Tests of symmetry based on transformed empirical processes

A probability distribution function F is said to be symmetric when 1 ‐ F(x) ‐ F(‐x) = 0 for all x∈ R. Given a sequence of alternatives contiguous to a certain symmetric F₀, the authors are concerned with testing for the null hypothesis of symmetry. The proposed tests are consistent against any nonsymmetric alternative, and their power with respect to the given sequence can easily be optimized. The tests are constructed by means of transformed empirical processes with an adequate selection of the underlying isometry, and the optimum power is obtained by suitably choosing the score functions. The test statistics are very easy to compute and their asymptotic distributions are simple.     

对称和对称方法

对称是指事物通过某种中介变化时出现的同一性,这种同一不是绝对的同一,是包含差异的同一;非对称则是事物通过某种中介而变化时出现的差异性,这种差异也不是绝对的差异,是包含同一的差异。对称和非对称,变化中的同一和变化中的差异,互相依存和互相转化,构成了自然界的生动图景,也构成了现代科学技术中的奇妙方法和富有启迪性的理论思考。     

A simple monotone process with application to radiocarbon-dated depth chronologies

Summary.  We propose a new and simple continuous Markov monotone stochastic process and use it to make inference on a partially observed monotone stochastic process. The process is piecewise linear, based on additive independent gamma increments arriving in a Poisson fashion. An independent increments variation allows very simple conditional simulation of sample paths given known values of the process. We take advantage of a reparameterization involving the Tweedie distribution to provide efficient computation. The motivating problem is the establishment of a chronology for samples taken from lake sediment cores, i.e. the attribution of a set of dates to samples of the core given their depths, knowing that the age–depth relationship is monotone. The chronological information arises from radiocarbon (¹⁴C) dating at a subset of depths. We use the process to model the stochastically varying rate of sedimentation.     

Kadomtsev-Petviashvili势方程的对称结构

本文将形式级数对称理论应用到Kadomtsev-Petviashvili势方程，得到了十二个截断的李点对称。     

论图案对称律形式及其构成方法

对称律是图案构成的基本形式规律之一,以对称的基本形式为出发点,从形态到空间利用对称律的构成形式和手法,根据这一构成形式规律,使对称在形态与空间两方面得到丰富的创造。     

Testing circular symmetry

The author addresses the problem of testing circular data for reflective symmetry about an unknown central direction and proposes a simple omnibus test based on the sample second sine moment about an estimation of this direction. Under quite general conditions, for an underlying distribution which is reflectively symmetric, the large‐sample asymptotic distribution of the test statistic is standard normal. Randomization and bootstrap variants of the test are also introduced, and the operating characteristics of different versions of the test are investigated in a Monte Carlo study. The large‐sample and bootstrap versions of the test are applied in the analysis of two illustrative examples drawn from the circular statistics literature.     

Concepts and measures for skewness with data-analytic implications

The decomposition of the distribution function into skewness and spread functions serves as a basis for conceptualizing the skewness of a random variable. Measures of skewness are also described in terms of the skewness and spread functions, thereby unifying the measures with the concepts. Results that relate these measures to whether one random variable is more skewed than another are reviewed and extended. Graphical displays are presented for uncovering the nature of the skewness of a variable. The measures are also linked to the issue of symmetrizing a variable.     

Category Distinguishability and Observer Agreement

It is common in the medical, biological, and social sciences for the categories into which an object is classified not to have a fully objective definition. Theoretically speaking the categories are therefore not completely distinguishable. The practical extent of their distinguishability can be measured when two expert observers classify the same sample of objects. It is shown, under reasonable assumptions, that the matrix of joint classification probabilities is quasi-symmetric, and that the symmetric matrix component is non-negative definite. The degree of distinguishability between two categories is defined and is used to give a measure of overall category distinguishability. It is argued that the kappa measure of observer agreement is unsatisfactory as a measure of overall category distinguishability.     

ON THE GENERALIZED RAO-RUBIN CONDITION and SOME VARIANTS

This paper gives a characterization of some members of the compound Poisson family of distributions based on the generalized Rao-Rubin condition. By considering some variants of this condition and using power series arguments, characterizations of the Poisson distribution are also obtained.     

一般动力学系统在相空间的Noether定理及其逆定理

根据一般动力学系统在相空间的作用积分在无穷小群变换下的变换性质,分别建立起完整和非完整非保守系统的Noether定理和逆定理.由给出的例子表明,在相空间中可以呈现出动力学系统新的对称性.