(𝑛𝑢−𝑖)𝕀_{(𝑘=𝑖+1)}, (4.9) it follows that 𝑃(𝑍^∗_𝑛≤𝑢)=𝑢, which ensures that 𝑍^∗_𝑛 is uniformly distributed in [0,1]. Similar arguments imply that 𝑊^∗_𝑛 is also uniformly distributed in [0,1], so that 𝐶^∗_𝑛 is a copula.
Now, we show the expression of 𝐶^∗_𝑛 given in (4.7). Let (𝑢,𝑣) be in the set [𝑖/𝑛,(𝑖+1)/𝑛]×[𝑗/𝑛,(𝑗+1)/𝑛], 𝑖,𝑗=0,…,𝑛−1. In view of relations (4.6), one has 𝐶^∗_𝑛(𝑢,𝑣)=𝑃(𝑍^∗_𝑛≤𝑢,𝑊^∗_𝑛=≤𝑣)_{𝑛𝑛𝑘=1𝑝=1}𝑃(𝑈_𝑛≥𝑘𝑛−𝑢)𝑃(𝑉_𝑛≥𝑝𝑛−𝑣)𝑃(𝑍_𝑛=𝑘,𝑊_𝑛==𝑝)_{𝑛𝑛𝑘=1𝑝=1}[𝕀_{(𝑘≤𝑖)}+(𝑛𝑢−𝑖)𝕀_{(𝑘=𝑖+1)}][𝕀_{(𝑝≤𝑗)}+(𝑛𝑣−𝑗)𝕀_{(𝑝=𝑗+1)}]𝑃(𝑍_𝑛=𝑘,𝑊_𝑛=𝑝). (4.10) After simplifications, one observes 𝐶^∗_𝑛(𝑢,𝑣)=𝐶_𝑛(𝑖𝑛,𝑗𝑛)+(𝑛𝑢−𝑖)[𝐶_𝑛(𝑖+1𝑛,𝑗𝑛)−𝐶_𝑛(𝑖𝑛,𝑗𝑛)]+(𝑛𝑣−𝑗)[𝐶_𝑛(𝑖𝑛,𝑗+1𝑛)−𝐶_𝑛(𝑖𝑛,𝑗𝑛)]+(𝑛𝑢−𝑖)(𝑛𝑣−𝑗)[𝐶_𝑛(𝑖+1𝑛,𝑗+1𝑛)−𝐶_𝑛(𝑖+1𝑛,𝑗𝑛)−𝐶_𝑛(𝑖𝑛,𝑗+1𝑛)+𝐶_𝑛(𝑖𝑛,𝑗𝑛)]. (4.11) which can be rewritten as 𝐶^∗_𝑛(𝑢,𝑣)=[1−𝑛𝑢+𝑖][1−𝑛𝑣+𝑗]𝐶_𝑛(𝑖𝑛,𝑗𝑛)+[𝑛𝑢−𝑖][1−𝑛𝑣+𝑗]𝐶_𝑛(𝑖+1𝑛,𝑗𝑛)+[1−𝑛𝑢+𝑖][𝑛𝑣−𝑗]𝐶_𝑛(𝑖𝑛,𝑗+1𝑛)+[𝑛𝑢−𝑖][𝑛𝑣−𝑗]𝐶_𝑛(𝑖+1𝑛,𝑗+1𝑛), (4.12) and hence the result is obtained, since 𝑖=⌊𝑛𝑢⌋ and 𝑗=⌊𝑛𝑣⌋.

Finally, one concludes that it will be convenient to estimate the theoretical copula 𝐶 by using the proposal estimator 𝐶^∗_𝑛 instead of the empirical copula. The reason is that 𝐶^∗_𝑛 is a copula which uses all the points (𝑖/𝑛,𝑗/𝑛), (𝑖/𝑛,(𝑗+1)/𝑛), (𝑖+1/𝑛,𝑗/𝑛), and ((𝑖+1)/𝑛,(1+𝑗)/𝑛) in order to estimate 𝐶 in [𝑖/𝑛,(𝑖+1)/𝑛]×[𝑗/𝑛,(𝑗+1)/𝑛].

5. Understanding Dependence Structure of the Bivariate Poisson Distribution

Our purpose in this section is to study dependence properties of the bivariate Poisson distribution 𝐻 of a random couple (𝑋,𝑌) and the relationship between 𝜏 and 𝜌 and the parameters of 𝐻. Several bivariate Poisson distributions have been proposed in the statistical literature, for example, S. Kocherlakota and K. Kocherlakota [20]. In applied statistics, however, the focus is on the trivariate reduction method described by Johnson et al. [21] who construct the Bivariate Poisson distribution using three independent random variables 𝑋₁, 𝑋₂, and 𝑍 all distributed as Poisson with parameters 𝜆₁, 𝜆₂, and 𝛼, respectively:

𝑋=𝑋₁+𝑍,𝑌=𝑋₂+𝑍. (5.1) The cumulative distribution of (𝑋,𝑌) is given by

𝐻_{𝛼,𝜆1,𝜆2}(𝑖,𝑗)=^𝑖⋀𝑗_𝑘=0𝐹_𝜆1(𝑖−𝑘)𝐹_𝜆2𝛼(𝑗−𝑘)^𝑘𝑒^−𝛼𝑘!, (5.2) where 𝐹_𝜆𝑖 denotes the cdf of 𝑋_𝑖, 𝑖=1,2. We notice that 𝑋 and 𝑌 are Poisson model with means 𝜆₁+𝛼 and 𝜆₂+𝛼, respectively. Note that the covariance and the correlation between 𝑋 and 𝑌 are expressed by

𝛼cov(𝑋,𝑌)=𝛼,corr(𝑋,𝑌)=(𝜆₁+𝛼)(𝜆₂+𝛼), (5.3) which are positive and nondecreasing functions of 𝛼.

To study further the relationships between 𝛼 and each of 𝜏 and 𝜌 for the bivariate Poisson model, we propose an alternative parametrization which consists in fixing the marginal parameters 𝛼+𝜆₁=𝑚₁ and 𝛼+𝜆₂=𝑚₂. In this context, the cdf (5.2) becomes

𝐻_𝛼(𝑖,𝑗)=^𝑖⋀𝑗_𝑘=0𝐹_𝑚1−𝛼(𝑖−𝑘)𝐹_𝑚2−𝛼𝛼(𝑗−𝑘)^𝑘𝑒^−𝛼.𝑘! (5.4) As a consequence of the above representation, we can see {𝐻_𝛼} as a family of bivariate Poisson models with fixed marginals which are univariate Poisson models with parameters 𝑚₁ and 𝑚₂, respectively. This means that the set {𝐻_𝛼}, 0≤𝛼≤𝑚₁⋀𝑚₂ is included in the particular Fréchet space Γ(𝐹_𝑚1,𝐹_𝑚2), where 𝐹_𝑚𝑖 denotes the cdf of a Poisson model with mean 𝑚_𝑖, 𝑖=1,2. The advantage of the parametrization (5.4) rather than (5.2) is that the coefficient 𝛼 may be interpreted as a dependence parameter in the family {𝐻_𝛼}.

Now, let 𝜏_𝛼 and 𝜌_𝛼 be Kendall's 𝜏 and Spearman's 𝜌 associated with the distribution 𝐻_𝛼. The result below provides the monotonicity of 𝜏_𝛼 and 𝜌_𝛼 as functions of 𝛼.

Proposition 5.1. Let 𝐻_𝛼1 and 𝐻_𝛼2 be two cdf of the set {𝐻_𝛼}. Then, 𝛼₁≤𝛼₂⇒𝐻_𝛼1≤𝐻_𝛼2, (5.5) and consequently, 𝛼₁≤𝛼₂⇒𝜏_𝛼1≤𝜏_𝛼2,𝜌_𝛼1≤𝜌_𝛼2. (5.6)

Proof. From (5.4), 𝜕𝐻_𝛼(𝑖,𝑗)=𝜕𝛼^𝑖⋀𝑗_𝑘=0𝜕𝐹_𝑚1−𝛼(𝑖−𝑘)𝐹𝜕𝛼^𝑚2−𝛼𝛼(𝑗−𝑘)^𝑘𝑒^−𝛼+𝑘!_{𝑖⋀𝑗𝑘=0}𝐹_𝑚1−𝛼(𝑖−𝑘)𝜕𝐹_𝑚2−𝛼(𝑗−𝑘)𝛼𝜕𝛼^𝑘𝑒^−𝛼+𝑘!_{𝑖⋀𝑗𝑘=0}𝐹_𝑚1−𝛼(𝑖−𝑘)𝐹_𝑚2−𝛼𝛼(𝑗−𝑘)[^𝑘−1𝑒^−𝛼−𝛼(𝑘−1)!^𝑘𝑒^−𝛼𝑘!], (5.7) and using the fact that 𝜕𝐹_𝑚1−𝛼(𝑖−𝑘)𝜕𝛼=𝐹_𝑚1−𝛼(𝑖−𝑘)−𝐹_𝑚1−𝛼(𝑖−𝑘−1),𝜕𝐹_𝑚2−𝛼(𝑗−𝑘)𝜕𝛼=𝐹_𝑚2−𝛼(𝑗−𝑘)−𝐹_𝑚2−𝛼(𝑗−𝑘−1), (5.8)(5.7) becomes, upon simplifications, 𝜕𝐻_𝛼(𝑖,𝑗)=𝜕𝛼^𝑖⋀𝑗_𝑘=0(𝑚₁−𝛼)^𝑖−𝑘𝑒^{−(𝑚1−𝛼)}(𝑖−𝑘)!(𝑚₂−𝛼)^𝑗−𝑘𝑒^{−(𝑚2−𝛼)}𝛼(𝑗−𝑘)!^𝑘𝑒^−𝛼𝑘!≥0. (5.9) Therefore (5.9) together with Proposition 2.1 provides (5.5) and (5.6).

Many statistical researches have focused on studying concepts of positive dependence for bivariate distributions, example right tail increasing, and positive quadrant dependence which are widely used in actuarial literature [22]. There are natural relationships between dependence properties and measures of concordance. An interesting property of positive dependence is the concept of positive quadrant dependence (PQD) defined as follows: let (𝑋,𝑌) be a random couple valued in ℝ×ℝ with joint cdf 𝐻, and marginals 𝐹 and 𝐺. These random variables are said to be positively quadrant dependent if, and only if, for all (𝑥,𝑦)∈ℝ²

𝐻(𝑥,𝑦)≥𝐹(𝑥)𝐺(𝑦). (5.10) The following corollary is a direct consequence of the previous result.

Corollary 5.2. The family {𝐻_𝛼} is positively quadrant dependent.

Proof. Since 𝐻_𝛼 is a nondecreasing function of 𝛼, then 𝐻₀≤𝐻_𝛼 for all 0≤𝛼≤𝑚₁⋀𝑚₂. Now, from (5.4), 𝐻₀(𝑖,𝑗)=𝐹_𝑚1(𝑖)𝐹_𝑚2(𝑗) for all 𝑖,𝑗. Therefore the family {𝐻_𝛼} is PQD. Consequently, 𝜏_𝛼≥0, 𝜌_𝛼≥0, and 3𝜏_𝛼≥𝜌_𝛼 for all 0≤𝛼≤𝑚₁⋀𝑚₂.

Remark 5.3. When 𝑚₁=𝑚₂=𝑚, the upper bound of the family {𝐻_𝛼} is given by the cdf 𝐻_𝑚, and using (5.4), we then obtain that 𝐻_𝑚(𝑖,𝑗)=𝐹_𝑚(𝑖⋀𝑗)=min[𝐹_𝑚(𝑖),𝐹_𝑚(𝑗)], for all 𝑖,𝑗, which is the upper Fréchet bound.

In order to appreciate the corrections of 𝜏 and 𝜌 given by (2.14), we consider the family of Poisson model {𝐻_𝛼} with marginal parameters 𝑚₁=𝑚₂=2. Using (3.2) and (3.12) with 𝐹_𝑚 instead of 𝐹, we obtain that 𝜌_max=0.951 and 𝜏_max=0.792. Table 1 provides 𝜏_𝛼 and 𝜌_𝛼 with their corrections 𝜏_𝛼,𝑐 and 𝜌_𝛼,𝑐 for chosen values of 𝛼.

From Table 1, we note that the differences 𝐷_𝜏,𝛼=𝜏_𝛼,𝑐−𝜏_𝛼 and 𝐷_𝜌,𝛼=𝜌_𝛼,𝑐−𝜌_𝛼 are increasing as function of the dependence parameter 𝛼. This constatation is true in general because 𝐷_𝜏,𝛼 and 𝐷_𝜌,𝛼 can be expressed as

Table 1 

𝜏_𝛼, 𝜌_𝛼, 𝜏_𝛼,𝑐 and 𝜌_𝛼,𝑐 for the Poisson model.

𝐷_𝜏,𝛼=(1−𝜏_max)𝜏_𝛼𝜏_max,𝐷_𝜌,𝛼=(1−𝜌_max)𝜌_𝛼𝜌_max, (5.11) which shows that these parameters are in fact increasing with 𝛼.

Acknowledgment

The second author acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada.