Distortion function

Abstract: We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank-dependent expected utility (RDEU) theory with a concave utility function and an inverse-S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.

Abstract: A number of more or less well-known, but quite complex, characterizations of stop-loss order are rewewed and proved m an elementary way. Two recent proofs of the stop-loss order preserwng property for the distortion pricing principle are invahdated through a simple counterexample A new proof is presented. It is based on the important Hardy-Littlewood transform, which ~s known to characterize the stop-loss order by reductmn to the usual stochastic order, and the dangerousness characterization of stop-loss order under a fimte crossing condmon Finally, we complete and summarize the main properties of the distortmn pricing principle.

Abstract: The cost function is crucial to the security of adaptive image steganography, However, some existing cost functions are heuristically designed and hard to be optimal in the undetectability against the evolving steganalyzer. In this letter, we propose a cost reassignment algorithm for adaptive steganography based on artificial immune system. Under the scenario of minimizing additive distortion, this method reassigns the cost by adjusting the modification probability distribution obtained by the cost function, and dynamically optimizes the adjustment mode through the immunity-based information hiding model, so that the modified pixels are more concentrated in the regions that are difficult to be detected. The experimental results show that the proposed method is suitable for a variety of the state-of-the-art cost functions and can achieve better performance on resisting the steganalysis.

Abstract: Throughout this article \({\mathbb X}\) and \({\mathbb Y}\) will be nonempty bounded domains in \({\mathbb {R}^n}\) , \({n \geqq 2}\) . The term deformation of \({{\mathbb X}\subset \mathbb {R}^n}\) onto \({{\mathbb Y}\subset \mathbb {R}^n}\) refers to an orientation preserving homeomorphism \({ h : {\mathbb X} \overset{\textnormal{\tiny{onto}}}{\longrightarrow} {\mathbb Y} }\) in the Sobolev class \({\fancyscript {W}^{1,1}} ({\mathbb X, {\mathbb Y})}\) whose inverse \({ f : {\mathbb Y} \overset{\textnormal{\tiny{onto}}}{\longrightarrow} {\mathbb X} }\) lies in \({{\fancyscript {W}^{1,1}} ({\mathbb Y, {\mathbb X})}}\) . The general law of hyperelasticity asserts that there exists an energy integral $${\mathcal E}_{_{\mathbb X}}[h]= \int_{\mathbb X} E(x,h,Dh)\, \rm dx$$ such that the elastic deformations have the smallest energy. We assume here that E is conformally coerced and polyconvex. Some additional regularity conditions are also imposed. Under those conditions we establish the existence and global invertibility of the minimizers. The key tools in obtaining an extremal deformation \({ h : {\mathbb X}\overset{\textnormal{\tiny{onto}}}{\longrightarrow} {\mathbb Y} }\) , regardless of its boundary values, are the free Lagrangians. Finding suitable free Lagrangians and using them for a specific stored-energy function E is truly a work of art. We have done it here for the so-called total harmonic energy and a pair of annuli in the plane. In fact this challenging problem illustrates rather clearly the strength of the concept of free Lagrangians.