Fréchet Classes and Nonmonotone Dependence

The analysis of the Fréchet class of bivariate distribution functions with fixed marginals is fundamental in the study of positive dependence. Several notions of positive dependence have been examined in the literature (see (1982), Kimeldorf and Sampson (1987, Kimeldorf and Sampson 1989), and (1996) for surveys of the field). All of these notions tend to capture the idea that, for a certain random pair, large values of one random variable go together with large values of the other random variable. Another way of expressing this concept is to say that the distribution of the random pair tends to concentrate its mass around the graph of an increasing function. The stronger the dependence, the more concentrated the probability mass will be around the graph of the function. If the marginal distributions are continuous, in the extreme case of a perfect positive dependence all the mass will lie exactly on the graph of an increasing function. Since the marginals are fixed, there exists only one increasing function on whose graph the probability mass can concentrate.