A measure of mutual complete dependence

The concept of measuring, by a scalar value, the strength of dependence between two random variables defined on a common probability space plays a major role in probability theory and statistics. However, despite abundant work on this problem, a measure of the degree of mutual complete dependence, defined as almost sure bijective functional dependence between the two random variables, does not exist. The main contribution of this dissertation consists in a new method to detect and measure mutual complete dependence of arbitrary form. The approach is based on copulas. By virtue of a fundamental result known as Sklar's theorem, the joint distribution function of any two continuous random variables on a common probability space can be decomposed into the marginal distribution functions and a unique copula. Therefore, the dependence between the random variables is fully captured by their copula. For example, they are independent if and only if their connecting copula is the so called product copula. Thus, a possible approach to measuring their stochastic dependence consists in measuring the distance between their copula and the product copula. This method for constructing a measure of dependence is not new. We argue, however, that it yields, in general, a measure of independence only. While independence in the variables can be detected using any distance function, the type of the “strongest” dependence detected depends heavily on the type of the distance function employed. It follows that the choice of the distance function cannot be arbitrary, but is predetermined by the desired properties of the resulting measure of dependence. We propose to measure the distance between two copulas by a (modified) Sobolev norm, introducing first a scalar product on the set of all copulas. This norm exploits the differentiability properties of copulas and turns out extremely advantageous since the degree of mutual complete dependence between two random variables can be determined by analytical and algebraic properties of their copula. Furthermore, with respect to the Sobolev norm, a sequence of copulas corresponding to mutual complete dependence can only converge to a copula which itself links mutually completely dependent random variables. Thus, mutual complete dependence cannot approximate any other kind of stochastic dependence. This resolves the counterintuitive phenomenon described in the literature that, with respect to convergence in distribution, mutual complete dependence can approximate any other kind of stochastic dependence, in particular independence. Using this Sobolev norm we define the first measure of mutual complete dependence for two random variables with continuous distribution functions, which is given by the (normalized) Sobolev distance between their unique copula and the product copula, corresponding to stochastic independence. We show that this measure has several appealing properties, e.g., it takes on its minimum and maximum precisely at independence and mutual complete dependence, respectively. Furthermore, since the measure is based on copulas, it is nonparametric and remains invariant under strictly monotone transformations of the random variables