Measures of Mutually Complete Dependence for Discrete Random Vectors

Abstract

In this paper, a marginal-free measure of mutually complete dependence for discrete random vectors through subcopulas is defined, which generalizes the corresponding results for discrete random variables. Properties of the measure are studied and an estimator of the measure is introduced. Several examples are given for illustration of our results.

1 Introduction

Complete dependence (or functional dependence) is an important concept in many aspects of our life, such as econometrics, insurance, finance, etc. Recently, measures of (mutually) complete dependence have been defined and studied by many authors, e.g. Rényi [8], Schweizer and Wolff [9], Lancaster [5], Siburg and Stoimenov [12], Trutschnig [16], Dette et al. [3], Tasena and Dhompongsa [14], Shan et al. [11], Tasena and Dhompongsa [15] and Boonmee and Tasena [2]. However, measures in above papers have some drawbacks. Some measures only work for continuous random variables or vectors and some measures rely on marginal distributions (See Sect. 2 for a summary of several important measures). To the best of our knowledge, none of previously proposed measures are marginal-free and can describe (mutually) complete dependence for discrete random vectors. To overcome this issue, in this paper, we define a marginal-free measure of (mutually) complete dependence for discrete random vectors by using subcopluas, which extends the corresponding results of discrete random variables given in [11] to multivariate cases.

This paper is organized as follows. Some necessary concepts and definitions, and several measures of (mutually) complete dependence are reviewed briefly in Sect. 2. A marginal-free measure of (mutually) complete dependence for discrete random vectors is defined and properties of this measure are studied in Sect. 3. An estimator of the measure is introduced in Sect. 4.

2 Preliminaries

Let $(\Omega , \mathscr {A}, P)$ be a probability space, where $\Omega$ is a sample space, $\mathscr {A}$ is a $\sigma$-algebra of $\Omega$ and P is a probability measure on $\mathscr {A}$. A random variable is a measurable function from $\Omega$ to the real line $\mathbb {R}$, and for any integer $n\ge 2$, an n -dimensional random vector is a measurable function from $\Omega$ to $\mathbb {R}^n$. For any $a=(a_1, \cdots , a_n)$ and $b=(b_1, \cdots , b_n)\in \mathbb {R}^n$, we say $a\le b$ if and only if $a_i\le b_i$ for all $i=1, \cdots , n$. Let X and Y be random vectors defined on the same probability space. X and Y are said to be independent if and only if $P(X\le x, Y\le y)=P(X\le x)P(Y\le y)$ for all x and y. Y is completely dependent (CD) on X if Y is a measurable function of X almost surely, i.e., there is a measurable function $\phi$ such that $P(Y=\phi (X))=1$. X and Y are said to be mutually completely dependent (MCD) if X and Y are completely dependent on each other.

Let $E_1, \cdots , E_n$ be nonempty subsets of $\mathbb {R}$ and Q a real-valued function with the domain $Dom(Q)=E_1\times \cdots \times E_n$. Let $[a, b]=[a_1, b_1]\times \cdots \times [a_n, b_n]$ such that all vertices of [a, b] belong to Dom(Q). The Q -volume of [a, b] is defined by

$$\begin{aligned} \mathscr {V}_Q([a, b])=\sum sgn(c)Q(c), \end{aligned}$$

where the sum is taken over all vertices $c=(c_1, \cdots , c_n)$ of [a, b], and

An n -dimensional subcopula (or n -subcopula for short) is a function C with the following properties [7].

• (i) The domain of C is $Dom(C)=D_1\times \cdots \times D_n$, where $D_1, \cdots , D_n$ are nonempty subsets of the unit interval $I=[0, 1]$ containing 0 and 1;

• (ii) C is grounded, i.e., for any $u=(u_1, \cdots , u_n)\in Dom(C)$, $C(u)=0$ if at least one $u_i=0$;

• (iii) For any $u_i\in D_i$, $C(1, \cdots , 1, u_i, 1, \cdots , 1)=u_i$, $i=1, \cdots , n$;

• (iv) C is n-increasing, i.e., for any u, $v\in Dom(C)$ such that $u\le v$, $\mathscr {V}_C([u, v])\ge 0.$

For any n random variables $X_1, \cdots , X_n$, by Sklar’s Theorem [13], there is a unique n-subcopula such that

$$\begin{aligned} H(x_1, \cdots , x_n)=C(F_1(x_1), \cdots , F_n(x_n)), \quad \text {for all } (x_1, \cdots , x_n)\in \overline{\mathbb {R}}^n, \end{aligned}$$

where $\overline{\mathbb {R}}=\mathbb {R}\cup \{-\infty , \infty \}$, H is the joint cumulative distribution function (c.d.f.) of $X_1, \cdots , X_n$, and $F_i$ is the marginal c.d.f. of $X_i$, $i=1, \cdots , n$. In addition, if ${X_1}, \cdots , X_n$ are continuous, then $Dom(C)=I^n$ and the unique C is called the n -copula of $X_1, \cdots , X_n$. For more details about the copula theory, see [7].

Next, we are going to recall some measures of MCD and CD, which are equal to 0 if and only if two random variables (or vectors) are independent, and equal to 1 if and only if they are MCD or CD. In 2010, Siburg and Stoimenov [12] defined an MCD measure for continuous random variables as

$$\begin{aligned} \omega (X, Y)=\left( 3\Vert C\Vert ^2-2\right) ^{\frac{1}{2}}, \end{aligned}$$

(1)

where X and Y are continuous random variables with the copula C and $\Vert \cdot \Vert$ is the Sobolev norm of bivariate copulas given by

$$\begin{aligned} \Vert C\Vert =\left( {\int \int }\left| \nabla C(u, v)\right| ^2dudv\right) ^{\frac{1}{2}}, \end{aligned}$$

where $\nabla C(u, v)$ is the gradient of C(u, v).

In 2013, Tasena and Dhompongsa [14] generalized Siburg and Stoimenov’s measure to multivariate cases as follows. Let $X_1, \cdots , X_n$ be continuous variables with the n-copula C. Define

$$\begin{aligned} \delta _i(X_1, \cdots , X_n)=\delta _i(C)=\dfrac{\int \cdots \int \left[ \partial _iC(u_1, \cdots , u_n)-\pi _iC(u_1, \cdots , u_n)\right] ^2du_1\cdots du_n}{\int \cdots \int \pi _iC(u_1, \cdots , u_n)(1-\pi _iC(u_1, \cdots , u_n))du_1\cdots du_n}, \end{aligned}$$

where $\partial _iC$ is the partial derivative on the ith coordinate of C and $\pi _iC: I^{n-1}\rightarrow I$ is defined by $\pi _iC(u_1, \cdots , u_{n-1}) =C(u_1, \cdots , u_{i-1}, 1, u_{i}, \cdots , u_{n-1})$, $i=1, 2, \cdots , n.$ Let

$$\begin{aligned} \delta (X_1, \cdots , X_n)=\delta (C)=\frac{1}{n}\overset{n}{\underset{i=1}{\sum }}\delta _i(C). \end{aligned}$$

(2)

Then $\delta$ is an MCD measure of $X_1, \cdots , X_n$.

In 2015, Shan et al. [11] considered discrete random variables. Let X and Y be two discrete random variables with the subcopula C. An MCD measure of X and Y is given by

$$\begin{aligned} \mu _t(X, Y)=\left( \frac{\Vert C\Vert _t^2-L_t}{U_t-L_t}\right) ^\frac{1}{2}, \end{aligned}$$

(3)

where $t\in [0, 1]$ and $\Vert C\Vert _t^2$ is the discrete norm of C defined by

$$\begin{aligned} \Vert C\Vert _t^2=\underset{i}{\sum }\underset{j}{\sum } \left\{ \left( tC^2_{\varDelta i, j}+(1-t)C^2_{\varDelta i, j+1}\right) \frac{\varDelta v_j}{\varDelta u_i} + \left( tC^2_{i, \varDelta j}+(1-t)C^2_{i+1, \varDelta j}\right) \frac{\varDelta u_i}{\varDelta v_j} \right\} , \end{aligned}$$

$$\begin{aligned} C_{\varDelta i, j}=C(u_{i+1}, v_j)-C(u_{i}, v_j), \quad C_{i, \varDelta j}=C(u_{i}, v_{j+1})-C(u_{i}, v_j),\end{aligned}$$

$$\begin{aligned} \varDelta u_i=u_{i+1}-u_i, \quad \varDelta v_j=v_{j+1}-v_j,\end{aligned}$$

$$\begin{aligned} L_t=\underset{i}{\sum }(tu_i^2+(1-t)u_{i+1}^2)\varDelta u_i+\underset{j}{\sum }(tv_j^2+(1-t)v_{j+1}^2)\varDelta v_j,\end{aligned}$$

and

$$\begin{aligned} U_t=\underset{i}{\sum }(tu_i+(1-t)u_{i+1})\varDelta u_i+\underset{j}{\sum }(tv_j+(1-t)v_{j+1})\varDelta v_j. \end{aligned}$$

In 2016 Tasena and Dhompongsa [15] defined a measure of CD for random vectors. Let X and Y be two random vectors. Define

$$\begin{aligned} \omega _k(Y|X)=\left[ \int \int \left| F_{Y|X}(y|x)-\frac{1}{2}\right| ^kdF_X(x)dF_Y(y)\right] ^\frac{1}{k}, \end{aligned}$$

where $k\ge 1$. The measure of Y CD on X is given by

$$\begin{aligned} \overline{\omega }_k(Y|X)=\left[ \frac{\omega ^k_k(Y|X)-\omega ^k_k(Y^\bot |X^\bot )}{\omega ^k_k(Y|Y)-\omega ^k_k(Y^\bot |X^\bot )}\right] ^{\frac{1}{k}}, \end{aligned}$$

(4)

where $X^\bot$ and $Y^\bot$ are independent random vectors with the same distributions as X and Y, respectively.

In the same period, Boonmee and Tasena [2] defined a measure of CD for continuous random vectors by using linkages which were introduced by Li et al. [6]. Let X and Y be two continuous random vectors with the linkage C. The measure of Y being completely dependent on X is defined by

$$\begin{aligned} \zeta _p(Y|X)=\left[ \int \int \Big |\frac{\partial }{\partial u}C(u, v)-\varPi (v)\Big |^pdudv\right] ^{\frac{1}{p}}, \end{aligned}$$

(5)

where $\varPi (v)=\overset{n}{\underset{i=1}{\varPi }}v_i$ for all $v=(v_1, \cdots , v_n)\in I^n$.

From above summaries we can see that measures given by (1), (2) and (5) only work for continuous random variables or vectors. The measure defined by (3) only works for bivariate discrete random variables. The measure given by (4) relies on marginal distributions of random vectors. Thus it is worth considering marginal-free measures of CD and MCD for discrete random vectors.

3 An MCD Measure for Discrete Random Vectors

In this section, we identify $\mathbb {R}^{n_1+\cdots +n_k}$ with $\mathbb {R}^{n_1}\times \cdots \times \mathbb {R}^{n_k}$, where $n_1, \cdots , n_k$ are positive integers. So any n-dimensional random vector X can be viewed as a tuple of n random variables, i.e., $X=(X_1, \cdots , X_n)$, where $X_1, \cdots , X_n$ are random variables. Also, if $Y=(Y_1, \cdots , Y_m)$ is an m-dimensional random vector, we use (X, Y) to denote the $(n+m)$-dimensional random vector $(X_1, \cdots , X_n, Y_1, \cdots , Y_m)$. Let $\psi =(\psi _1, \cdots , \psi _n): \mathbb {R}^n\rightarrow \mathbb {R}^n$ be a function. $\psi$ is said to be strictly increasing if and only if each component $\psi _i: \mathbb {R}\rightarrow \mathbb {R}$ is strictly increasing, i.e., for any $a_i$ and $b_i \in \mathbb {R}$ such that $a_i<b_i$, we have $\psi _i(a_i)<\psi _i(b_i)$, $i=1, \cdots , n$.

We will focus on discrete random vectors in this section, i.e., they can take on at most a countable number of possible values. Let $X=(X_1, \cdots , X_n)\in \mathcal{L}_1\subseteq \mathbb {R}^n$ and $Y=(Y_1, \cdots , Y_m)\in \mathcal{L}_2\subseteq \mathbb {R}^m$ be two discrete random vectors defined on the same probability space $(\Omega , \mathscr {A}, P)$. Their joint c.d.f. H, marginal c.d.f.’s F and G, and marginal probability mass functions (p.m.f.) f and g are defined, respectively, as follows.

$$\begin{aligned} H(x, y)=P(X\le x, Y\le y), \quad F(x)=P(X\le x), \quad G(y)=P(Y\le y), \end{aligned}$$

$$\begin{aligned} f(x)=P(X=x), \quad \text {and}\quad g(y)=P(Y=y),\quad \text {for all} x\in \mathbb {R}^n \text {and}y\in \mathbb {R}^m. \end{aligned}$$

Also, we use $F_i$ and $f_i$ to denote the marginal c.d.f. and p.m.f. of the ith component $X_i$ of X, $i=1, \cdots , n$, and use $G_j$ and $g_j$ to denote the marginal c.d.f. and p.m.f. of the jth component $Y_j$ of Y, $j=1, \cdots , m$, respectively. For simplicity, we assume that $f(x)\ne 0$ and $g(y)\ne 0$ for all $x\in \mathcal{L}_1$ and $y\in \mathcal{L}_2$. If C is the subcopula of the $(n+m)$-dimensional random vector (X, Y), i.e.,

$$\begin{aligned} H(x, y)=C(u(x), v(y)), \quad \text {for all} x\in \overline{\mathbb {R}}^n\ \text {and} y\in \overline{\mathbb {R}}^m, \end{aligned}$$

where

$$\begin{aligned} u(x)=(u_1(x_1), \cdots , u_n(x_n))=(F_1(x_1), \cdots , F_n(x_n))\in I^n, \end{aligned}$$

and

$$\begin{aligned} v(y)=(v_1(y_1), \cdots , v_m(y_m))=(G_1(y_1), \cdots , G_m(y_m))\in I^m, \end{aligned}$$

for all $x\in \mathcal{L}_1$ and $y\in \mathcal{L}_2$, then C is said to be the subcopula of X and Y. In addition, for each vector $e\in E_1\times \cdots \times E_n$, where $E_i$ is a countable subset of $\overline{\mathbb {R}}$, let $e_L$ be the greatest lower bound of e with respect to the coordinate-wise order, i.e., $e_L=(e_1^\prime , \cdots , e_n^\prime )$ such that if there exists some element in $E_i$ that is less than $e_i$, then $e_i^\prime$ is the greatest element in $E_i$ so that $e_i^\prime < e_i$, otherwise $e_i^\prime =e_i$, $i=1, \cdots , n$. We use ${1}_n$ and $\infty _n$ to denote the n-dimensional constant vector $(1, \cdots , 1)$ and $(\infty , \cdots , \infty )\in \overline{\mathbb {R}}^n$.

To construct desired measures, a distance between two discrete random vectors is defined as follows.

Definition 1

Let X and Y be discrete random vectors. The distance between the conditional distribution of Y given X and marginal distribution of Y is defined by

$$\begin{aligned} \omega ^2(Y|X)=\underset{y\in \mathcal{L}_2,}{\sum }\underset{x\in \mathcal{L}_1}{\sum }[P(Y\le y|X=x)-G(y)]^2f(x)g(y). \end{aligned}$$

(6)

From the above definition, we can obtain the following two results.

Lemma 1

For any discrete random vectors X and Y, we have $\omega ^2(Y|X)\le \omega _{max}^2(Y|X)$, where

$$\begin{aligned} \omega _{max}^2(Y|X)=\underset{y\in \mathcal{L}_2}{\sum }\left[ G(y)-(G(y))^2\right] g(y). \end{aligned}$$

Proof

By the definition,

$$\begin{aligned} \omega ^2(Y|X)&=\underset{y\in \mathcal{L}_2,}{\sum }\underset{x\in \mathcal{L}_1}{\sum }[P(Y\le y|X=x)-G(y)]^2f(x)g(y)\\&=\underset{y\in \mathcal{L}_2,}{\sum }\underset{x\in \mathcal{L}_1}{\sum }[P(Y\le y|X=x)^2-2P(Y\le y|X=x)G(y)+(G(y))^2]f(x)g(y)\\&\le \underset{y\in \mathcal{L}_2,}{\sum }\underset{x\in \mathcal{L}_1}{\sum }[P(Y\le y|X=x)-2P(Y\le y|X=x)G(y)+(G(y))^2]f(x)g(y)\\&=\underset{y\in \mathcal{L}_2,}{\sum }\underset{x\in \mathcal{L}_1}{\sum }[P(X=x, Y\le y)-2P(X=x, Y\le y)G(y)+(G(y))^2f(x)]g(y)\\&=\underset{y\in \mathcal{L}_2}{\sum }\left[ G(y)-(G(y))^2\right] g(y). \end{aligned}$$

   $\square$

Lemma 2

Let X and Y be discrete random vectors. There is a function $\phi : \mathcal{L}_1\rightarrow \mathcal{L}_2$ such that $\phi (X)=Y$ if and only if $\omega ^2(Y|X)=\omega _{max}^2(Y|X)$, i.e., $P(Y\le y|X=x)= 0$ or 1 for all $(x, y)\in \mathcal{L}_1\times \mathcal{L}_2$.

Proof

For “if” part, suppose that $P(Y\le y|X=x)= 0$ or 1 for all $(x, y)\in \mathcal{L}_1\times \mathcal{L}_2$. Then $\underset{t\le y}{\sum }P(Y=t|X=x)= 0$ or 1. So $\underset{t\le y}{\sum }P(X=x, Y=t)= 0$ or $P(X=x)$. Thus there exists a unique $y(x)\in \mathcal{L}_2$, which depends on x, such that $P(X=x, Y=y(x))=P(X=x)$, and $P(X=x, Y=y)=0$ for all $y\in \mathcal{L}_2$ with $y\ne y(x)$. Now if we define $\phi (x)=y(x)$ for all $x\in \mathcal{L}_1$, then $\phi (X)=Y$.

For “only if” part, suppose that $\phi (X)=Y$. Fix $x\in \mathcal{L}_1$. It is sufficient to show that $P(X=x, Y=y)=0$ for all $y\ne \phi (x)$. Suppose that, on the contrary, there is $y^\prime \in \mathcal{L}_2$ such that $y^\prime \ne \phi (x)$ and $P(X=x, Y=y^\prime )\ne 0$, then there exists $\omega \in \Omega$ so that $X(\omega )=x$ and $Y(\omega )=y^\prime$. So we have $\phi (X)(\omega )\ne Y(\omega )$. It’s a contradiction.    $\Box$

Now we can define a measure of CD for two discrete random vectors as follows.

Definition 2

For any discrete random vectors X and Y, the measure of Y being completely dependent on X is given by

$$\begin{aligned} \mu (Y|X)=\left[ \frac{\omega ^2(Y|X)}{\omega _{max}^2(Y|X)}\right] ^\frac{1}{2} =\left[ \frac{\underset{y\in \mathcal{L}_2,}{\sum }\underset{x\in \mathcal{L}_1}{\sum }[P(Y\le y|X=x)-G(y)]^2f(x)g(y)}{\underset{y\in \mathcal{L}_2}{\sum }\left[ G(y)-(G(y))^2\right] g(y)}\right] ^\frac{1}{2}. \end{aligned}$$

(7)

Properties of the measure $\mu (Y|X)$ are given as follows.

Theorem 1

For any discrete random vectors X and Y, the measure $\mu (Y|X)$ has the following properties:

• (i) $0\le \mu (Y|X)\le 1;$

• (ii) $\mu (Y|X)=0$ if and only if X and Y are independent;

• (iii) $\mu (Y|X)=1$ if and only if Y is a function of X;

• (iv) $\mu (Y|X)$ is invariant under strictly increasing transformations of X and Y, i.e., if $\psi _1$ and $\psi _2$ are strictly increasing functions defined on $\mathcal{L}_1$ and $\mathcal{L}_2$, respectively, then $\mu \left( \psi _2(Y)|\psi _1(X)\right) =\mu (Y|X)$.

Proof

Property (i) is obvious by Lemma 1. For Property (ii), note that $\mu (Y|X)=0$ if and only if $\omega ^2(Y|X)=0$. It is equivalent to $P(Y\le y|X=x)=G(y)$ for all $(x, y)\in \mathcal{L}_1\times \mathcal{L}_2$, i.e., X and Y are independent. Property (iii) follows Lemma 2. Lastly, since $\psi _1$ and $\psi _2$ are strictly increasing, we have $P(\psi _1(X)\le \psi _1(x))=P(X\le x)$ and $P(\psi _2(Y)\le \psi _2(y))=P(Y\le y)$. Thus Property (iv) holds.    $\Box$

Remark 1

• (i) It can be shown that the measure given by (7) is a discrete version of the measure $\overline{\omega }_k(Y|X)$ for random vectors given by (4) with $k=2$. The difference here is that based on (7), we are going to define a marginal-free measure by using subcopulas for discrete random vectors.

• (ii) The above measure may be simplified into

  $$\begin{aligned} \mu ^\prime (Y|X)=\left[ \frac{\underset{y\in \mathcal{L}_2,}{\sum }\underset{x\in \mathcal{L}_1}{\sum }[P(Y\le y|X=x)-G(y)]^2f(x)}{\underset{y\in \mathcal{L}_2}{\sum }\left[ G(y)-(G(y))^2\right] }\right] ^\frac{1}{2}. \end{aligned}$$

  (8)

  As indicated by Shan [10], $\mu ^\prime (Y|X)$ is well defined only if Y is a finite discrete random vector, i.e., if $\mathcal{L}_2$ is a finite set. Otherwise, $\underset{y\in \mathcal{L}_2}{\sum }\left[ G(y)-(G(y))^2\right]$ may diverge. However, the measure $\mu (Y|X)$ given by (7) is well defined for all discrete random vectors.

• (iii) It is easy to see that $\omega _{max}^2(Y|X)=0$ if and only if Y is a constant random vector, i.e., if and only if there is $y\in \mathbb {R}^m$ such that $P(Y=y)=1$. In this case, Y is clearly a function of X. Thus, without loss of generality, we assume that X and Y are not constant random vectors.

Since most multivariate dependence properties of random variables can be determined by their subcopula C, we are going to redefine the measure $\mu (Y|X)$ by using subcopulas such that $\mu (Y|X)$ is free of marginal distributions of X and Y. First, note that for any $x\in \mathcal{L}_1$ and $y\in \mathcal{L}_2$, we have

$$\begin{aligned} G(y) = H(\infty _n, y)=C(1_n, v(y)), \end{aligned}$$

(9)

$$\begin{aligned} f(x)= \mathscr {V}_H([(x_L, \infty _m), (x, \infty _m) ])=\mathscr {V}_C([(u(x)_L, 1_m), (u(x), 1_m)]), \end{aligned}$$

(10)

$$\begin{aligned} g(y)= \mathscr {V}_H([(\infty _n, y_L), (\infty _n, y)])=\mathscr {V}_C([(1_n, v(y)_L), (1_n, v(y))]), \end{aligned}$$

(11)

and

$$\begin{aligned} P(Y\le y|X=x)&=\dfrac{P(X=x, Y\le y)}{P(X=x)} \nonumber \\&= \dfrac{\mathscr {V}_H([(x_L, y), (x, y)])}{\mathscr {V}_H([(x_L, \infty _m), (x, \infty _m) ])} \nonumber \\&= \dfrac{\mathscr {V}_C([(u(x)_L, v(y)), (u(x), v(y))])}{\mathscr {V}_C([(u(x)_L, 1_m), (u(x), 1_m)])} \end{aligned}$$

(12)

Thus, from Eqs. (9)–(12), we can redefine $\mu (Y|X)$ as follows.

Definition 3

Let X and Y be two discrete random vectors with the subcopula C. Suppose that the domain of C is $Dom(C)=\mathcal{L}_1^\prime \times \mathcal{L}_2^\prime$, where $\mathcal{L}_1^\prime \subseteq I^n$ and $\mathcal{L}_2^\prime \subseteq I^m$. The measure of Y being completely dependent on X based on C is given by

$$\begin{aligned}&\mu _{C}(Y|X) = \left[ \frac{\omega ^2(Y|X)}{\omega _{max}^2(Y|X)}\right] ^\frac{1}{2} \nonumber \\ =&\left[ \frac{\underset{v\in \mathcal{L}_2^\prime }{\sum }\underset{u\in \mathcal{L}_1^\prime }{\sum } \left[ \frac{\mathscr {V}_C([(u_L, v), (u, v)])}{\mathscr {V}_C([(u_L, 1_m), (u, 1_m)])}-C(1_n, v)\right] ^2\mathscr {V}_C([(u_L, 1_m), (u, 1_m)])\mathscr {V}_C([(1_n, v_L), (1_n, v)])}{\underset{v\in \mathcal{L}_2^\prime }{\sum }\left[ C(1_n, v)-(C(1_n, v))^2\right] \mathscr {V}_C([(1_n, v), (1_n, v_L])}\right] ^\frac{1}{2}. \end{aligned}$$

(13)

Remark 2

Based on the same idea, if X and Y are continuous random vectors with the unique copula C, the measure $\mu _{C}(Y|X)$ given by (13) can be rewritten as

$$\begin{aligned} \mu _{C}(Y|X))=\left[ \frac{\int \int \left( \frac{\partial C}{\partial C_X}-C_Y\right) ^2\frac{\partial C_X}{\partial u}\frac{\partial C_Y}{\partial v}dudv}{\int C_Y(1-C_Y)\frac{\partial C_Y}{\partial v}dv}\right] ^{\frac{1}{2}}, \end{aligned}$$

where $C_X$ and $C_Y$ are copulas of X and Y. This is a marginal-free measure of CD for continuous random vectors.

By using $\mu _C(Y|X)$ defined in Definition 3, we can define a marginal-free measure of mutual complete dependence for two discrete random vectors as follows.

Definition 4

For any discrete random vectors X and Y with the subcopula C, the MCD measure of X and Y is defined by

$$\begin{aligned} \mu _{C}(X, Y) = \left[ \frac{\omega ^2(Y|X)+\omega ^2(X|Y)}{\omega _{max}^2(Y|X)+\omega _{max}^2(X|Y)}\right] ^\frac{1}{2}, \end{aligned}$$

(14)

where $\omega ^2(\cdot |\cdot )$ and $\omega _{max}^2(\cdot |\cdot )$ are the same as those given in Definition 3.

The properties of the measure $\mu _{C}(X, Y)$ are given in the following theorem. The proof is straightforward.

Theorem 2

Let X and Y be two discrete random vectors with the subcopula C. The measure $\mu _{C}(X, Y)$ has following properties,

• (i) $\mu _{C}(X, Y)=\mu _{C}(Y, X)$;

• (ii) $0\le \mu _{C}(X, Y)\le 1$;

• (iii) $\mu _{C}(X, Y)=0$ if and only if X and Y are independent;

• (iv) $\mu _{C}(X, Y)=1$ if and only if X and Y are MCD;

• (v) $\mu _{C}(X, Y)$ is invariant under strictly increasing transformations of X and Y.

Remark 3

• (i) The insufficiency of 2-copulas to describe joint distributions with given multivariate marginal distributions was discussed by Genest et al. [4]. Let C be a 2-copula. They showed that

  $$\begin{aligned} H(x_1, \cdots , x_{n_1}, x_{n_1+1}, \cdots , x_{n_1+n_2})=C(H_1(x_1, \cdots , x_{n_1}), H_2(x_{n_1+1}, \cdots , x_{n_1+n_2}))\end{aligned}$$

  defines a $(n_1+n_2)$ dimensional c.d.f., where $n_1+n_2\ge 3$, for all marginal c.d.f.’s $H_1$ and $H_2$ with dimensions $n_1$ and $n_2$, respectively, only if C is the bivariate independence copula, i.e.,

  $$\begin{aligned} C(u, v)=uv, \quad \text {for all } (u, v)\in I^2. \end{aligned}$$

  Thus, in this work, we have to use the $(n+m)$-subcopula of (X, Y) to construct a marginal-free measure.

• (ii) Both Shan et al. [11] and Tasena and Dhompongsa [15] tried to use copulas to construct measures of functional dependence for discrete random variables or vectors. However, we do not think that copulas should be used to construct measures for discrete random variables or vectors because, for fixed discrete random variables or vectors, the corresponding copulas may not be unique. Thus, as shown in their papers, if we have different copulas for two fixed discrete random variables, copula-based measures may give us different results.

• (iii) Boonmee and Tasena [2] used linkages to construct a marginal-free measure of CD for continuous random vectors, but linkages have some defects. First, linkages are defined for continuous random vectors. Second, to find the linkage of two random vectors, they need to be transformed to uniform random vectors. It is not convenient in applications (See Li et al. [6] for more details of linkages). Thus, in this work, we prefer to use the subcopula of (X, Y) to construct marginal-free measures, since subcopulas are not only good for discrete random vectors but also more popular than linkages.

• (iv) If both X and Y are discrete random variables with the 2-subcopula C, then we have

  $$\begin{aligned} \omega ^2(Y|X)= \underset{v\in \mathcal{L}_2^\prime }{\sum }\underset{u\in \mathcal{L}_1^\prime }{\sum } \left[ \frac{C(u, v)-C(u_L, v)^2}{u-u_L}-v\right] ^2(u-u_L)(v-v_L), \end{aligned}$$
  $$\begin{aligned} \omega ^2(X|Y)= \underset{u\in \mathcal{L}_1^\prime }{\sum }\underset{v\in \mathcal{L}_2^\prime }{\sum } \left[ \frac{C(u, v)-C(u, v_L)^2}{v-v_L}-u\right] ^2(u-u_L)(v-v_L), \end{aligned}$$
  $$\begin{aligned} \omega ^2_{max}(Y|X)=\underset{v\in \mathcal{L}_2^\prime }{\sum }(v-v^2)(v-v_L) \quad \text {and}\quad \omega ^2_{max}(X|Y)=\underset{u\in \mathcal{L}_1^\prime }{\sum }(u-u^2)(u-u_L). \end{aligned}$$

  In this case, the measure $\mu _C(X, Y) = \left[ \frac{\omega ^2(Y|X)+\omega ^2(X|Y)}{\omega _{max}^2(Y|X)+\omega _{max}^2(X|Y)}\right] ^\frac{1}{2}$ is identical to the measure given by (3) with $t=0$.

• (v) If both X and Y are continuous random variables, i.e., $\max \{u-u_L, v-v_L\}\rightarrow 0$, then it can be show that

  $$\begin{aligned} \mu _C(X, Y)&= \left[ \frac{\omega ^2(Y|X)+\omega ^2(X|Y)}{\omega _{max}^2(Y|X)+\omega _{max}^2(X|Y)}\right] ^\frac{1}{2}\\&=\left\{ 3\int \int \left[ \left( \dfrac{\partial C}{\partial u}\right) ^2+\left( \frac{\partial C}{\partial v}\right) ^2\right] dudv-2\right\} ^\frac{1}{2}, \end{aligned}$$

  which is identical to the measure given by (1).

Next, we use two examples to illustrate our above results.

Example 1

Let $X=(X_1, X_2)$ be a random vector with the distribution given in Table 1. Let $Y=(Y_1, Y_2)=(X^2_1, X_2^2)$. Then the distribution of Y, the joint distribution of X and Y, and the corresponding subcopula are given in Tables 2, 3 and 4, respectively. It is easy to show that $\omega ^2(Y|X)=\omega ^2_{max}(Y|X)=161/1458$, $\omega ^2(X|Y)=2699/38880$, $\omega ^2_{max}(X|Y)=469/2916$. So $\mu _{C}(Y|X)=1$. $\mu _{C}(X|Y)=0.6569$ and $\mu _{C}(X, Y)=0.8142$.

Table 1. Distribution of X.

Table 2. Distribution of Y.

Table 3. Joint distribution of X and Y.

Table 4. Subcopula of X and Y.

Table 5. Joint distribution of X and Y.

Example 2

Let $X=(X_1, X_2)$ be a discrete random vector, where $X_1$ is a geometric random variable with the success rate $p=\frac{1}{2}$, $X_2$ is a binomial random variable with the number of trails $n=2$ and the success rate $p=\frac{1}{2}$, and $X_1$ and $X_2$ are independent. Let $Y=X_1-X_2$. Then the joint distribution and subcopula of X and Y are given in Tables 5 and 6. By calculation, $\omega ^2(Y|X)=\omega ^2_{max}(Y|X)=1223/7168$, $\omega ^2(X|Y)=3407543/30965760$ and $\omega ^2_{max}(X|Y)=1/3$. So $\mu _{C}(Y|X)=1$, $\mu _{C}(X|Y)=0.3301$ and $\mu _{C}(X, Y)=0.5569$.

Table 6. Subcopula of X and Y.

4 Estimators of $\mu (Y|X)$ and $\mu (X, Y)$

In this section, we are going to construct estimators of measures $\mu (Y|X)$ and $\mu (X, Y)$. Let $X\in \mathcal{L}_1$ and $Y \in \mathcal{L}_2$ be two discrete random vectors and $[n_{xy}]$ be their observed multi-way contingency table. Suppose that the total number of observation is n. For every $x\in \mathcal{L}_1$ and $y\in \mathcal{L}_2$, let $n_{xy}$, $n_{x\cdot }$ and $n_{\cdot y}$ be numbers of observations of (x, y), x and y, respectively, i.e., $n_{x\cdot }=\underset{y\in \mathcal{L}_2}{\sum }n_{xy}$ and $n_{\cdot y}=\underset{x\in \mathcal{L}_1}{\sum }n_{xy}$. If we define $\hat{p}_{xy}=n_{xy}/n$, $\hat{p}_{x\cdot }=n_{x\cdot }/n$, $\hat{p}_{\cdot y}=n_{\cdot y}/n$, $\hat{p}_{y|x}=\hat{p}_{xy}/\hat{p}_{x\cdot }=n_{xy}/n_{x\cdot }$ and $\hat{p}_{x|y}=\hat{p}_{xy}/\hat{p}_{\cdot y}=n_{xy}/n_{\cdot y}$, then estimators of measures $\mu (Y|X)$ and $\mu (X, Y)$ can be defined as follows.

Definition 5

Let $X\in \mathcal{L}_1$ and $Y \in \mathcal{L}_2$ be two discrete random vectors with a multi-way contingency table $[n_{xy}]$. Estimators of $\mu (Y|X)$ and $\mu (X, Y)$ are given by

$$\begin{aligned} \hat{\mu }(Y|X)\left[ \frac{\hat{\omega }^2(Y|X)}{\hat{\omega }_{max}^2(Y|X)}\right] ^\frac{1}{2}, \end{aligned}$$

and

$$\begin{aligned} \hat{\mu }(X, Y) = \left[ \frac{\hat{\omega }^2(Y|X)+\hat{\omega }^2(X|Y)}{\hat{\omega }_{max}^2(Y|X)+\hat{\omega }_{max}^2(X|Y)}\right] ^\frac{1}{2}, \end{aligned}$$

where

$$\begin{aligned} \hat{\omega }^2(Y|X)=\underset{y\in \mathcal{L}_2,}{\sum }\underset{x\in \mathcal{L}_1}{\sum }\left[ \underset{y^\prime \le y,}{\sum }\left( \hat{p}_{y^\prime |x}- \hat{p}_{\cdot y^\prime }\right) \right] ^2\hat{p}_{x\cdot }\hat{p}_{\cdot y}, \end{aligned}$$

$$\begin{aligned} \hat{\omega }_{max}^2(Y|X)=\underset{y\in \mathcal{L}_2}{\sum }\left[ \underset{y^\prime \le y,}{\sum }\hat{p}_{\cdot y^\prime }- \left( \underset{y^\prime \le y,}{\sum }\hat{p}_{\cdot y^\prime }\right) ^2\right] \hat{p}_{\cdot y}, \end{aligned}$$

and $\hat{\omega }^2(X|Y)$ and $\hat{\omega }_{max}^2(X|Y)$ are similarly defined as $\hat{\omega }^2(Y|X)$ and $\hat{\omega }_{max}^2(Y|X)$ by interchanging X and Y.

From the above definition, we have the following result. The proof is trivial.

Theorem 3

Let X and Y be discrete random vectors. Estimators $\hat{\mu }(Y|X)$ and $\hat{\mu }(X, Y)$ have following properties,

• (i) $0\le \hat{\mu }(Y|X), \hat{\mu }(X, Y)\le 1$;

• (ii) X and Y are empirically independent, i.e., $\hat{p}_{xy}=\hat{p}_{x\cdot }\hat{p}_{\cdot y}$ for all $(x, y)\in \mathcal{L}_1\times \mathcal{L}_2$ if and only if $\hat{\mu }(Y|X)=\hat{\mu }(X, Y)=0$;

• (iii) $\hat{\mu }(Y|X)=1$ if and only if Y is a function of X. And $\hat{\mu }(X, Y)=1$ if and only if X and Y are functions of each other.

Table 7. Contingency table of X and Y.

Table 8. Contingency table of X and Y.

Next, we use two example to illustrate our results.

Example 3

Suppose that we have the following multi-way contingency tables. Then from Table 7, we have $\hat{\mu }(Y|X)=0.0516$, $\hat{\mu }(X|Y)=0.0762$ and $\hat{\mu }(X, Y)=0.0642$, so X and Y have very weak functional relations. However, from Table 8, we have $\hat{\mu }(Y|X)=0.5746$, $\hat{\mu }(X|Y)=0.0465$ and $\hat{\mu }(X, Y)=0.3485$, so the functional dependence of Y on X is much stronger than the functional dependence of X on Y.

Example 4

The data given in Table 9 [1] is from a survey conducted by the Wright State University School of Medicine and the United Health Services in Dayton, Ohio. The survey asked students in their final year of a high school near Dayton, Ohio, whether they had ever used alcohol, cigarettes, or marijuana. Denote the variables by A for alcohol use, C for cigarette use, and M for marijuana use. By Pearson’s Chi-squared test (A, C) and M are not independent. The estimations of functional dependence between M and (A, C) are $\hat{\mu }(M|(A, C))=0.3097$, $\hat{\mu }((A, C)|M)=0.2776$ and $\hat{\mu }((A, C), M)=0.2893$.

Table 9. Alcohol (A), Cigarette (C), and Marijuana (M) use for high school seniors.