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. Other figures-of-merit have been introduced in numerical analysis. The objective of the project is to investigate the behavior of these measures of uniformity using tools at the interface between statistics and numerical analysis.

##### Raffaello Seri

###### Professor of Econometrics

My research interests include statistics, numerical analysis, operations research, psychology, economics and management.

### Related

- Approximation of Stochastic Programming Problems
- Asymptotic Distributions of Covering and Separation Measures on the Hypersphere
- Computing the Asymptotic Distribution of Second-order $U$- and $V$-statistics
- The asymptotic distribution of Riesz' energy
- Computational aspects of discrepancies for equidistribution on the hypercube

## Publications

### Asymptotic Distributions of Covering and Separation Measures on the Hypersphere

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.

### Computing the Asymptotic Distribution of Second-order $U$- and $V$-statistics

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.

### Statistical Properties of $b$-adic Diaphonies

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.

### Quadrature Rules and Distribution of Points on Manifolds

We study the error in quadrature rules on a compact manifold. Our estimates are in the same spirit of the Koksma-Hlawka inequality and they depend on a sort of discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

### Computational Aspects of Cui-Freeden Statistics for Equidistribution on the Sphere

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.### Numerical Properties of Generalized Discrepancies on Spheres of Arbitrary Dimension

Quantifying uniformity of a configuration of points on the sphere is an interesting topic that is receiving growing attention in numerical analysis. An elegant solution has been provided by Cui and Freeden [J. Cui, W. Freeden, Equidistribution on the sphere,

*SIAM J. Sci. Comput.*18 (2) (1997) 595-609], where a class of discrepancies, called generalized discrepancies and originally associated with pseudodifferential operators on the unit sphere in R3, has been introduced. The objective of this paper is to extend to the sphere of arbitrary dimension this class of discrepancies and to study their numerical properties. First we show that generalized discrepancies are diaphonies on the hypersphere. This allows us to completely characterize the sequences of points for which convergence to zero of these discrepancies takes place. Then we discuss the worst-case error of quadrature rules and we derive a result on tractability of multivariate integration on the hypersphere. At last we provide several versions of Koksma-Hlawka type inequalities for integration of functions defined on the sphere.### Statistical Properties of Generalized Discrepancies

When testing that a sample of n points in the unit hypercube $[0,1]^d$ comes from a uniform distribution, the Kolmogorov-Smirnov and the Cramér-von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell introduced the so-called generalized $\mathscr{L}^p$-discrepancies. These discrepancies can be used in numerical integration through Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity testing and goodness-of-fit tests. The aim of this paper is to derive the statistical asymptotic properties of these statistics under Monte Carlo sampling. In particular, we show that, under the hypothesis of uniformity of the sample of points, the asymptotic distribution is a complex stochastic integral with respect to a pinned Brownian sheet. On the other hand, if the points are not uniformly distributed, then the asymptotic distribution is Gaussian.

## Talks

### The asymptotic distribution of Riesz' Energy

International conference

Jul 15, 2019 — Jul 19, 2019
Valencia, Spain

### The asymptotic distribution of Riesz' energy

International conference

Jul 1, 2018 — Jul 6, 2018
Rennes, France

### Universal HD robustness of uniformity tests on the hypersphere

International conference

Feb 26, 2018 — Mar 2, 2018
Providence, USA

### Infinite weighted sums of chi square random variables: econometric examples and approximations

National conference

Nov 12, 2015 — Nov 13, 2015
Bologna, Italy

### Computing Weighted Chi-Squared Distributions and Related Quantities

International conference

Aug 25, 2015 — Aug 28, 2015
Oxford, United Kingdom

### Computational aspects of the distribution of generalized discrepancies

International conference

Jul 6, 2015 — Jul 10, 2015
Linz, Austria

### Comparison of quadrature rules for the Wielandt-Nyström method with statistical applications

International conference

Sep 19, 2012 — Sep 25, 2012
Kos, Greece

### Quadrature rules and distribution of points on manifolds

International conference

Sep 9, 2012 — Sep 14, 2012
Alba di Canazei, Italy

### Computational aspects of discrepancies for equidistribution on the hypercube

International conference

Aug 27, 2012 — Aug 31, 2012
Limassol, Cyprus

### Formule di quadratura e distribuzione di punti su varietà compatte

National conference

Sep 12, 2011 — Sep 17, 2011
Bologna, Italy

### Quadrature rules and distribution of points on manifolds

National conference

May 30, 2011 — Jun 4, 2011
Roma, Italy

### Statistical tests of uniformity on the hypersphere

International conference

Aug 17, 2010 — Aug 22, 2010
Piraeus, Greece

### Statistical tests of uniformity on the hypersphere

International conference

Jul 12, 2010 — Jul 15, 2010
Coventry, United Kingdom

### Computing Weighted Chi-square Distributions and Related Quantities

International conference

Jun 19, 2008 — Jun 21, 2008
Neuchåtel, Switzerland

### Diaphonies on Sobolev classes of functions

International conference

Jul 16, 2007 — Jul 20, 2007
Zürich, Switzerland

### Diaphonies on Sobolev classes of functions

International conference

Jun 18, 2007 — Jun 22, 2007
Varenna, Italy

### Computing Weighted Chi-square Distributions and Related Quantities

International conference

Aug 14, 2006 — Aug 18, 2006
Ulm, Germany

### Generalized Discrepancies on Spheres of Arbitrary Dimension

International conference

Aug 14, 2006 — Aug 18, 2006
Ulm, Germany

### Computing Weighted Chi-square Distributions and Related Quantities

International conference

Jun 15, 2006 — Jun 17, 2006
Wien, Austria

### Approximation of the Asymptotic Distribution of Quadratic Discrepancies

National conference

Apr 18, 2006 — Apr 19, 2006
Monte Porzio Catone, Italy

### Statistical Properties of Generalized Discrepancies and Related Quantities

International conference

Jul 24, 2005 — Jul 28, 2005
Oslo, Norway

### Statistical Properties of Generalized Discrepancies and Related Quantities

International conference

Aug 20, 2004 — Aug 24, 2004
Madrid, Spain

### Statistical Properties of Generalized Discrepancies and Related Quantities

International conference

Jul 26, 2004 — Jul 31, 2004
Barcelona, Spain

### Statistical Properties of Generalized Discrepancies and Related Quantities

International conference

Jul 4, 2004 — Jul 9, 2004
Perugia, Italy

### Statistical Properties of Generalized and Quadratic Discrepancies

International conference

Jun 7, 2004 — Jun 10, 2004
Antibes, France

### Statistical Properties of Generalized Discrepancies and Related Quantities

National conference

Sep 4, 2003 — Sep 6, 2003
Treviso, Italy

### Statistical Properties of Generalized Discrepancies

International conference

Dec 13, 2002 — Dec 14, 2002
Bologna, Italy

### Statistical Properties of Generalized Discrepancies

International conference

Aug 25, 2002 — Aug 28, 2002
Venezia, Italy

### Statistical Properties of Generalized Discrepancies

International conference

May 13, 2002 — May 17, 2002
Bruxelles and Louvain-la-Neuve, Belgium