Risk excess measures induced by hemi-metrics

Abstract

which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. AbstractThe main aim of this paper is to introduce the notion of risk excess measure, to analyze its properties, and to describe some basic construction methods. To compare the risk excess of one distribution Q w.r.t. a given risk distribution P, we apply the concept of hemi-metrics on the space of probability measures. This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures. Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings. Our view towards measuring risk excess adds to the usually considered method to compare risks of Q and P by the values ρ(Q), ρ(P) of a risk measure ρ. We argue that the difference ρ(Q) − ρ(P) neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure. We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.

A Unifying Framework for Some Directed Distances in Statistics

“…" ş X rpxq ¨Sx pP q ¨log `SxpP q SxpQq ˘¨1 s0,8r `Sx pP q ¨Sx pQq ˘dλpxq `şX rpxq ¨pS x pQq ´Sx pP qq dλpxq `8 ¨şX rpxq ¨Sx pP q ¨1t0u `Sx pQq ˘dλpxq, (68) 0 ď D φ0,SpQq,SpQq,r¨SpQq,λ pSpP q, SpQqq " ş X " ´log `SxpP q SxpQq ˘`SxpP q…”

“…of ( 67), (68). For the special subsetup of nonnegative random variables (and thus Y " X "s0, 8r) with finite expectations and strictly positive cdf, (76) simplifies to the so-called "cumulative Kullback-Leibler information" of Park et al [156] (see also Park et al [155] for an extension to the whole real line, Di Crescenzo & Longobardi [59] for an adaption to possibly smaller support as well as for an adaption to a dynamic form analogously to the explanations in the following lines).…”

“…of ( 67), (68). This has been employed by Liu [116] for the special case of P " P emp N and Q " Q θ in order to obtain corresponding minimum-divergence parameter estimator of θ (see e.g.…”

Risk Quantization by Magnitude and Propensity

“…Among the numerous approaches encountered in the literature, let us mention the classical premium calculation principles based on probabilistic models (see e.g. Mikosch (2009), Asmussen and Hansjörg Albrecher (2010), Bühlmann (1996)), the axiomatic approach where premium principles are subject to a set of desirable properties (see Artzner et al (1999)), the abstract/functional analytic approach where risk measures are derived from an acceptance set and a set of scenario measures (see Föllmer and Schied (2002)), distortion-based measures Wang (1996), and eventually the approach to risk excess measures induced by hemi-metrics (see Olivier P. Faugeras and Rüschendorf (2018)). These numerous approaches have lead to considerable debate on the pros and cons of the risk measures available in the literature, see e.g.…”

“…• And so on for other statistical functionals. See also Olivier P. Faugeras and Rüschendorf (2018) for optimal transportation induced by cost functions which are hemi-metrics encoding an order.…”