Asymptotic expansions for bivariate von Mises functionals

Summary A Berry-Essen result and asymptotic expansions are derived for the distribution of bivariate von Mises functionals under moment and smoothness conditions.The results apply to the Cramér-von Mises ² — statistic as well as to the Central Limit Theorem in Hilbert space, yielding a convergence rate O(n _–1+) for every >0 on centered ellipsoids.Herrn Professor Dr. Leopold Schmetterer zu Ehren seines sechzigsten Geburtstages gewidmet     

Large deviations of degenerate Mises functionals

Ya. Yu. Nikitin's method of obtaining asymptotics of large deviations is extended to a more general situation. Bibliography:14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 256–261.     

The von Mises girder

The von Mises girder is the simplest example of an elastic system that exhibits nonlinear behavior, bifurcates, and has nonunique equilibrium states. The classical von Mises girder consists of two joined elastic compressible rods with small slopes. A typical manifestation of stability loss under vertical forces is buckling into the mirror-reflected shape. The critical load is a limiting point on the force-deflection curve.     

Invariance principles for von Mises and U-statistics

Summary The almost sure approximation of von Mises-statistics and U-statistics by appropriate stochastic integrals with respect to Kiefer processes is obtained. In general these integrals are non-Gaussian processes. As applications we get almost sure versions for the estimator of the variance and for the ²-test of goodness of fit.This work was done while the last author was a visiting professor at the Institut für Mathematische Stochastik at the University of Göttingen during the Spring of 1982. He thanks the Institut and its members for their hospitality     

Parameter estimation for von Mises–Fisher distributions

When analyzing high-dimensional data, it is often appropriate to pay attention only to the direction of each datum, disregarding its norm. The von Mises–Fisher (vMF) distribution is a natural probability distribution for such data. When we estimate the parameters of vMF distributions, parameter κ which corresponds to the degree of concentration is difficult to obtain, and some approximations are necessary. In this article, we propose an iterative algorithm using fixed points to obtain the maximum likelihood estimate (m.l.e.) for κ. We prove that there is a unique local maximum for κ. Besides, using a specific function to calculate the m.l.e., we obtain the upper and lower bounds of the interval in which the exact m.l.e. exists. In addition, based on these bounds, a new and good approximation is derived. The results of numerical experiments demonstrate the new approximation exhibits higher precision than traditional ones.     

关于Von Mises统计量Berry-Esseen定理的一点注记

Let V_n be a von Mises Statistice with kernel φ(x₁,x₂,… ,x_m).     

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation \(T\) of a probability space \((X,\mathcal{F },\mu )\) and some \(d \ge 1\) we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d where \(C_1, C_2,\ldots \) are normalizing constants and the kernel \(f\) belongs to an appropriate subspace in some \(L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)\) . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with \(T\) and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and \(d=2\) , we also show that the convergence holds in distribution towards a quadratic form \(\sum _{m=1}^{\infty } \lambda _m\eta ^2_m\) in independent standard Gaussian variables \(\eta _1, \eta _2, \ldots \) .     

Limit theorems for the canonical von Mises statistics with dependent data

We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert noises generated by Gaussian processes with nonorthogonal increments.     

Georg Hamel and Richard von Mises in Brno

The article focuses on two outstanding mathematicians who worked at Brno German Technical University, Georg Hamel and Richard von Mises. After their biographies are summarized, their Brno stays are described in detail. A major part of the article is devoted to the process of appointing professors at Brno German Technical University and the difficulties Richard von Mises as a candidate, or those who wanted him to become professor, had to face: he participated in four consecutive competitions for a chair, but without success. © 2002 Elsevier Science (USA).Der Beitrag ist Georg Hamel und Richard von Mises gewidmet—zwei Mathematikern von Weltrang, welche an der deutschen technischen Hochschule in Brünn tätig waren. Auf eine kurze Zusammenfassung ihrer Lebensläufe folgt eine ausführliche Beschreibung ihrer Aufenthalte in Brünn unter besonderer Beachtung der erfolglosen Bemühungen um die Ernennung von Richard von Mises zum Professor für Mathematik und Mechanik. © 2002 Elsevier Science (USA).Tento článek je věnován dvěma mimořádným osobnostem světové matematiky, které působily na německé technice v Brně—Georgu Hamelovi a Richardu von Misesovi. V úvodu jsou krátce zmı́něny jejich životnı́ osudy a poté je podrobně zachycen jejich brněnský pobyt. Je připomenuto úsilı́ o Misesovo jmenovánı́ profesorem matematiky a mechaniky v Brně. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 01A70, 01A73, 01A60.     

Stability of a characterization of the von Mises distribution

Translated from Problemy Ustoichivpsti Stokhasticheskikh Modelei, Trudy Seminara, pp. 73–84, 1986.     

An efficient simulation algorithm for the generalized von Mises distribution of order two

In this article we propose an exact efficient simulation algorithm for the generalized von Mises circular distribution of order two. It is an acceptance-rejection algorithm with a piecewise linear envelope based on the local extrema and the inflexion points of the generalized von Mises density of order two. We show that these points can be obtained from the roots of polynomials and degrees four and eight, which can be easily obtained by the methods of Ferrari and Weierstrass. A comparative study with the von Neumann acceptance-rejection, with the ratio-of-uniforms and with a Markov chain Monte Carlo algorithms shows that this new method is generally the most efficient.     

Inferences based on a bivariate distribution with von Mises marginals

There is very little literature concerning modeling the correlation between paired angular observations. We propose a bivariate model with von Mises marginal distributions. An algorithm for generating bivariate angles from this von Mises distribution is given. Maximum likelihood estimation is then addressed. We also develop a likelihood ratio test for independence in paired circular data. Application of the procedures to paired wind directions is illustrated. Employing simulation, using the proposed model, we compare the power of the likelihood ratio test with six existing tests of independence.     

Average projection type weighted Cramér- von Mises statistics for testing some distributions

This paper addresses the problem of testing goodness-of-fit for several important multivariate distributions: (I) Uniform distribution on p-dimensional unit sphere; (II) multivariate standard normal distribution; and (III) multivariate normal distribution with unknown mean vector and covariance matrix. The average projection type weighted Cramér-von Mises test statistic as well as estimated and weighted Cramér-von Mises statistics for testing distributions (I), (II) and (III) are constructed via integrating projection direction on the unit sphere, and the asymptotic distributions and the expansions of those test statistics under the null hypothesis are also obtained. Furthermore, the approach of this paper can be applied to testing goodness-of-fit for elliptically contoured distributions.     

Expansions for Singular Perturbations

The author applies the two-variable expansion procedure to anon-linear problem in singular perturbations, and compares theresults with those obtained using the composite expansion method.     

The distribution of length of N random unit vectors for a von Mises population

Greenwood and Durand (1955) have expressed the distribution function (d.f.) of the length $\text{R}$ of N random unit vectors for a von Mises population as a double integral. This double integral is simplified herein by calculating analytically one of the integrals. Hence a numerical calculation for large parameter values now becomes possible.     

Hardy Spaces for Laguerre Expansions

Let be the standard Laguerre functions of type a. We denote . Let and be the semigroups associated with the orthonormal systems and . We say that a function f belongs to the Hardy space associated with one of the semigroups if the corresponding maximal function belongs to . We prove special atomic decompositions of the elements of the Hardy spaces.     

Two-Variable Expansions for Singular Perturbations

The author applies the two-variable expansion procedure to anexample in singular perturbations and shows how the validityof this procedure can be established in many cases (includingsome non-linear problems).