We show that for every nondecreasing concave function ω : [0,∞) → [0,∞) with ω(0) = 0, either every finite metric space embeds with distortion arbitrarily close to 1 into a metric space of the form (X,ω◦d) for some metric d on X, or there exists α = α(ω) > 0 and n0 = n0(ω) ∈ N such that for all n > n0, any embedding of {0, . . . , n} ⊆ R into a metric space of the form (X,ω ◦ d) incurs distorti...

We consider a semilinear parametric Neumann problem driven by the negative Laplacian plus an indefinite and unbounded potential. The reaction is asymptotically linear and exhibits a negative concave term near the origin. Using variational methods together with truncation and perturbation techniques and critical groups, we show that for all small values of the parameter the problem has at least ...

Bounding hull, such as convex hull, concave hull, alpha shapes etc. has vast applications in different areas especially in computational geometry. Alpha shape and concave hull are generalizations of convex hull. Unlike the convex hull, they construct non-convex enclosure on a set of points. In this paper, we introduce another generalization of convex hull, named alpha-concave hull, and compare ...

We study nonparametric maximum likelihood estimation of a log–concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least (log(n)/n) and typically (log(n)/n...

Learning a Gaussian graphical model with latent variables is ill posed when there is insufficient sample complexity, thus having to be appropriately regularized. A common choice is convex ℓ1 plus nuclear norm to regularize the searching process. However, the best estimator performance is not always achieved with these additive convex regularizations, especially when the sample complexity is low...

We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, i.e. a density of the form f(0) = exp varphi(0) where varphi(0) is a concave function on R. Existence, form, characterizations and uniform rates of convergence of the MLE are given by Rufibach (2006) and Dümbgen and Rufibach (2007). The characterization of the log-concave MLE in term...

Production functions are estimated for South African industries using a novel industry panel dataset that combines education and employment data from a series of household surveys with output and physical capital data from the South African Reserve Bank Quarterly Bulletins. We are particularly interested in estimating the effect of education on worker productivity. The analysis starts by explor...

Economic theory often provides information on the variables to be included in economic relationships (e.g., demands are functions of prices) and sometimes provides information on the signs and magnitudes of firstand second-order derivatives (e.g., homogeneity and concavity information). However, it rarely provides information concerning functional forms. In the absence of this information, it i...

This paper proposes a tractable and consistent estimator of the (possibly multi-equation) nonparametric regression model. The estimator is based on least squares over sets of functions bounded in Sobolev norm and is closely related to penalized least squares. We establish consistency and rate of convergence results as well as asymptotic normality of the (suitably standardized) sum of squared re...

Recently the problem of determining the best, in the least-squares sense, rank-1 approximation to a higher-order tensor was studied and an iterative method that extends the wellknown power method for matrices was proposed for its solution. This higher-order power method is also proposed for the special but important class of supersymmetric tensors, with no change. A simplified version, adapted ...