A test for elliptical symmetry

This paper presents a statistic for testing the hypothesis of elliptical symmetry. The statistic also provides a specialized test of multivariate normality. We obtain the asymptotic distribution of this statistic under the null hypothesis of multivariate normality, and give a bootstrapping procedure for approximating the null distribution of the statistic under an arbitrary elliptically symmetric distribution. We present simulation results to examine the accuracy of the asymptotic distribution and the performance of the bootstrapping procedure. Finally, for selected alternatives, we compare the power of our test statistic with that of recently proposed tests for elliptical symmetry given by Manzotti et al. [A statistic for testing the null hypothesis of elliptical symmetry, J. Multivariate Anal. 81 (2002) 274-285] and Schott [Testing for elliptical symmetry in covariance-matrix-based analyses, Statist. Probab. Lett. 60 (2002) 395-404], and with that of the well known tests for multivariate normality of Mardia [Measures of multivariate skewness and kurtosis with applications, Biometrika 57 (1970) 519-530] and Baringhaus and Henze [A consistent test for multivariate normality based on the empirical characteristic function, Metrika 35 (1988) 339-348].     

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-yon 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.     

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.     

随机PP Kolmogorov多维拟合优度检验乃其Bootstrap逼近

本文提出了一种基于随机选择投影方向的PP型棉球等高分布族的拟合优度检验，其特点是计算上较通常的PP检验统计量简单．得到了其检验统计量在零假设下的极限分布，讨论了其Bootstrap逼近及逼近的相容性．     

On consistent testing for serial correlation of unknown form in vector time series models

Multivariate autoregressive models with exogenous variables (VARX) are often used in econometric applications. Many properties of the basic statistics for this class of models rely on the assumption of independent errors. Using results of Hong (Econometrica 64 (1996) 837), we propose a new test statistic for checking the hypothesis of non-correlation or independence in the Gaussian case. The test statistic is obtained by comparing the spectral density of the errors under the null hypothesis of independence with a kernel-based spectral density estimator. The asymptotic distribution of the statistic is derived under the null hypothesis. This test generalizes the portmanteau test of Hosking (J. Amer. Statist. Assoc. 75 (1980) 602). The consistency of the test is established for a general class of static regression models with autocorrelated errors. Its asymptotic slope is derived and the asymptotic relative efficiency within the class of possible kernels is also investigated. Finally, the level and power of the resulting tests are also studied by simulation.     

HAC estimation and strong linearity testing in weak ARMA models

In the framework of ARMA models, we consider testing the reliability of the standard asymptotic covariance matrix (ACM) of the least-squares estimator. The standard formula for this ACM is derived under the assumption that the errors are independent and identically distributed, and is in general invalid when the errors are only uncorrelated. The test statistic is based on the difference between a conventional estimator of the ACM of the least-squares estimator of the ARMA coefficients and its robust HAC-type version. The asymptotic distribution of the HAC estimator is established under the null hypothesis of independence, and under a large class of alternatives. The asymptotic distribution of the proposed statistic is shown to be a standard χ² under the null, and a noncentral χ² under the alternatives. The choice of the HAC estimator is discussed through asymptotic power comparisons. The finite sample properties of the test are analyzed via Monte Carlo simulation.     

完全区组设计下基于Aligned秩的有方向检验问题

本文基于Aligned秩给出了用于解完全区组设计有方向检验问题的, 我们称之为$C$-检验的检验方法. 本文分别对每个试验单元仅有一个观测值以及等重复观测值和不等重复观测值各种情形下的$C$检验进行了讨论, 并在原假设$H_0$成立时计算了上述各种情形下$C$检验统计量的数学期望和方差, 且证明了$C$检验统计量的渐近分布为正态分布.     

A Statistic for Testing the Null Hypothesis of Elliptical Symmetry

We present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika65, 657–658) of the d-dimensional data on the unit sphere of ^d. We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives.     

On Mardia's and Song's measures of kurtosis in elliptical distributions

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure β_2,p and Song's measure S(f). In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t-distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Rényi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.     

序约束下多元正态均值的检验问题

设有k组均值有简单半序约束,协方差阵未知的p维正态分布．Sasabuchi等在2003年研究了均值是否相等的检验问题．考虑到似然比检验统计量的临界点难以获得,以致于它不容易实施,Sasabuchi提出了一个检验方法．称为Sasabuchi检验．Sasabuchi检验的一个不足之处在于,它并不优于经典的MANOVA检验．作者提出了一个新的检验方法,它比Sasabuchi检验有一致优的势,而且形式更为简单．通过模拟发现这个检验方法还优势于MANOVA．最后导出了这个检验统计量的渐近零分布．     

A Partial Empirical Likelihood Based Score Test Under a Semiparametric Finite Mixture Model

We propose a score statistic to test the null hypothesis that the two-component density functions are equal under a semiparametric finite mixture model. The proposed score test is based on a partial empirical likelihood function under an I-sample semiparametric model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to a partial empirical likelihood ratio test and a Wald test. We present some results on a simulation study.     

Some high-dimensional tests for a one-way MANOVA

A statistic is proposed for testing the equality of the mean vectors in a one-way multivariate analysis of variance. The asymptotic null distribution of this statistic, as both the sample size and the number of variables go to infinity, is shown to be normal. Thus, this test can be used when the number of variables is not small relative to the sample size. In particular, it can be used when the number of variables exceeds the degrees of freedom for error, a situation in which standard MANOVA tests are invalid. A related statistic, also having an asymptotic normal distribution, is developed for tests concerning the dimensionality of the hyperplane formed by the population mean vectors. The finite sample size performances of the normal approximations are evaluated in a simulation study.     

Permutation Tests for Multivariate Location Problems

The paper presents some permutation test procedures for multivariate location. The tests are based on projected univariate versions of multivariate data. For one-sample cases, the tests are affine invariant and strictly distribution-free for the symmetric null distribution with elliptical direction and their permutation counterparts are conditionally distribution-free when the underlying null distribution of the sample is angularly symmetric. For multi-sample cases, the tests are also affine invariant and permutation counterparts of the tests are conditionally distribution-free for any null distribution with certain continuity. Hence all of the tests in this paper are exactly valid. Furthermore, the equivalence, in the large sample sense, between the tests and their permutation counterparts are established. The power behavior of the tests and of their permutation counterparts under local alternative are investigated. A simulation study shows the tests to perform well compared with some existing tests in the literature, particularly when the underlying null distribution is symmetric whether light-tailed or heavy-tailed. For revealing the influence of data sparseness on the effect of the test, some simulations with different dimensions are also performed.     

线性度量误差模型的序列相关检验

回归模型的序列相关检验是经济和金融数据分析中的一项重要的工作.基于最小二乘残差,作者提出了一个检验统计量以检验线性度量误差模型的误差序列是否存在序列相关性.在零假设下,得到了检验统计量的渐近分布.数值模拟结果表明,这里提出的检验统计量具有良好的有限样本性质.     

Approximation by randomly weighting method in censored regression model

Censored regression (“Tobit”) models have been in common use, and their linear hypothesis testings have been widely studied. However, the critical values of these tests are usually related to quantities of an unknown error distribution and estimators of nuisance parameters. In this paper, we propose a randomly weighting test statistic and take its conditional distribution as an approximation to null distribution of the test statistic. It is shown that, under both the null and local alternative hypotheses, conditionally asymptotic distribution of the randomly weighting test statistic is the same as the null distribution of the test statistic. Therefore, the critical values of the test statistic can be obtained by randomly weighting method without estimating the nuisance parameters. At the same time, we also achieve the weak consistency and asymptotic normality of the randomly weighting least absolute deviation estimate in censored regression model. Simulation studies illustrate that the performance of our proposed resampling test method is better than that of central chi-square distribution under the null hypothesis. This work was supported by National Natural Science Foundation of China (Grant No. 10471136), PhD Program Foundation of the Ministry of Education of China, and Special Foundations of the Chinese Academy of Sciences and University of Science and Technology of China     

Partial penalized empirical likelihood ratio test under sparse case

A consistent test via the partial penalized empirical likelihood approach for the parametric hypothesis testing under the sparse case, called the partial penalized empirical likelihood ratio (PPELR) test, is proposed in this paper. Our results are demonstrated for the mean vector in multivariate analysis and regression coefficients in linear models, respectively. And we establish its asymptotic distributions under the null hypothesis and the local alternatives of order n^?1/2 under regularity conditions. Meanwhile, the oracle property of the partial penalized empirical likelihood estimator also holds. The proposed PPELR test statistic performs as well as the ordinary empirical likelihood ratio test statistic and outperforms the full penalized empirical likelihood ratio test statistic in term of size and power when the null parameter is zero. Moreover, the proposed method obtains the variable selection as well as the p-values of testing. Numerical simulations and an analysis of Prostate Cancer data confirm our theoretical findings and demonstrate the promising performance of the proposed method in hypothesis testing and variable selection.     

Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ² when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.     

Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions

In this paper, the authors derived asymptotic expressions for the null distributions of the likelihood ratio test statistics for multiple independence and multiple homogeneity of the covariance matrices when the underlying distributions are complex multivariate normal. Also, asymptotic expressions are obtained in the non-null cases for the likelihood ratio test statistics for independence of two sets of variables and the equality of two covariance matrices. The expressions obtained in this paper are in terms of beta series. In the null cases, the accuracy of the first terms alone is sufficient for many practical purposes.     

Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator

The minimum covariance determinant (MCD) scatter estimator is a highly robust estimator for the dispersion matrix of a multivariate, elliptically symmetric distribution. It is relatively fast to compute and intuitively appealing. In this note we derive its influence function and compute the asymptotic variances of its elements. A comparison with the one step reweighted MCD and with S-estimators is made. Also finite-sample results are reported.     

删截分布未知的多维随机删截数据的PP拟合优度检验

文[5]在假定删截分布已知的条件下,用投影寻踪（PP）技巧讨论了多维随机删截数据的PP拟合优度检验问题．本文讨论截尾分布未知时,多维随机删截数据的拟合优度检验问题,得到了检验统计量在零假设成立时的渐近分布,并讨论了其Bootstrap逼近．