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

Abstract

We study the problem of maximizing the probability that (i) an electric component or financial institution XX does not default before another component or institution YY and (ii) that XX and YY default jointly within the class of all random variables X,YX,Y with given univariate continuous distribution functions FF and GG, respectively, and show that the maximization problems correspond to finding copulas maximizing the mass of the endograph Γ(T)\Gamma^\leq(T) and the graph Γ(T)\Gamma(T) of T=GFT=G \circ F^-, respectively. After providing simple, copula-based proofs for the existence of copulas attaining the two maxima mT\overline{m}_T and wT\overline{w}_T we generalize the obtained results to the case of general (not necessarily monotonic) transformations T:[0,1][0,1]T:[0,1] \rightarrow [0,1] and derive simple and easily calculable formulas for mT\overline{m}_T and wT\overline{w}_T involving the distribution function FTF_T of TT (interpreted as random variable on [0,1][0,1]). The latter are then used to charac\-terize all non-decreasing transformations T:[0,1][0,1]T:[0,1] \rightarrow [0,1] for which mT\overline{m}_T and wT\overline{w}_T coincide. A strongly consistent estimator for the maximum probability that XX does not default before YY 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.

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