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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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Journal ArticleDOI
Lionel Dupont1
TL;DR: In this paper, a new type of facility location model is introduced, in which the global cost incurred for each established facility is a concave function of the quantity q delivered by this facility.

50 citations

Posted Content
TL;DR: In this article, the authors provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013), along with connections between log-concavity and other areas of mathematics and statistics.
Abstract: We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

50 citations

Journal ArticleDOI
TL;DR: This paper considers the following model, described in terms of an investment problem, where D units available for investment are D, and how much to invest at each opportunity is decided so as to maximize total expected profit.
Abstract: This paper considers the following model, described in terms of an investment problem. We have D units available for investment. During each of N time periods an opportunity to invest will occur with probability p. As soon as an opportunity presents itself, we must decide how much of our available resources to invest. If we invest y, then we obtain an expected profit P(y), where P is a nondecreasing continuous function. The amount y then becomes unavailable for future investment. The problem is to decide how much to invest at each opportunity so as to maximize total expected profit. When P(y) is a concave function, the structure of the optimal policy is obtained (§1). Bounds on the optimal value function and asymptotic results are presented in §2. A closed-form expression for the optimal value to invest is found in §3 for the special cases of P(y) = log y and P(y) = yα, for 0 < α < 1. §4 presents a continuous-time version of the model, i.e., we assume that opportunities occur in accordance with a Poisson ...

50 citations

Journal ArticleDOI
TL;DR: In this paper, the Neyman-Pearson lemma for 2-alternating capacities is applied to test problems between noncompact neighbourhoods of probability measures, and it is shown that the Radon-Nikodym derivative between the special capacities is usually a nondecreasing function of the truncated likelihood ratio of some probability measures.
Abstract: Solutions to minimax test problems between neighbourhoods generated by specially defined capacities are discussed. The capacities are superpositions of probability measures and concave functions, so the paper covers most of the earlier results of Huber and Rieder concerning minimax testing between ɛ-contamination and total variation neighbourhoods. It is shown that the Neyman-Pearson lemma for 2-alternating capacities, proved by Huber and Strassen, can be applied to test problems between noncompact neighbourhoods of probability measures. It turns out that the Radon-Nikodym derivative between the special capacities is usually a nondecreasing function of the truncated likelihood ratio of some probability measures.

49 citations

Journal ArticleDOI
TL;DR: An upper bound is derived for the largest Lyapunov exponent of a Markovian product of nonnegative matrices usingMarkovian type counting arguments as the maximum of a nonlinear concave function over a finite-dimensional convex polytope of probability distributions.

49 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202316
202240
202158
202049
201952
201860