arXiv subjects

Thomas Mroz

Publications and source records attributed to Thomas Mroz.

3 recordsLinked to original sources

arXiv2021

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

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 $Γ^\leq(T)$ and the graph $Γ(T)$ 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.

math.PR↗

arXiv2021

How simplifying and flexible is the simplifying assumption in pair-copula constructions -- analytic answers in dimension three and a glimpse beyond

Motivated by the increasing popularity and the seemingly broad applicability of pair-copula constructions underlined by numerous publications in the last decade, in this contribution we tackle the unavoidable question on how flexible and simplifying the commonly used `simplifying assumption' is from an analytic perspective and provide answers to two related open questions posed by Nagler and Czado in 2016. Aiming at a simplest possible setup for deriving the main results we first focus on the three-dimensional setting. We prove that the family of simplified copulas is flexible in the sense that it is dense in the set of all three-dimensional co\-pulas with respect to the uniform metric $d_\infty$ - considering stronger notions of convergence like the one induced by the metric $D_1$, by weak conditional convergence, by total variation, or by Kullback-Leibler divergence, however, the family even turn out to be nowhere dense and hence insufficient for any kind of flexible approximation. Furthermore, returning to $d_\infty$ we show that the partial vine copula is never the optimal simplified copula approximation of a given, non-simplified copula $C$, and derive examples illustrating that the corresponding approximation error can be strikingly large and extend to more than 28\% of the diameter of the metric space. Moreover, the mapping $ψ$ assigning each three-dimensional copula its unique partial vine copula turns out to be discontinuous with respect to $d_\infty$ (but continuous with respect to $D_1$ and to weak conditional convergence), implying a surprising sensitivity of partial vine copula approximations. The afore-mentioned main results concerning $d_\infty$ are then extended to the general multivariate setting.

math.ST↗

arXiv2018

A sharp inequality for Kendall's $τ$ and Spearman's $ρ$ of Extreme-Value Copulas

We derive a new (lower) inequality between Kendall's tau? and Spearman's rho? for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.

math.ST↗