On distributions with fixed marginals maximizing the joint or the prior default probability, estimation, and related results

Abstract

Motivated by (random) lifetimes of electronic components or financial institutions we study the problem of maximizing the probability that (i) a random variable X is not smaller than another random object Y and (ii) that X and Y coincide within the class of all random variables X,Y with given univariate continuous distribution functions F and G, respectively. We show that the maximization problems correspond to finding copulas maximizing the mass of the endograph Γ≤(T)={(x,y)∈[0,1]2:y≤T(x)} and the graph Γ(T)={(x,T(x)):x∈[0,1]} of T=G∘F−, respectively. After providing simple, copula-based proofs for the existence of copulas attaining the two maxima m¯T and w¯T we generalize the obtained results to the case of general (not necessarily monotonic) transformations T:[0,1]→[0,1] and derive simple and easily calculable formulas for m¯T and w¯T involving the distribution function FT of T (interpreted as random variable on [0,1]). The latter are then used to characterize all non-decreasing transformations T:[0,1]→[0,1] for which m¯T and w¯T coincide. A strongly consistent estimator for m¯T is derived and proven to be asymptotically normal under very mild regularity conditions. Several examples and graphics illustrate the main results and falsify some seemingly natural conjectures, an application of some of the obtained results to the seemingly unrelated topic of relative effects indicates the importance of the tackled questions.