1. On distributions with fixed marginals maximizing the joint or the prior default probability, estimation, and related results
- Author
-
Thomas Mroz, Juan Fernández Sánchez, Sebastian Fuchs, and Wolfgang Trutschnig
- Subjects
Statistics and Probability ,Optimization and Control (math.OC) ,Applied Mathematics ,Probability (math.PR) ,FOS: Mathematics ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Statistics, Probability and Uncertainty ,Mathematics - Optimization and Control ,60E05, 28A50, 91G70 ,Mathematics - Probability - Abstract
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., Comment: 27 pages, 4 figures
- Published
- 2023
- Full Text
- View/download PDF