arXiv subjects

Norbert Henze

Publications and source records attributed to Norbert Henze.

At least 19 recordsLinked to original sources

arXiv2026

The limit law of the largest interpoint distance in a $d$-dimensional ellipsoid

We consider the largest interpoint distance $M_n=\max_{1\le i<j\le n}\|X_i-X_j\|$ among independent random points $X_1,\ldots,X_n$, uniformly distributed on a $d$-dimensional ellipsoid. We assume that the largest semi-axis has length 1 and multiplicity $k\ge 2$, whereas the remaining semi-axes are strictly smaller. In this situation, the diameter is attained on a manifold of dimension $k-1$, and the extremal points are no longer isolated. We establish a weak limit law for the diameter deficit $2-M_n$. Writing $q=d-k$ and $\alpha=q+(k+3)/2$, we show that $n^{2/\alpha}(2-M_n)$ converges in distribution to a Weibull random variable. The proof is based on a local analysis near the diameter manifold, a sharp asymptotic formula for the two-point tail probability, and a Chen--Stein Poisson approximation for rare nearly diametral pairs.

math.PR↗

arXiv2026

The location of the largest exponential spacing and Euler's generalized pentagonal numbers

We study the location of the largest spacing generated by the order statistics of a sample from the standard exponential distribution. Although the asymptotic behaviour of the largest spacing itself is well understood, considerably less is known about the index at which it is attained. Using the independence and unequal rates of exponential spacings, we derive exact finite-sample formulas and show that, when measured relative to the right endpoint, the location of the largest spacing converges in distribution to a non-degenerate probability distribution on the positive integers. We obtain explicit integral representations for the limiting probabilities and, using Euler's pentagonal number theorem, derive a series representation involving the generalized pentagonal numbers. This reveals that the Euler product appearing in the limiting distribution of the largest exponential spacing also governs the distribution of its location. The convergence result is also extended to the location of the largest $m$-spacing for every fixed $m\geq 1$, despite the dependence among overlapping $m$-spacings. Numerical values illustrate the concentration of the limiting distribution near the right endpoint and the rapid convergence of the finite-sample probabilities.

math.PR↗

arXiv2026

Logarithmic energy distances and Gini covariance for Hilbert-valued random elements

For $\alpha\in(0,2)$, the generalized energy distance and the Gini covariance statistic are based on kernels of the form $(x,y)\mapsto \|x-y\|^\alpha$, where $\|\cdot\|$ denotes the norm in a real separable Hilbert space. This paper investigates the boundary regime $\alpha\downarrow 0$. After suitable normalization, the corresponding energy distance converges to a logarithmic energy distance involving the kernel $(x,y)\mapsto\log\|x-y\|$. We establish that the resulting logarithmic energy distance retains the fundamental characterization property of ordinary energy distances in separable Hilbert spaces and derive a representation in terms of Gaussian-kernel maximum mean discrepancies. Motivated by this representation, we introduce a logarithmic Gini covariance for the $k$-sample problem and investigate its structural and asymptotic properties. In particular, we derive a representation in terms of pairwise logarithmic energy distances, establish a characterization theorem for equality of distributions, develop asymptotic null and alternative theory for the corresponding empirical statistic, and discuss permutation-based implementation. The logarithmic framework reveals a new boundary phenomenon within the family of energy-type statistics and provides connections with kernel methods, functional data analysis, and high-dimensional inference.

stat.ME↗

arXiv2026

A 4/7-limit law for the largest interpoint distance in a rotational ellipsoid

Let $M_n$ denote the largest interpoint distance among independent random points $X_1,\dots,X_n$ uniformly distributed in a compact set in $\mathbb{R}^d$. Weak limit laws for $M_n$ are known in several geometric settings, in particular for ellipsoids with a unique major axis. In this paper we treat the simplest nontrivial case in which the largest semi-axis is not unique, namely the rotational ellipsoid $\{(x_1,x_2,x_3)\in\mathbb{R}^3: (x_1^2+x_2^2)/h^2 + x_3^2/a^2 \le 1\}$, where $0<a<h$. The diameter of this ellipsoid is attained by all antipodal pairs on the equatorial circle, so the extremal points are not isolated. We prove that $n^{4/7}(2h-M_n)$ converges in distribution to a Weibull-type limit law with explicit parameter. The proof combines geometric localization arguments with a Chen--Stein Poisson approximation for rare nearly diametral pairs.

math.PR↗

arXiv2023

A unified approach to goodness-of-fit testing for spherical and hyperspherical data

We propose a general and relatively simple method for the construction of goodness-of-fit tests on the sphere and the hypersphere. The method is based on the characterization of probability distributions via their characteristic function, and it leads to test criteria that are convenient regarding applications and consistent against arbitrary deviations from the model under test. We emphasize goodness-of-fit tests for spherical distributions due to their importance in applications and the relative scarcity of available methods.

math.ST↗

arXiv2022

A genuine test for hyperuniformity

We introduce a rigorous and sensitive significance test for hyperuniformity that yields reliable results even from a single sample. Our approach is based on a detailed analysis of the empirical Fourier transform of a stationary point process in $\mathbb{R}^d$. For large system sizes, we derive the asymptotic covariances and establish a multivariate central limit theorem (CLT) for these empirical Fourier transforms. Their absolute square value, the scattering intensity, is then used as the standard estimator of the structure factor. The above CLT holds for a preferably large class of point processes, and whenever this is the case, the scattering intensity satisfies a multivariate limit theorem as well. Hence, we can use the likelihood ratio principle to test for hyperuniformity. Remarkably, the asymptotic distribution of the resulting test statistic is universal under the null hypothesis of hyperuniformity. We obtain its explicit form from simulations with very high accuracy. The novel test precisely keeps a nominal significance level for hyperuniform models, and it rejects non-hyperuniform examples with high power even in borderline cases. Moreover, it does so given only a single sample with a practically relevant system size.

math.ST↗

arXiv2022

Weibull or not Weibull?

We propose novel goodness-of-fit tests for the Weibull distribution with unknown parameters. These tests are based on an alternative characterizing representation of the Laplace transform related to the density approach in the context of Stein's method. Asymptotic theory of the tests is derived, including the limit null distribution, the behaviour under contiguous alternatives, the validity of the parametric bootstrap procedure, and consistency of the tests against a large class of alternatives. A Monte Carlo simulation study shows the competitiveness of the new procedure. Finally, the procedure is applied to real data examples taken from the materials science.

math.ST↗

arXiv2021

On the eigenvalues associated with the limit null distribution of the Epps-Pulley test of normality

The Shapiro--Wilk test (SW) and the Anderson--Darling test (AD) turned out to be strong procedures for testing for normality. They are joined by a class of tests for normality proposed by Epps and Pulley that, in contrary to SW and AD, have been extended by Baringhaus and Henze to yield easy-to-use affine invariant and universally consistent tests for normality in any dimension. The limit null distribution of the Epps--Pulley test involves a sequences of eigenvalues of a certain integral operator induced by the covariance kernel of the limiting Gaussian process. We solve the associated integral equation and present the corresponding eigenvalues.

math.ST↗

arXiv2021

Bahadur efficiencies of the Epps--Pulley test for normality

The test for normality suggested by Epps and Pulley (1983) is a serious competitor to tests based on the empirical distribution function. In contrast to the latter procedures, it has been generalized to obtain a genuine affine invariant and universally consistent test for normality in any dimension. We obtain approximate Bahadur efficiencies for the test of Epps and Pulley, thus complementing recent results of Milo\v{s}evi\'c et al. (2021). For certain values of a tuning parameter that is inherent in the Epps--Pulley test, this test outperforms each of its competitors considered in Milo\v{s}evi\'c et al. (2021), over the whole range of six close alternatives to normality.

math.ST↗

arXiv2021

Limit laws for large kth-nearest neighbor balls

Let $X_1,\ldots,X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density $f$. Under some conditions on $f$, we obtain a Poisson limit theorem, as $n \to \infty$, for the number of large probability $k$th-nearest neighbor balls of $X_1,\ldots,X_n$. Our result generalizes Theorem 2. of [10], which refers to the special case $k=1$. Our proof is completely different since it employs the Chen-Stein method instead of the method of moments. Moreover, we obtain a rate of convergence for the Poisson approximation.

math.PR↗

arXiv2020

Analysis of the Matrix Event Graph Replicated Data Type

Matrix is a new kind of decentralized, topic-based publish-subscribe middleware for communication and data storage that is getting popular particularly as a basis for secure instant messaging. In comparison to traditional decentralized communication systems, Matrix replaces pure message passing with a replicated data structure. This data structure, which we extract and call the Matrix Event Graph (MEG), depicts the causal history of messages. We show that this MEG represents an interesting and important replicated data type for general decentralized applications that are based on causal histories of publish-subscribe events: we show that a MEG possesses strong properties with respect to consistency, byzantine attackers, and scalability. First, we show that the MEG provides Strong Eventual Consistency (SEC), and that it is available under partition, by proving that the MEG is a Conflict-Free Replicated Data Type for causal histories. While strong consistency is impossible here as shown by the famous CAP theorem, SEC is among the best known achievable trade-offs. Second, we discuss the implications of byzantine attackers on the data type's properties. We note that the MEG, as it does not strive for consensus, can cope with $n > f$ environments with $n$ total participants of which $f$ show byzantine faults. Furthermore, we analyze scalability: Using Markov chains we study the width of the MEG, defined as the number of forward extremities, over time and observe an almost optimal evolution. We conjecture that this property is inherent to the underlying spatially inhomogeneous random walk.

cs.DC↗

arXiv2020

Tests for circular symmetry of complex-valued random vectors

We propose tests for the null hypothesis that the law of a complex-valued random vector is circularly symmetric. The test criteria are formulated as $L^2$-type criteria based on empirical characteristic functions, and they are convenient from the computational point of view. Asymptotic as well as Monte-Carlo results are presented. Applications on real data are also reported. An R package called CircSymTest is available from the authors.

math.ST↗

arXiv2020

Testing normality in any dimension by Fourier methods in a multivariate Stein equation

We study a novel class of affine invariant and consistent tests for multivariate normality. The tests are based on a characterization of the standard $d$-variate normal distribution by means of the unique solution of an initial value problem connected to a partial differential equation, which is motivated by a multivariate Stein equation. The test criterion is a suitably weighted $L^2$-statistic. We derive the limit distribution of the test statistic under the null hypothesis as well as under contiguous and fixed alternatives to normality. A consistent estimator of the limiting variance under fixed alternatives as well as an asymptotic confidence interval of the distance of an underlying alternative with respect to the multivariate normal law is derived. In simulation studies, we show that the tests are strong in comparison with prominent competitors, and that the empirical coverage rate of the asymptotic confidence interval converges to the nominal level. We present a real data example, and we outline topics for further research.

math.ST↗

arXiv2020

Tests for multivariate normality -- a critical review with emphasis on weighted $L^2$-statistics

This article gives a synopsis on new developments in affine invariant tests for multivariate normality in an i.i.d.-setting, with special emphasis on asymptotic properties of several classes of weighted $L^2$-statistics. Since weighted $L^2$-statistics typically have limit normal distributions under fixed alternatives to normality, they open ground for a neighborhood of model validation for normality. The paper also reviews several other invariant tests for this problem, notably the energy test, and it presents the results of a large-scale simulation study. All tests under study are implemented in the accompanying R-package mnt.

math.ST↗

arXiv2019

A new test of multivariate normality by a double estimation in a characterizing PDE

This paper deals with testing for nondegenerate normality of a $d$-variate random vector $X$ based on a random sample $X_1,\ldots,X_n$ of $X$. The rationale of the test is that the characteristic function $ψ(t) = \exp(-\|t\|^2/2)$ of the standard normal distribution in $\mathbb{R}^d$ is the only solution of the partial differential equation $Δf(t) = (\|t\|^2-d)f(t)$, $t \in \mathbb{R}^d$, subject to the condition $f(0) = 1$. By contrast with a recent approach that bases a test for multivariate normality on the difference $Δψ_n(t)-(\|t\|^2-d)ψ(t)$, where $ψ_n(t)$ is the empirical characteristic function of suitably scaled residuals of $X_1,\ldots,X_n$, we consider a weighted $L^2$-statistic that employs $Δψ_n(t)-(\|t\|^2-d)ψ_n(t)$. We derive asymptotic properties of the test under the null hypothesis and alternatives. The test is affine invariant and consistent against general alternatives, and it exhibits high power when compared with prominent competitors.

math.ST↗

arXiv2019

Curiosities regarding waiting times in Pólya's urn model

Consider an urn initially containing $b$ black and $w$ white balls. Select a ball at random and observe its color. If it is black, stop. Otherwise, return the white ball together with another white ball to the urn. Continue selecting at random, each time adding a white ball, until a black ball is selected. Let $T_{b,w}$ denote the number of draws until this happens. Surprisingly, the expectation of $T_{b,w}$ is infinite for the "fair" initial scenario $b =w=1$, but finite if $b=2$ and $w=10^9$. In fact, $\mathbb{E}[T_{b,w}]$ is finite if and only if $b\ge 2$, and the variance of $T_{b,w}$ is finite if and only if $b \ge 3$, regardless of the number $w$ of white balls. These observations extend to higher moments.

math.PR↗

arXiv2019

A test for Gaussianity in Hilbert spaces via the empirical characteristic functional

Let $X_1,X_2, \ldots$ be independent and identically distributed random elements taking values in a separable Hilbert space $\mathbb{H}$. With applications for functional data in mind, $\mathbb{H}$ may be regarded as a space of square-integrable functions, defined on a compact interval. We propose and study a novel test of the hypothesis $H_0$ that $X_1$ has some unspecified non-degenerate Gaussian distribution. The test statistic $T_n=T_n(X_1,\ldots,X_n)$ is based on a measure of deviation between the empirical characteristic functional of $X_1,\ldots,X_n$ and the characteristic functional of a suitable Gaussian random element of $\mathbb{H}$. We derive the asymptotic distribution of $T_n$ as $n \to \infty$ under $H_0$ and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of $T_n$ as well as a normal limit distribution of $T_n$ under alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.

math.ST↗

arXiv2019

Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

We study a novel class of affine invariant and consistent tests for normality in any dimension. The tests are based on a characterization of the standard $d$-variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schrödinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood-of-model validation procedure.

stat.ME↗