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Working with shuffles, we establish a close link between Kendall’s τ\tau , the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well known that Spearman’s ρ\rho of a bivariate copula AA is a rescaled version of the volume of the area under the graph of AA, in this contribution we show that the other famous concordance measure, Kendall’s τ\tau , allows for a simple geometric interpretation as well – it is inextricably linked to the surface area of AA.

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 F T 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. • Simple formulas for maximum/minimum prior/joint default probability are derived. • The situation of the two probabilities coinciding is characterized. • A strongly consistent estimator for the prior default probability is established. • The estimator is shown to be asymptotically normal. • An application to relative effects is given. [ABSTRACT FROM AUTHOR]

In this article, we focus on copulas underlying maximal non-exchangeable pairs (X,Y)\left(X,Y) of continuous random variables X,YX,Y either in the sense of the uniform metric d∞{d}_{\infty } or the conditioning-based metrics Dp{D}_{p}, and analyze their possible extent of dependence quantified by the recently introduced dependence measures ζ1{\zeta }_{1} and ξ\xi . Considering maximal d∞{d}_{\infty }-asymmetry we obtain ζ1∈56,1{\zeta }_{1}\in \left[\frac{5}{6},1\right] and ξ∈23,1\xi \in \left[\frac{2}{3},1\right], and in the case of maximal D1{D}_{1}-asymmetry we obtain ζ1∈34,1{\zeta }_{1}\in \left[\frac{3}{4},1\right] and ξ∈12,1\xi \in \left(\frac{1}{2},1\right], implying that maximal asymmetry implies a very high degree of dependence in both cases. Furthermore, we study various topological properties of the family of copulas with maximal D1{D}_{1}-asymmetry and derive some surprising properties for maximal Dp{D}_{p}-asymmetric copulas.

Despite the fact that copulas are commonly considered as analytically smooth/regular objects, derivatives of copulas have to be handled with care. Triggered by a recently published result characterizing multivariate copulas via (d − 1) -increasingness of their partial derivative we study the bivariate setting in detail and show that the set of non-differentiability points of a copula may be quite large. We first construct examples of copulas C whose first partial derivative ∂ 1 C (x , y) is pathological in the sense that for almost every x ∈ (0 , 1) it does not exist on a dense subset of y ∈ (0 , 1) , and then show that the family of these copulas is dense. Since in commonly considered subfamilies more regularity might be typical, we then focus on bivariate Extreme Value copulas (EVCs) and show that a topologically typical EVC is not absolutely continuous but has degenerated discrete component, implying that in this class typically ∂ 1 C (x , y) exists in full (0 , 1) 2. Considering that regularity of copulas is closely related to their mass distributions we then study mass distributions of topologically typical copulas and prove the surprising fact that topologically typical bivariate copulas are mutually completely dependent with full support. Furthermore, we use the characterization of EVCs in terms of their associated Pickands dependence measures ϑ on [ 0 , 1 ] , show that regularity of ϑ carries over to the corresponding EVC and prove that the subfamily of all EVCs whose absolutely continuous, discrete and singular component has full support is dense in the class of all EVCs. [ABSTRACT FROM AUTHOR]

We describe a theory of probabilistic Morse decomposition for continuous-state, probabilistic dynamical systems. Morse decompositions, studied in the topological theory of dynamical systems, allow reasoning about asymptotic behaviors of dynamical systems. We generalize notions of attractors, repellers, and invariant sets to the probabilistic context, and show how these can be used to describe the topological structure of the probabilistic dynamics. Additionally, we show a Lyapunov function characterization for probabilistic Morse decompositions. Our probabilistic Morse decompositions enable an abstraction-based verification methodology for asymptotic specifications such as “the trajectories asymptotically converge to a set with positive probability,” which are not expressible in usual linear temporal logics. We describe computational approaches to computing probabilistic Morse decompositions, using state-space gridding as well as Lyapunov functions. Interestingly, the construction of Morse decompositions is crucial: we show that existing abstraction-based techniques based on gridding the state space are not sufficiently powerful to verify asymptotic specifications. [ABSTRACT FROM AUTHOR]

The separate tasks of denoising, least squares expectation, and manifold learning can often be posed in a common setting of finding the conditional expectations arising from a product of two random variables. This paper focuses on this more general problem and describes an operator theoretic approach to estimating the conditional expectation. Kernel integral operators are used as a compactification tool, to set up the estimation problem as a linear inverse problem in a reproducing kernel Hilbert space. This equation is shown to have solutions that allow numerical approximation, thus guaranteeing the convergence of data-driven implementations. The overall technique is easy to implement, and their successful application to some real-world problems is also shown. [ABSTRACT FROM AUTHOR]

Motivated by the nice characterization of copulas A for which d∞(A, At) is maximal as established independently by Nelsen [11] and Klement & Mesiar [7], we study maximum asymmetry with respect to the conditioning-based metric D1 going back to Trutschnig [12]. Despite the fact that D1(A, At) is generally not straightforward to calculate, it is possible to provide both, a characterization and a handy representation of all copulas A maximizing D1(A, At). This representation is then used to prove the existence of copulas with full support maximizing D1(A, At). A comparison of D1- and d∞-asymmetry including some surprising examples rounds off the paper.

Aiming at the simplest possible representation of maxmin copulas in terms of two generators Košir and Omladič recently introduced the class of so-called reflected maxmin (RMM) copulas in [10]. All results for RMM copulas can easily be translated to results on maxmin copulas. Building upon the afore-mentioned article we provide alternative simplified (mainly Markov kernel based) proofs for some of their theorems and, more importantly, derive numerous new results. In particular, we show that the family of all RMM copulas is a compact subset of the metric space of all copulas (with the standard uniform metric), characterize absolutely continuous RMM copulas, and show that the limit of absolutely continuous RMM copulas is absolutely continuous too. Finally, we determine Kendall's τ and Spearman's ρ of RMM copulas, prove some first inequalities for τ and ρ , and state a conjecture on sharp inequalities. Some examples illustrating the main results round off the paper. [ABSTRACT FROM AUTHOR]

Markov kernels play a decisive role in probability and mathematical statistics theories, and are an extension of the concepts of σ -field and statistic. Concepts such as independence, sufficiency, completeness, ancillarity or conditional distribution have been extended previously to Markov kernels. In this paper, the concept of conditional expectation of a Markov kernel given another is introduced, setting its first properties. An application to clinical diagnosis is provided, obtaining an optimality property of the predictive values of a diagnosis test. In a statistical framework, this new probabilistic tool is used to extend to Markov kernels the theorems of Rao-Blackwell and Lehmann-Scheffé. A result about the completeness of a sufficient statistic is obtained in passing by properly enlarging the family of probabilities. As a final statistical scholium, a generalization of a result about the completeness of the family of nonrandomized estimators is given. [ABSTRACT FROM AUTHOR]

Two known results on the relationship between conditional and unconditional independence are obtained as a consequence of the main result of this paper, a theorem that uses independence of Markov kernels to obtain a minimal condition, which, added to conditional independence, implies independence. Some examples, counterexamples, and representation results are provided to clarify the concepts introduced and the propositions of the statement of the main theorem. Moreover, conditional independence and the mentioned results are extended to the framework of Markov kernels. [ABSTRACT FROM AUTHOR]

Models are often defined through conditional rather than joint distributions, but it can be difficult to check whether the conditional distributions are compatible, i.e. whether there exists a joint probability distribution which generates them. When they are compatible, a Gibbs sampler can be used to sample from this joint distribution. When they are not, the Gibbs sampling algorithm may still be applied, resulting in a "pseudo-Gibbs sampler". We show its stationary probability distribution to be the optimal compromise between the conditional distributions, in the sense that it minimizes a mean squared misfit between them and its own conditional distributions. This allows us to perform Objective Bayesian analysis of correlation parameters in Kriging models by using univariate conditional Jeffreys-rule posterior distributions instead of the widely used multivariate Jeffreys-rule posterior. This strategy makes the full-Bayesian procedure tractable. Numerical examples show it has near-optimal frequentist performance in terms of prediction interval coverage. [ABSTRACT FROM AUTHOR]