On the consistency of the spacings test for multivariate uniformity, including on manifolds

2018 ◽  
Vol 55 (2) ◽  
pp. 659-665
Author(s):  
Norbert Henze
Keyword(s):  
Riemannian Manifolds ◽  
Convex Set ◽  
Asymptotic Results ◽  
Bounded Set ◽  
Conceptual Proof ◽  
Independent And Identically Distributed

Abstract We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set K ⊂ ℝd that is based on the maximal spacing generated by independent and identically distributed points X1, . . ., Xn in K, i.e. the volume of the largest convex set of a given shape that is contained in K and avoids each of these points. Since asymptotic results for the d > 1 case are only availabe under uniformity, a key element of the proof is a suitable coupling. The proof is general enough to cover the case of testing for uniformity on compact Riemannian manifolds with spacings defined by geodesic balls.

On resampling schemes for polytopes

2019 ◽  
Vol 56 (4) ◽  
pp. 959-980
Author(s):  
Weinan Qi ◽  
Mahmoud Zarepour
Keyword(s):  
Convex Hull ◽  
Asymptotic Theory ◽  
Convex Set ◽  
Real Life ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Underlying Distribution ◽  
Asymptotic Consistency ◽  
Computational Intractability ◽  
Practical Implications

AbstractThe convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. To approximate the distribution of the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately most of the asymptotic results are computationally intractable. To address this computational intractability, we consider consistent bootstrapping schemes for certain cases. Let $S_n=\{X_i\}_{i=1}^{n}$ be a sequence of independent and identically distributed random points uniformly distributed on an unknown convex set in $\mathbb{R}^{d}$ ($d\ge 2$ ). We suggest a bootstrapping scheme that relies on resampling uniformly from the convex hull of $S_n$ . Moreover, the resampling asymptotic consistency of certain functionals of convex hulls is derived under this bootstrapping scheme. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. For $d=2$ , we investigate the asymptotic consistency of the suggested bootstrapping scheme for the area of the symmetric difference and the perimeter difference between the actual convex set and its estimate. In all cases the consistency allows us to rely on the suggested resampling scheme to study the actual distributions, which are not computationally tractable.

A Note on Hyperconvexity in Riemannian Manifolds

1959 ◽  
Vol 11 ◽  
pp. 576-582
Author(s):  
Albert Nijenhuis
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Shortest Path ◽  
Riemannian Manifolds ◽  
Convex Set ◽  
Geodesic Segment ◽  
Positive Definite ◽  
Classical Theorem ◽  
Arc Length ◽  
Class C

Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.

scholarly journals A SPHERICAL MAPPING AND BORSUK CONJECTURE IN RIEMANNIAN AND NON-EUCLIDEAN SPACES

1994 ◽  
Vol 25 (2) ◽  
pp. 149-155
Author(s):  
BORIS V. DEKSTER
Keyword(s):  
Convex Body ◽  
Riemannian Manifolds ◽  
Additional Assumption ◽  
Convex Bodies ◽  
Smooth Convex ◽  
Euclidean Spaces ◽  
Smooth Convex Body ◽  
Bounded Set

We introduce an analog of the spherical mapping for convex bodies in a Riemannian $n$-manifold, and then use this construction to prove the Borsuk conjecture for some types of such bodies. The Borsuk conjecture is that each bounded set $X$ in the Euclidean $n$-space can be covered by $n +1$ sets of smaller diameter. The conjecture was disproved recently by Kahn and Kalai. However Hadwiger proved the Borsuk conjecture under the additional assumption that the set $X$ is a smooth convex body. Here we extend this result to convex bodies in Riemannian manifolds under some further restrictions.

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

2013 ◽  
Vol 56 (2) ◽  
pp. 272-282 ◽  
Author(s):  
Lixin Cheng ◽  
Zhenghua Luo ◽  
Yu Zhou
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Lower Semicontinuous ◽  
Weakly Compact ◽  
Bounded Set ◽  
Fréchet Differentiable ◽  
Bounded Convex

AbstractIn this note, we first give a characterization of super weakly compact convex sets of a Banach space X: a closed bounded convex set K ⊂ X is super weakly compact if and only if there exists a w* lower semicontinuous seminorm p with p ≥ σK ≌ supxєK 〈.,x〉 such that p2 is uniformly Fréchet differentiable on each bounded set of X*. Then we present a representation theoremfor the dual of the semigroup swcc(X) consisting of all the nonempty super weakly compact convex sets of the space X.

A competitive best-choice problem with Poisson arrivals

1990 ◽  
Vol 27 (02) ◽  
pp. 333-342 ◽  
Author(s):  
E. G. Enns ◽  
E. Z. Ferenstein
Keyword(s):  
Continuous Distribution ◽  
Random Variables ◽  
Optimal Strategies ◽  
Choice Problem ◽  
Asymptotic Results ◽  
Decision Scheme ◽  
Best Choice Problem ◽  
Poisson Stream ◽  
Poisson Arrivals ◽  
Independent And Identically Distributed

Two competitors observe a Poisson stream of offers. The offers are independent and identically distributed random variables from some continuous distribution. Each of the competitors wishes to accept one offer in the interval [0,T] and each aims to select an offer larger than that of his competitor. Offers are observed sequentially and decisions to accept or reject must be made when the offers arrive. Optimal strategies and winning probabilities are obtained for the competitors under a priorized decision scheme. The time of first offer acceptance is also analyzed. In all cases the asymptotic results are obtained.

A competitive best-choice problem with Poisson arrivals

1990 ◽  
Vol 27 (2) ◽  
pp. 333-342 ◽  
Author(s):  
E. G. Enns ◽  
E. Z. Ferenstein
Keyword(s):  
Continuous Distribution ◽  
Random Variables ◽  
Optimal Strategies ◽  
Choice Problem ◽  
Asymptotic Results ◽  
Decision Scheme ◽  
Best Choice Problem ◽  
Poisson Stream ◽  
Poisson Arrivals ◽  
Independent And Identically Distributed

Two competitors observe a Poisson stream of offers. The offers are independent and identically distributed random variables from some continuous distribution. Each of the competitors wishes to accept one offer in the interval [0, T] and each aims to select an offer larger than that of his competitor. Offers are observed sequentially and decisions to accept or reject must be made when the offers arrive. Optimal strategies and winning probabilities are obtained for the competitors under a priorized decision scheme. The time of first offer acceptance is also analyzed. In all cases the asymptotic results are obtained.

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