Wilcoxon‐type rank‐sum statistics for selecting the best population: Some advances

In this article, we introduce three new nonparametric procedures for testing the equality of two lifetime distributions. The proposed testing processes are based on appropriately modified Wilcoxon‐type rank‐sum statistics. The exact null distribution of these statistics is studied and closed formulae for the corresponding exact probability of correct selection of the best population are derived for the class of Lehmann alternatives. A detailed numerical study is carried out to elucidate the performance of the proposed testing schemes. For illustration purposes, a real data application is presented in some detail.     

On some testing problems for sparse contingency tables

A class of test statistics are proposed for sparse tables with ordered categories. It is shown that for different testing situations these test statistics asymptotically more powerful tests than Pearson's chi-square.     

基于投影的多元正态性相关与回归检验

考虑多元分布的正态性检验问题,用投影寻踪方法,给出了两个检验统计量,模拟计算也表明所提出的检验统计量具有良好的功效.     

带有误差变量的投影偏度和峰度的多元正态性检验--用投影寻踪自助法

考虑带有误差变量的多元正态检验问题,给出了新的投影偏度和峰度检验统计量,证明了在零假设成立时,所给的检验统计量的极限分布为一高斯过程的上界,为计算机模拟计算提供了有力的手段和依据.     

Compact Matrix Expressions for Generalized Wald Tests of Equality of Moment Vectors

Asymptotic chi-squared test statistics for testing the equality of moment vectors are developed. The test statistics proposed are generalized Wald test statistics that specialize for different settings by inserting an appropriate asymptotic variance matrix of sample moments. Scaled test statistics are also considered for dealing with nonstandard conditions. The specialization will be carried out for testing the equality of multinomial populations, and the equality of variance and correlation matrices for both normal and nonnormal data. When testing the equality of correlation matrices, a scaled version of the normal theory chi-squared statistic is proven to be an asymptotically exact chi-squared statistic in the case of elliptical data.     

带有误差变量的偏度和峰度正态性检验

考虑带有误差变量偏度和峰度正态性检验问题 ,提出了新的偏度和峰度正态性检验统计量 ,证明了在零假设时 ,所提出的偏度和峰度检验统计量具有渐近正态的优良性质     

Asymptotic expansions of the distributions of some test statistics

Summary A modified Wald statistic for testing simple hypothesis against fixed as well as local alternatives is proposed. The asymptotic expansions of the distributions of the proposed statistic as well as the Wald and Rao statistics under both the null and alternative hypotheses are obtained. The powers of these statistics are compared and its is shown that for special structures of parameters some statistics have same power in the sence of order . The results obtained are applied for testing the hypothesis about the covariance matrix of the multivariate normal distribution and it is shown that none of the tests based on the above statistics is uniformly superior. Research supported by the National Science Foundation Grant MCS 830149.     

A generalized Hollander-Proschan type test for NBUE alternatives

In this note we develop a family of test statistics for testing exponentiality against NBUE alternatives. The asymptotic distribution of the test statistics is derived. The test statistics are shown to be asymptotically normal and consistent. This family of test statistics includes the test proposed by Hollander and Proschan (1975) as a special case. Efficiency studies have also been done.     

Testing conditional independence with data missing at random

It is known that conditional independence is a quite basic assumption in many fields of statistics. How to test its validity is of great importance and has been extensively studied by the literature. Nevertheless, all of the existing methods focus on the case that data are fully observed, but none of them seems having taken into account of the scenario when missing data are present. Motivated by this, this paper develops two testing statistics to handle such a situation relying on the idea of inverse probability weighted and augmented inverse probability weighted techniques. The asymptotic distributions of the proposed statistics are also derived under the null hypothesis. The simulation studies indicate that both testing statistics perform well in terms of size and power.     

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??Kolmogorov-Smirnov (KS), Cramer-von Mises (CM) and Anderson-Darling (AD) test, which are based on empirical distribution function (EDF), are well-known statistics in testing univariate normality. In this paper, we focus on the high dimensional case and propose a family of generalized EDF based statistics to test the high-dimensional normal distribution by reducing the dimension of the variable. Not only can we approximate the corresponding critical values of three statistics by Monte Carlo method, we also can investigate the approximate distributions of proposed statistics based on approximate formulas in univariate case under null hypothesis. The Monte Carlo simulation is carried out to demonstrate that the performance of proposed statistics is more competitive than existing methods under some alternative hypotheses. Finally, the proposed tests are applied to real data to illustrate their utility.     

The empirical characteristic function and large sample hypothesis testing

The empirical characteristic function is considered as a tool for large sample testing of a hypothesis that can be characterized in terms of the characteristic function. Two test statistics based upon the empirical characteristic function are proposed. The limiting distributions of these test statistics are obtained and methods are suggested for using these limiting distributions to calculate critical regions.     

Goodness-of-fit tests for multivariate Laplace distributions

Consistent goodness-of-fit tests are proposed for symmetric and asymmetric multivariate Laplace distributions of arbitrary dimension. The test statistics are formulated following the Fourier-type approach of measuring the weighted discrepancy between the empirical and the theoretical characteristic function, and result in computationally convenient representations. For testing the symmetric Laplace distribution, and in the particular case of a Gaussian weight function, a limit value of these test statistics is obtained when this weight function approaches a Dirac delta function. Interestingly, this limit value is related to a couple of well-known measures of multivariate skewness. A Monte Carlo study is conducted in order to compare the new procedures with standard tests based on the empirical distribution function. A real data application is also included.     

椭球检验与统计量的渐近尾部性

In this paper,some test statistics of Kolmogorov type and Cramervon Mises type based on projection pursuit technique are proposed for testing the sphericity problem of a high-dimensional distribution. The limiting distributions of the test statistics are derived under the null hypothesis. The asymptotic properties of Bootstrap approximation are investigated and the tail behaviors of the statistics are studied.     

Nonparametric tests for homogeneity of scale against ordered alternatives

Summary In this paper the nonparametric several sample scale problem is considered and some tests are proposed for the hypothesis of homogeneity versus ordered alternatives. These tests are based on statistics that are weighted linear combinations of Sugiura (1965,Osaka J. Math.,2, 385–426) type statistics proposed for testing homogeneity of scale against the omnibus alternative. For each class of test statistics suggested, the member with maximum Pitman efficiency is identified. The optimal statistics are compared with their parametric and nonparametric competitors.     

Tests of elliptical symmetry and the asymptotic tail behavior of the statistics

In this paper, some test statistics Of Kolmogorov type and Cramervon Mises type based on projection pursuit technique are proposed for testing the sphericity problem of a high-dimensional distribution. The limiting distributions of the test statistics are derived under the null hypothesis. The asymptotic properties of Bootstrap approximation are investigated and the tail behaviors of the statistics are studied.     

ON SOME TESTS FOR ELLIPTICAL SYMMETRY

In this paper, some test statistics of Kolmogorov type and Cramer-von Mises type based on projection pursuit technique are proposed for testing the sphericity problem of a high-dimensional distribution. The limiting distributions of the test statistics are derived under the null hypothesis and any fixed alternative. The asymptotic properties of Bootstrap approximation are investigated. Furthermore, for computational reasons, an approximation for the statistics, based on number theoretic method, is suggested.     

Dimensionality in Manova Tested by a Closed Testing Procedure

The decision on dimensionality of the space spanned by general linear functions of the parameter matrix of a MANOVA model is considered. This problem is related to the investigation, whether graphically or analytically, of significant empirical departures from the overall null hypothesis on these functions. A closed testing procedure for a sequence of relevant hypotheses is proposed. Unlike the classical procedures based on asymptotic distributions of the likelihood ratio statistics, the proposed method ensures that the Type I familywise error rate does not exceed the nominalα-level. Also, it is consistent with testing the overall null hypothesis, while relying on tests of subsequent linear hypotheses implied by the former. Examples are given to compare the proposed procedure with a classical one.     

Some test statistics for the structural coefficients of the multivariate linear functional relationship model

Summary For the testing problem concerning the coefficients of the multivariate linear functional relationship model, the distribution of a statistic previously proposed by A. P. Basu depends on the unknown covariance matrixV of errors, so limiting its applicability. This article proposes new test statistics with sampling distributions which are independent of the unknown parameters for the cases whereV is either unknown or known only up to a proportionality factor. The exact distributions of the test statistics are also discussed.     

Nonuniform bounds on the convergence rate in the invariance principle

A general method is proposed for obtaining both uniform and nonuniform bounds on the convergence rate in the invariance principle. Nonuniform bounds on the convergence rate are derived for Cramer-von Mises-Smirnov type statistics used in testing hypotheses associated with circular distributions.Translated from Statisticheskie Metody, pp. 135–156, 1980.