On the rate of convergence to asymptotic independence between order statistics under power normalization with extension to the generalized order statistics

Based on the notion of the associated copula for order statistics and some L ₁-distances, we derive a symmetric nonparametric measure of asymptotic dependence between the order statistics under power normalization. This nonparametric criterion of dependence is extended to the generalized, as well as the dual generalized, order statistics.     

Introducing the Emerging Discipline of Statistics Education

Increasing attention has been given over the last decade by the statistics, mathematics and science education communities to the development of statistical literacy and numeracy skills of all citizens and the enhancement of statistics education at all levels. This paper introduces the emerging discipline of statistics education and considers its role in the development of these important skills. The paper begins with information on the growing importance of statistics in today's society, schools and colleges, summarizes unique challenges students face as they learn statistics, and makes a case for the importance of collaboration between mathematicians and statisticians in preparing teachers to teach students how to understand and reason about data. We discuss the differences and interrelations between statistics and mathematics, recognizing that mathematics is the discipline that has traditionally included instruction in statistics. We conclude with an argument that statistics should be viewed as a bridge between mathematics and science and should be taught in both disciplines.     

On the joint distribution of Grubbs' statistics

In this paper, the joint pdf's of Grubbs' statistics for normal and exponential populations are obtained; and relationship between the two pdf's is established. New formulations of the first marginal pdf of Grubbs' statistics for these two populations are given; the formulation of the exponential population case is a new one. Iterative formulas for the pdf of Grubbs' statistics are also obtained.     

On the rate of convergence of the distributions of certain statistics to the Laplace and student distributions

This work deals with asymptotically normal statistics. If the size of sample is replaced by a random variable that is the maximum of n independent random variables with Pareto discrete distribution, then the distribution of these statistics is asymptotically Laplacian. If the size of the sample is replaced by a random variable with negative binomial distribution, then the distribution of these statistics is asymptotically a Student distribution. In the present study, the rates of convergences of these statistics’ distributions to the corresponding limiting distributions are estimated and the constants from expressions for estimating the rates are determined more accurately.     

Some test statistics based on the martingale term of the empirical distribution function

Summary It is proved that the martingale term of the empirical distribution function converges weakly to a Gaussian process inD[0, 1]. Some statistics for goodness-of-fit tests based on the martingale term of the empirical distribution function are proposed. Asymptotic distributions of these statistics under the null hypothesis are given. The approximate Bahadur efficiencies of the statistics to the Kolmogorov-Smirnov statistic and to the Cramér-von Mises statistic are also calculated. The Institute of Statistical Mathematics     

泛函型统计量的随机加权逼近

§ 1 引言及一般结果设X_,X_2,…X_n是来自分布为F(未知)的总体的n个iid.样本。它们的经验分布为 F_n(x)=1/n sum from i=1 to n I{X_ix}。 (1)常用的许多统计量都是通过经验分布而依赖于样本的,即能写成经验分布的泛函形式     

Symmetrized approximate score rank tests for the two-sample case

Rank test statistics for the two-sample problem are based on the sum of the rank scores from either sample. However, a critical difference can occur when approximate scores are used since the sum of the rank scores from sample 1 is not equal to minus the sum of the rank scores from sample 2. By centering and scaling as described in Hajek and Sidak (1967, Theory of Rank Tests, Academic Press, New York) for the uncensored data case the statistics computed from each sample become identical. However such symmetrized approximate scores rank statistics have not been proposed in the censored data case. We propose a statistic that treats the two approximate scores rank statistics in a symmetric manner. Under equal censoring distributions the symmetric rank tests are efficient when the score function corresponds to the underlying model distribution. For unequal censoring distributions we derive a useable expression for the asymptotic variance of our symmetric rank statistics.     

On testing for homogeneity of the covariance matrices

Testing equality of covariance matrix ofk populations has long been an interesting issue in statistical inference. To overcome the sparseness of data points in a high-dimensional space and deal with the general cases, we suggest several projection pursuit type statistics. Some results on the limiting distributions of the statistics are obtained. Some properties of Bootstrap approximation are investigated. Furthermore, for computational reasons an approximation which is based on Number theoretic method for the statistics is adopted. Several simulation experiments are performed.     

On the asymptotic efficiency of normality tests based on the Shepp property

New invariant tests of normality based on Shepp property are constructed and studied. Test statistics are functions of a U-empirical process. The limiting distributions and large deviations of test statistics under the null-hypothesis are described. Local Bahadur efficiency of new tests is calculated under simplest parametric alternatives (shift, skew, and contamination alternatives).     

均值混合正态分布统计量的性质

这篇文章基于基因遗传背景,提出了一类均值混合正态分布,它不同于通常所讨论的方差混合正态分布. 作者研究了这类均值混合正态分布统计量的性质,给出了平移变换群下不变量的稳健性,即它与正态分布下该统计量有相同的性质, 并且讨论了其它统计量的分布.     

A constructive hypothesis test for the single-index models with two groups

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.     

On multidimensional expansions of distributions of sums by the saddle point method

Many problems of applied statistics, in particular, hypotheses testing, contain no information about true distributions of the statistics under consideration. Therefore, limit distributions of those statistics are used, which may give inadequate results in actual practice. Problems on the refinement of the distributions thus arise. In the present paper, we extend the results of H. Daniels, R. Lugannani, and S. Rice to the multidimensional case. We prefer to use the Lugannani-Rice method because it yields approximations which are as good as those of other forms and do not depend on variation of statistics. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 195–202, Perm, 1993.     

Asymptotic ergodicity of the eigenvalues of random operators in the localized phase

We prove that, for a general class of random operators, the family of the unfolded eigenvalues in the localization region is asymptotically ergodic in the sense of Minami (Spectra of random operators and related topics, 2011). Minami conjectured this to be the case for discrete Anderson model in the localized regime. We also provide a local analogue of this result. From the asymptotics ergodicity, one can recover the statistics of the level spacings as well as a number of other spectral statistics. Our proofs rely on the analysis developed in Germinet and Klopp (Spectral statistics for random Schrödinger operators in the localized regime, 2010).     

Limit distributions and comparative asymptotic efficiency of the Smirnov — Kolmogorov statistics with a random index

Limit distributions for certain statistics of Smirnov — Kolmogorov type are obtained which consider the weak convergence of the corresponding empirical process. Approximate and precise asymptotic efficiencies of these statistics are computed. It is shown that they are worse in a certain sense than the classical Kolmogorov — Smirnov statistics.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 185–194, 1976.     

20 Years of Statistics at the National Center for Atmospheric Research

Almost every U.S.-based statistician working on problems motivated by atmospheric science is connected to the statistics program at the National Center for Atmospheric Research (NCAR). Through its permanent staff scientists, postdoctoral researchers, visitors, seminars, workshops, and published work, NCAR has made a profound impact on the community of statisticians working in the atmospheric and climate sciences. This past year saw a reorganization of statistics at NCAR. This article looks back at more than 20 years of statistics there.     

On the skewness of order statistics with applications

Order statistics from heterogeneous samples have been extensively studied in the literature. However, most of the work focused on the effect of heterogeneity on the magnitude or dispersion of order statistics. In this paper, we study the skewness of order statistics from heterogeneous samples according to star ordering. The main results extend the corresponding results in Kochar and Xu (J. Appl. Probab. 46:342–352, 2009; J. Appl. Probab. 48:271–284, 2011). Examples and applications are highlighted as well.     

On the introduction of the temperature standard in the undistinguishing parastatistics of objectively distinguishable objects

We consider various normalizations enabling us to change the scale of the graphs of isotherms and isochores. The relationship between parastatisticsal value of the maximal number of particles, corresponding to a given energy, and temperature allows us to pass in the domain of positive chemical potentials from the parastatisticsal number K (K = ∞ corresponds to Bose statistics and K = 0 to Fermi statistics) to the temperature, which changes the scaling of the pressure in this domain.     

Toward Understanding the Role of Technological Tools in Statistical Learning

This article begins with some context setting on new views of statistics and statistical education. These views are reflected, in particular, in the introduction of exploratory data analysis (EDA) into the statistics curriculum. Then, a detailed example of EDA learning activity in the middle school is introduced, which makes use of the power of the spreadsheet to mediate students' construction of meanings for statistical conceptions. Through this example, I endeavor to illustrate how an attempt at serious integration of computers in teaching and learning statistics brings about a cascade of changes in curriculum materials, classroom praxis, and students' ways of learning. A theoretical discussion follows that underpins the impact of technological tools on teaching and learning statistics by emphasizing how the computer lends itself to supporting cognitive and sociocultural processes. Subsequently, I present a sample of educational technologies, which represents the sorts of software that have typically been used in statistics instruction: statistical packages (tools), microworlds, tutorials, resources (including Internet resources), and teachers' metatools. Finally, certain implications and recommendations for the use of computers in the statistical educational milieu are suggested.     

An Algebraic Method for the Exact Solution of the Partition Function of the Canonical Ensemble in Nuclear Multifragmentation

We propose a unified method for deducing recursive relations for the canonical partition function of a system of noninteracting particles with charge conservation if the particles follow the Bose–Einstein, Fermi–Dirac, or Maxwell–Boltzmann statistics or parastatistics. For all these types of statistics, we find recursive relations for the partition function of a new statistical model of nuclear multifragmentation with electric charge and baryon number conservation, accounting for the internal degrees of freedom of the nuclear fragments.     

On the anomalous behaviour of a class of locality statistics

A scan statistic methodology for detecting anomalies has been developed for application to graphs, where “anomalies” are equated with vertices that exhibit distinctive local connectivity properties. We present an “anomaly graph” construction that illustrates the capabilities of these scan statistics via the behaviour of their associated locality statistics on our anomaly graphs.