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Abstract. The aim of this paper is to introduce some classes of aggregation functionals when the evaluation scale is a complete lattice. We focus on the notion of quantile of a lattice-valued function which have several properties of its real-valued counterpart and we study a class of aggregation functionals that generalizes Sugeno integrals to the setting of complete lattices. Then we introduce in the real-valued case some classes of aggregation functionals that extend Choquet and Sugeno integrals by considering a multiple quantile model generalizing the approach proposed in [3].
Abstract. In this paper we are interested in functionals defined on completely distributive lattices and which are invariant under mappings preserving arbitrary joins and meets. We prove that the class of nondecreasing invariant functionals coincides with the class of Sugeno integrals associated with {0, 1}-valued capacities, the so-called term functionals, thus extending previous results both to the infinitary case as well as to the realm of completely distributive lattices. Furthermore, we show that, in the case of functionals over complete chains, the nondecreasing condition is redundant. Characterizations of the class of Sugeno integrals, as well as its superclass comprising all polynomial functionals, are provided by showing that the axiomatizations (given in terms of homogeneity) of their restriction to finitary functionals still hold over completely distributive lattices. We also present canonical normal form representations of polynomial functionals on completely distributive lattices, which appear as the natural extensions to their finitary counterparts, and as a by-product we obtain an axiomatization of complete distributivity in the case of bounded lattices.
Since Shalit and Yitzhaki (1984) the Mean-Extended Gini (MEG) has been proposed as a workable alternative to the classical Markowitz mean-variance Capital Asset Pricing Model (CAPM). Although the MEG controls the risk belonging to the left-tail of the return distribution, little attention is given to potential gains belonging to the right tail of the return distribution. A generalization of the MEG able to select personalized optimal mean-risk and/or mean-gain portfolios is proposed. We give evidence that if the portfolio distributions are symmetrical and/or the investor has a moderate risk-gain profile, then the efficient meanrisk portfolio always coincides with a not efficient mean-gain portfolio. In more realistic scenarios admitting the existence of asymmetrically distributed assets and/or investors with very defensive or very aggressive investment profiles, portfolios which are optimal under both criteria may exist. J S C : G10, G11, G12,G29 K : Extended Gini Index; MEG; risk-aversion and gain-propensity.
We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lovász extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory.
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