ASYMPTOTIC DISTRIBUTION OF LIKELIHOOD RATIO STATISTIC FOR TESTING SPHERICTY IN GROWTH CURVE MODELS

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TESTING SPHERICITY IN A GMANOVA-MANOVA MODEL WITH NORMAL ERROR

This article presents a statistic for testing the sphericity in a GMANOVA- MANOVA model with normal error. It is shown that the null distribution of this statistic is beta and its nonnull distribution is given in series form of beta distributions.     

Tests for sphericity under correlated multivariate regression equations model

Summary In this paper, the authors investigated the asymptotic distribution theory connected with the likelihood ratio test (LRT)-like test statistic for sphericity under correlated multivariate regression equations (CMRE) model. An asymptotic expression is obtained for the null distribution of the above test statistic. Asymptotic nonnull distribution of the above test statistic under fixed alternatives is also derived. The above results are derived when the underlying distribution is multivariate normal. It was also shown that the above results are valid even when the joint distribution of the observations is elliptically symmetric. The authors also derived the asymptotic null distribution of the LRT-like test statistic when the observations on each variable are elliptically symmetric. This work was supported by the Air Force Office of Scientific Research under Contract F49620-82-K-0001. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Tests for independence of two multivariate regression equations with different design matrices

In this paper, the authors considered various procedures for testing for the independence of two multivariate regression equations with different design matrices. Asymptotic null distributions as well as nonnull distributions under local alternatives of the test statistics associated with the above procedures are also derived.     

The likelihood ratio tests for the dimensionality of regression coefficients

In this paper we give a unified derivation of the likelihood ratio (LR) statistics for testing the hypothesis on the dimensionality of regression coefficients under a usual MANOVA model. We also derive the LR statistics under a general MANOVA model and study their asymptotic null and nonnull distributions. Further it is shown that the test statistic used by Bartlett [4] for testing the hypothesis that the last p?k canonical correlations are all zero is the LR statistic.     

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.     

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.     

生长曲线模型中的球性检验问题

本文给出了生长曲线模型中球性检验似然比准则在原假设相接近的两类备择假设下的非零渐近分布。     

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.     

Sphericity Test for High Dimensional Data Based on Random Matrix Theory

In this article we study test of sphericity for high-dimensional covariance matrix in the general population based on random matrix theory. When the sample size is less than data dimension, the classical likelihood ratio test has poor performance for test of sphericity. Thus, we propose a new statistic for test of sphericity by using the higher moments of spectral distribution of the sample covariance matrix, and derive the asymptotic distribution of the statistic under the null hypothesis. Simulation results show that the proposed statistics can effectively improve the power of the test of sphericity for high dimensional data, and have especially significant effects for Spiked model, on the basis of controlling the type-one error probability.     

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??In this paper, we focus on the tests for covariance matrices in panel data model with interactive fixed effects. For the problem of testing identity and sphericity of covariance matrices, we first propose test statistics based on the estimators of the trace of covariance matrices. Under both the null hypothesis and the alternatives, we establish the asymptotic distributions of the proposed test statistics under some regularity conditions, and we further show that the proposed tests are distribution free. Subsequently simulation studies suggest that the proposed tests perform well under the high dimensional panel data.     

Testing for Covariance Matrices in Panel Data Models with Interactive Fixed Effect

In this paper, we focus on the tests for covariance matrices in panel data model with interactive fixed effects. For the problem of testing identity and sphericity of covariance matrices, we first propose test statistics based on the estimators of the trace of covariance matrices. Under both the null hypothesis and the alternatives, we establish the asymptotic distributions of the proposed test statistics under some regularity conditions, and we further show that the proposed tests are distribution free. Subsequently simulation studies suggest that the proposed tests perform well under the high dimensional panel data.     

Some asymptotic properties of a progressively censored nonparametric test for multiple regression

The large sample null distribution of a progressively censored nonparametric test for multiple regression proposed by Majumdar and Sen is computed. Also the asymptotic nonnull distribution for the test based on Savage scores is computed for local alternatives when the underlying distribution is exponential. The power of this test is compared with the power of the corresponding fixed sample tests. The stopping properties are also investigated. A short table of critical values is included.     

A locally most powerful invariant test for the equality of means associated with covariate discriminant analysis

In this paper, the authors propose a locally most powerful invariant test for the equality of means in the presence of covariate variables. Also the null and nonnull distributions associated with the above test are developed. This problem arises in covariate discriminant analysis and has been treated by various authors, notably Cochran and Bliss 1948, Ann. Math. Statist.19, 151–176 and Rao 1949, Sankhyã9 343–366; 1966. The test derived here locally dominates in power the tests proposed so far. It is also shown that the Cochran-Bliss test is uniformly most powerful in the class of conditional invariant tests.     

The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption

This paper examines asymptotic distributions of the likelihood ratio criteria, which are proposed under normality, for several hypotheses on covariance matrices when the true distribution of a population is a certain nonnormal distribution. It is well known that asymptotic distributions of test statistics depend on the fourth moments of the true population's distribution. We study the effects of nonnormality on the asymptotic distributions of the null and nonnull distributions of likelihood ratio criteria for covariance structures.     

Tests for standardized generalized variances of multivariate normal populations of possibly different dimensions

In many practical problems, one needs to compare variabilities of several multidimensional populations. The concept of standardized generalized variance (SGV) is introduced as an extension of the concept of GV. Considering multivariate normal populations of possibly different dimensions and general covariance matrices, LRTs are derived for SGVs. The criteria turn out to be elegant multivariate analogs to those for tests for variances in the univariate cases. The null and nonnull distributions of the test criteria are deducdd in computable forms in terms of Special Functions, e.g., Pincherle'sH-function, by exploiting the theory of calculus of residues (Mathai and Saxena,Ann. Math. Statist.40, 1439–1448).     

The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

Using a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its hth null moment, and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a generalized integer gamma distribution with the sum of a given number of independent logbeta distributions, taking the form of a single generalized integer gamma distribution when each set of variables has two variables. The development of near‐exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent‐induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near‐exact distributions proposed and to compare them with an asymptotic approximation on the basis of Box's method. Copyright © 2011 John Wiley & Sons, Ltd.     

Locally Conformally Flat Lorentzian Gradient Ricci Solitons

It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson–Walker warped product, if the gradient of the potential function is nonnull, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.     

Monotonicity of the power functions of modified likelihood ratio criterion for the homogeneity of variances and of the sphericity test

The modified likelihood ratio criterion for testing the homogeneity of variances of p univariate normal populations, and the sphericity test, are both shown in this paper to have a monotone nondecreasing power function.     

Asymptotic nonnull distributions for tests for reality of a covariance matrix

In this paper asymptotic nonnull distributions are derived for two statistics used in testing for the reality of the covariance matrix in a complex Gaussian distribution.