We consider measures of covering and separation that are expressed through maxima and minima of distances between points of an hypersphere. We investigate the behavior of these measures when applied to a sample of independent and uniformly distributed points. In particular, we derive their asymptotic distributions when the number of points diverges. These results can be useful as a benchmark against which deterministic point sets can be evaluated. Whenever possible, we supplement the rigorous derivation of these limiting distributions with some heuristic reasonings based on extreme value theory. As a by-product, we provide a proof for a conjecture on the hole radius associated to a facet of the convex hull of points distributed on the hypersphere.
Under general conditions, the asymptotic distribution of degenerate second-order $U$- and $V$-statistics is an (infinite) weighted sum of $\chi^2$ random variables whose weights are the eigenvalues of an integral operator associated with the kernel of the statistic. Also the behavior of the statistic in terms of power can be characterized through the eigenvalues and the eigenfunctions of the same integral operator. No general algorithm seems to be available to compute these quantities starting from the kernel of the statistic. An algorithm is proposed to approximate (as precisely as needed) the asymptotic distribution and the power of the test statistics, and to build several measures of performance for tests based on $U$- and $V$-statistics. The algorithm uses the Wielandt–Nyström method of approximation of an integral operator based on quadrature, and can be used with several methods of numerical integration. An extensive numerical study shows that the Wielandt–Nyström method based on Clenshaw–Curtis quadrature performs very well both for the eigenvalues and the eigenfunctions.
Quantifying uniformity of a configuration of points on a space is a topic that is receiving growing attention in computer science, physics and mathematics. The problem has interesting connections with statistics, where several tests of uniformity have been introduced.
The aim of this paper is to derive the asymptotic statistical properties of a class of discrepancies on the unit hypercube called $b$-adic diaphonies. They have been introduced to evaluate the equidistribution of quasi-Monte Carlo sequences on the unit hypercube. We consider their properties when applied to a sample of independent and uniformly distributed random points. We show that the limiting distribution of the statistic is an infinite weighted sum of chi-squared random variables, whose weights can be explicitly characterized and computed. We also describe the rate of convergence of the finite-sample distribution to the asymptotic one and show that this is much faster than in the classical Berry-Esséen bound. Then, we consider in detail the approximation of the asymptotic distribution through two truncations of the original infinite weighted sum, and we provide explicit and tight bounds for the truncation error. Numerical results illustrate the findings of the paper, and an empirical example shows the relevance of the results in applications.
In this paper, we derive the asymptotic statistical properties of a class of generalized discrepancies introduced by Cui and Freeden (*SIAM J. Sci. Comput.*, 1997) to test equidistribution on the sphere. We show that they have highly desirable properties and encompass several statistics already proposed in the literature. In particular, it turns out that the limiting distribution is an (infinite) weighted sum of chi-squared random variables. Issues concerning the approximation of this distribution are considered in detail and explicit bounds for the approximation error are given. The statistics are then applied to assess the equidistribution of Hammersley low discrepancy sequences on the sphere and the uniformity of a dataset concerning magnetic orientations.