Additional Topics

Abstract

In this chapter we consider three additional applications of copulas. The first, distributions with fixed margins, dates back to the early history of the subject. The original question whose answer leads to the Fréchet-Hoeffding bounds (2.5.1) is: Of all joint distribution functions H constrained to have fixed margins F and G, which is the “largest,” and which the “smallest”? Another example, which also involves optimization when the margins are fixed, is the following. In studying “distances” between distributions, Dall’Aglio (1956, 1991) considered the following problem (see Exercise 6.5): What is the minimum value of

$$E{\left| {X - Y} \right|^\alpha } = \iint {_{{R^2}}}{\left| {x - y} \right|^\alpha }dH\left( {x,y} \right),$$

given that the margins of H are fixed to be F and G,respectively?