A Measure of Monotonicity of Two Random Variables

Problem statement: When analyzing random variables it is useful to measure the degree of their monotone dependence or compare pairs of random variables with respect to their monotonicity. Existing coefficients measure general or linear dependence of random variables. Developing a measure of monotonicity is useful for practical applications as well as for general theory, since monotonicity is an important type of dependence. Approach: Existing measures of dependence are briefly reviewed. The Reimann coefficient is generalized to arbitrary random variables with finite variances. Results: The article describes criteria for monotone dependence of two random variables and introduces a measure of this dependence-monotonicity coefficient. The advantages 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 measure and that it has properties similar to the properties of the Pearson correlation; in particular, it equals 1 (-1) if and only if the pair X, Y is comonotonic (counter-monotonic). The monotonicity coefficient is calculated for some bivariate distributions and the sample version of the coefficient 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 monotone dependence. In the problems where the monotone relation of two variables has a random noise, 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.