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Concave function
About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.
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TL;DR: It is shown that the minimum value of a sample of feasible points uniformly distributed over a linear constraint set is, for concave functions, asymptotically Weibull distributed with shape parameter equal to the dimension of the feasible region.
Abstract: We show that the minimum value of a sample of feasible points uniformly distributed over a linear constraint set is, for concave functions, asymptotically Weibull distributed with shape parameter equal to the dimension of the feasible region.
12 citations
TL;DR: This paper forms the convexity and the continuity axioms for preference relations on @s-algebras with a metric topology and shows the existence of utility functions for convex continuous preference relations.
12 citations
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TL;DR: In this paper, the authors developed two methods for imposing curvature conditions globally in the context of cost function estimation, based on a generalization of a functional form first proposed by McFadden.
Abstract: Empirically estimated flexible functional forms frequently fail to satisfy the appropriate theoretical curvature conditions. Lau and Gallant and Golub have worked out methods for imposing the appropriate curvature conditions locally, but those local techniques frequently fail to yield satisfactory results. We develop two methods for imposing curvature conditions globally in the context of cost function estimation. The first method adopts Lau's technique to a generalization of a functional form first proposed by McFadden. Using this Generalized McFadden functional form, it turns out that imposing the appropriate curvature conditions at one data point imposes the conditions globally. The second method adopts a technique used by McFadden and Barnett, which is based on the fact that a non-negative sum of concave functions will be concave. Our various suggested techniques are illustrated using the U.S. Manufacturing data utilized by Berndt and Khaled
12 citations
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TL;DR: In this paper, the authors prove that a finite family of convex sets in R^d$ has a large intersection. But they do not characterize conditions that are sufficient for the intersection of such sets to contain a "witness set" under some concave or log-concave measure.
Abstract: We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colorful and fractional variants of all our Helly-type theorems.
12 citations
TL;DR: In this paper, the authors consider minimax shrinkage estimation of location for spherically symmetric distributions under a concave function of the usual squared error loss and consider scale mixtures of normal distributions and losses with completely monotone derivatives.
12 citations