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On distributions with fixed marginals ma...
Mroz, Thomas...

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On distributions with fixed marginals maximizing the joint or the prior default probability, estimation, and related results by Mroz, Thomas ( Author )
Lemonjar Open Access
Australian National University
08-08-2023
We study the problem of maximizing the probability that (i) an electric component or financial institution $X$ does not default before another component or institution $Y$ and (ii) that $X$ and $Y$ default jointly within the class of all random variables $X,Y$ with given univariate continuous distribution functions $F$ and $G$, respectively, and show that the maximization problems correspond to finding copulas maximizing the mass of the endograph $\Gamma^\leq(T)$ and the graph $\Gamma(T)$ of $T=G \circ F^-$, respectively. After providing simple, copula-based proofs for the existence of copulas attaining the two maxima $\overline{m}_T$ and $\overline{w}_T$ we generalize the obtained results to the case of general (not necessarily monotonic) transformations $T:[0,1] \rightarrow [0,1]$ and derive simple and easily calculable formulas for $\overline{m}_T$ and $\overline{w}_T$ involving the distribution function $F_T$ of $T$ (interpreted as random variable on $[0,1]$). The latter are then used to charac\-terize all non-decreasing transformations $T:[0,1] \rightarrow [0,1]$ for which $\overline{m}_T$ and $\overline{w}_T$ coincide. A strongly consistent estimator for the maximum probability that $X$ does not default before $Y$ 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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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1602.05807
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