A Measure of Monotonicity of Two Random Variables

Problem statement: When analyzing random variables it is useful to mea sure the degree of their monotone dependence or compare pairs of rando m variables with respect to their monotonicity. Existing coefficients measure general or linear dep endence of random variables. Developing a measure of monotonicity is useful for practical application s as well as for general theory, since monotonicity s an important type of dependence. Approach: Existing measures of dependence are briefly reviewe d. The Reimann coefficient is generalized to arbitrary random variables with finite variances. Results: The article describes criteria for monotone depende nce of two random variables and introduces a measure of this dependence-monotonicity coefficient. The ad vantages of this coefficient are shown in comparison with other global measures of dependence . It is shown that the monotonicity coefficient satisfies natural conditions for a monotonicity mea sure and that it has properties similar to the properties of the Pearson correlation; in particula r, it equals 1 (-1) if and only if the pair X, Y is comonotonic (counter-monotonic). The monotonicity c oefficient is calculated for some bivariate distributions and the sample version of the coeffic i nt is defined. Conclusion/Recommendations: The monotonicity coefficient should be used to compare pairs of random variables (such as returns from financial assets) with respect to their degree of m notone dependence. In the problems where the monotone relation of two variables has a random noi se, the monotonicity coefficient can be used to estimate variance and other central moments of the noise. By calculating the sample version of the coefficient one will quickly find pairs of monotone dependent variables in a big dataset.


INTRODUCTION
Dependence of random variables is studied and estimated in various applications (see, for example, Tularam et al., 2010). Two most important types of dependence are linear and monotone dependence. Other types include positive and negative quadrant dependence (see Kimeldorf and Sampson, 1987).
Random variables X and Y are called Positively Quadrant Dependent (PQD) if for any x, y ∈R, F X,Y (x, y) ≥ ≥ ≥ ≥ F X (x) ⋅ F Y (y). "Negative Quadrant Dependence (NQD) is defined by reversing the concept of PQD" (Kimeldorf and Sampson, 1987).
Suppose H 1 and H 2 are the joint distribution functions of <X 1 , Y 1 > and <X 2 , Y 2 >, respectively, with the same marginals; then <X 1 , Y 1 > is called more PQD than <X 2 , Y 2 > if for any x, y∈ R, H 1 (x, y) ≥ ≥ ≥ ≥ H 2 (x, y). Kimeldorf and Sampson (1978) introduced the concept of monotone dependence as follows: two continuous random variables X and Y are called monotone dependent if there exists a monotone function g, in which Y = g (X) with probability 1.
Global measures of dependence of two random variables include the Pearson correlation ρ, the coefficients of Spearman, Kendall, Schweizer and Wolff and others (see, for example, Scarsini, 1984;Schweizer and Wolff, 1981) Kimeldorf and Sampson (1978) introduced monotone correlation ρ* and showed that if X and Y are monotone dependent, then ρ* (X, Y) = 1; but the converse is false. Also ρ* does not distinguish between the increasing and decreasing types of monotonicity.
The Spearman, Kendall and Schweizer-Wolff coefficients can be used to measure monotone dependence but they are more appropriate for ordinal variables rather than continuous variables, since they depend only on the ranks of the observations. Schweizer and Wolff (1981) used copulas and metrics to generate several nonparametric measures of dependence. Copulas are a useful technique developed earlier for Pearson, Spearman and Kendall correlations. Reimann (1992) introduced a measure λ** for PQD random variables X and Y. In this article it is generalized to some random variables X and Y with finite variances; it is shown to measure monotonicity.

MONOTONICITY CONDITIONS
In this article we will consider only non-degenerate random variables (a degenerate random variable is constant almost surely).
As usual, for a random variable X with distribution function F X , the quantile function The quantile function 1 X F − is non-decreasing and leftcontinuous. Other simple properties of the quantile function are listed in the following lemma.

Lemma 1:
1) For any u∈[0, 1], x∈R: 1 -u) for all u∈(0, 1), except a countable or finite set of points The concepts of comonotonicity and countermonotonicity were studied by Bauerle and Muller (1998); Denuit and Dhaene (2003); Dhaene et al. (2002); Dempster (2002);Rachev (2003). Comonotonicity of a pair of random variables X and Y means their monotone increasing dependence, i.e., their values change in the same direction. Countermonotonicity of the pair of X and Y means their monotone decreasing dependence, i.e., their values change in opposite directions. Dhaene et al. (2002) give a mathematically accurate definition of comonotonicity for n random variables, which we reproduce here for n = 2.
2) A pair <X, Y > of random variables is said to be comonotonic if it has a comonotonic support.
A counter-monotonic pair <X, Y > is defined by changing the second row in 1) to: either (x 1 ≤ x 2 and y 1 ≥ y 2 ) or (x 1 ≥ x 2 and y 1 ≤ y 2 ) holds.
Three criteria for comonotonicity were proven in (Dhaene et al., 2002). In case n = 2 they have the following form.
Theorem 1: A pair <X, Y > of random variables is comonotonic if and only if one of the following equivalent conditions holds: There exist a random variable Z and nondecreasing functions g, h, such that Here F X, Y denotes the joint distribution function of X, Y and = d denotes equality in distribution. A similar theorem for counter-monotonicity follows.
Theorem 2: The pair <X, Y > is counter-monotonic if and only if one of the following equivalent conditions holds: (4) (∀x, y∈R)[F X,Y (x, y) = max{F X (x) + F Y (y) −1, 0}] (5) For U ~ Uniform (0, 1) There exist a random variable Z, a non-decreasing function g and a non-increasing function h, such that: In the following theorem we prove other criteria for comonotonicity and counter-monotonicity. We assume that the random variables X and Y are defined on the same probability space < Ω, ∑, P >, where ∑ is the collection of all events in this space.

2) <X, Y > is counter-monotonic if and only if <X,
−Y> is comonotonic. Hence part 2) of the Theorem follows from part 1) Note: Clearly X and Y can be interchanged with each of the conditions (7) and (8).
Sometimes the condition (1) in Theorem 1 is taken as the definition for comonotonicity and the condition (4) in Theorem 2 for counter-monotonicity. The criteria in Theorem 3 are more suitable for the definitions of comonotonicity and countermonotonicity, since they reflect their meaning and are similar to the definitions of increasing and decreasing functions.
In the case of continuous marginals F X , F Y , the second inequality in the formulas (7) and (8) can be made more strict. Also in the following two theorems the conditions (2), (3), (5) and (6) are made stronger (the proofs follow from Theorem 3).
Theorem 4: Suppose the marginal distribution functions F X and F Y are continuous. The pair <X, Y > is comonotonic if and only if one of the following equivalent conditions holds: with probability 1. • There is a non-decreasing function g such that Y = g (X) with probability 1.
According to Theorems 4 and 5, in the case of continuous marginals, comonotonicity (countermonotonicity) is equivalent to monotone increasing (decreasing) dependence defined by Kimeldorf and Sampson (1978), which was described in our introduction.

MONOTONICITY COEFFICIENT
We will fix a random variable U with the uniform distribution on (0, 1). For a random variable X with distribution function F X denote: By the quantile transfer theorem, X * = d X and X′ = d X (see Dhaene et al., 2002).
In the rest of the article we will consider only nondegenerate random variables with finite variances. The following theorem presents some well-known results (see, for example, Denuit and Dhaene, 2003) using the aforenamed notations.
So the second pair is more comonotonic, which is also obvious from the graphs in Fig. 1. Theorem 7: Properties of the monotonicity coefficient: For the random variables X and Y the following holds: The degree of linear dependence is not greater than the degree of monotone dependence. 4) −1 ≤ ρm (X, Y) ≤ 1. 5) ρm (Y, X) = ρm (X, Y). 6) If X and Y are independent, then ρm (X, Y) = 0. 7) For any α∈R: ρm (X +α, Y) = ρm (X, Y).

13)
Proof: 1)-10) follows from Lemma 2 and Theorem 6. 11) and 12) follow from the formula of Hoeffding: The linear properties in Theorem 7.7), 8) are stated only for one argument but they also hold for the other argument due to symmetry.
Thus, the properties of the monotonicity coefficient ρm are similar to the properties of the Pearson correlation ρ but with respect to monotone dependence. The measure ρm is not entirely new. For PQD random variables X and Y, Reimann (1992) defined a measure λ ** by the formula: The coefficient ρm is a generalization of λ ** to any random variables with finite variances; for PQD variables X and Y, ρm (X, Y) = λ ** (X, Y) by the Hoeffding formula. Reimann (1992) did not study the properties of λ ** except the property λ ** ≥ ρ. He described λ ** as a measure of association of two random variables rather than a measure of monotone dependence. He defined λ ** in terms of double integrals and ρm has a simpler definition in terms of covariances.

COMPARISON TO OTHER COEFFICIENTS AND APPLICATIONS
The relation between ρm and ρ is described in the following theorem.

Theorem 8: Relation to the Pearson correlation:
Suppose Cov (X, Y) ≠ 0. Then: there exist numbers a and b, such that Y = d a + bX.
In the following examples we compare the values of ρm and ρ for some bivariate distributions.
Example 2: Suppose the joint density function of X and Y is given by: Then the monotonicity coefficient is: ρm (X, Y) = 2 32 9 − π ≈ 0.5368 versus the Pearson correlation ρ (X, Y) = 0.5.
The Pearson correlation is:  Scarsini (1984) introduced some conditions for a measure of dependence of two random variables. Theorems 7 and 8 shows that ρm satisfies a reasonable modification of these conditions. In particular, if each of random variables X and Y has a normal distribution, then ρm(X, Y) = ρ(X, Y) by Theorem 8.
One of the Scarsini's conditions is the invariance of a measure of concordance under increasing transformations of X and Y. This condition might be useful for a measure of general dependence but not monotone dependence, since the result of increasing transformations of two variables can get closer to or further from a monotonic relation than the original pair, i.e., increasing transformations can change the degree of monotone dependence. The Pearson correlation does not satisfy this condition; it measures linear dependence, which is a particular case of monotone dependence. The Spearman, Kendall and Schweizer-Wolff coefficients satisfy this condition; they depend only on the ranks of the observations. The coefficient ρm is only invariant under changes of scale and location in X and Y. We believe that ρm is a more appropriate measure of monotone dependence of two variables and illustrate this with the following two examples.
Since the pairs <X, Y > and <X, Z > have the same ranks, their Kendall coefficients are equal: τ (X, Y) = = τ (X, Z) = 11 15 and so are their Spearman coefficients: . But the second pair is more comonotonic: it is closer to an increasing relation as the graphs in Fig. 3 shows; this is reflected by its higher monotonicity coefficient: ρm (X, Y) ≈ 0.766 and ρm (X, Z) ≈ 0.991.
The coefficient ρm can be applied to the problems where Y is a monotone function of X with a random noise ε included; the monotonicity coefficient can be used to estimate some characteristics of the noise. It is natural to assume that ε has a normal distribution, as usual. The following two examples illustrate some cases when ρm(X, Y) is used to estimate the variance and central moments of the noise. Clearly, ρm cannot be used to estimate the mean of ε, since this mean is a constant shift of X or Y and it does not affect the degree of their monotone dependence. In the following examples we also assume the normality of X or Y, which simplifies the calculations.
Example 7: Suppose Y = g (X) + ε, where g is a monotone function, noise ε has a normal distribution with variance τ 2 , variables X and ε are independent and g (X) has a normal distribution with variance β 2 . Then:   The result holds for the particular case when g(x) = ln x, so X has a lognormal distribution with the second parameter β. The following example is a generalization of Example 3.
Example 8: Suppose Y = g(X+ε), where g is a monotone function, noise ε has a normal distribution with variance τ 2 , variables X and ε are independent and X has a normal distribution with variance β 2 . Then For the random variables from Examples 7 and 8, the variance τ 2 of the noise ε can be expressed in terms of the monotonicity coefficient ρm = ρm(X, Y): Since ε has a normal distribution, this also defines its central moments: where s(x, y) is the sample covariance, x* is the sample x with its values in ascending order and y′ is the sample y with its values in descending order. The properties of rm are similar to the properties of ρm. Details are given in (Kachapova and Kachapov, 2010).

CONCLUSION
This article introduced the monotonicity coefficient ρm, a new measure of the monotone dependence of random variables with finite variances. It was proven that ρm satisfies reasonable conditions for such a measure: • ρm has linear properties • it is invariant under changes of scale and location • ρm(X, Y) = 0 for independent random variables X, Y • ρm(X, Y) = 1 for a comonotonic pair X, Y • ρm(X, Y) = −1 for a counter-monotonic pair X, Y The coefficient ρm is a more sensitive measure of monotonicity than the coefficients depending only on the ranks of observations.
The sample version rm of the monotonicity coefficient was defined.
We recommend using ρm to compare pairs of random variables with respect to their degree of monotonicity. For example, in portfolio analysis the monotonicity coefficient can be used to assess the degree of increasing or decreasing monotone dependence between two asset returns and to do respective comparison of pairs of assets. In the problems where the monotone relation of two variables has a random noise, the coefficient ρm can be used to estimate variance and other central moments of the noise.
We recommend to use the sample monotonicity coefficient rm to find monotonic relationships in big datasets.