Scheffés mixed model for multivariate repeated measures:a relative efficiency evaluation

Scheffé’s mixed model, generalized for application to multivariate repeated measures, is known as the multivariate mixed model (MMM). The primary advantages the MMM are (1) the minimum sample size required to conduct an analysis is smaller than for competing procedures and (2) for certain covariance structures, the MMM analysis is more powerful than its competitors. The primary disadvantage is that the MMM makes a very restrictive covariance assumption; namely multivariate sphericity. This paper shows, first, that even minor departures from multivariate sphericity inflate the size of MMM based tests. Accordingly, MMM analyses, as computed in release 4.0 of SPSS MANOVA (SPSS Inc., 1990), can not be recommended unless it is known that multivariate sphericity is satisfied. Second, it is shown that a new Box-type (Box, 1954) Δ-corrected MMM test adequately controls test size unless departure from multivariate sphericity is severe or the covariance matrix departs substantially from a multiplicative-Kronecker structure. Third, power functions of adjusted MMM tests for selected covariance and noncentrality structures are compared to those of doubly multivariate methods that do not require multivariate sphericity. Based on relative efficiency evaluations, the adjusted MMM analyses described in this paper can be recommended only when sample sizes are very small or there is reason to believe that multivariate sphericity is nearly satisfied. Neither the e-adjusted analysis suggested in the SPSS MANOVA output (release 4.0) nor the adjusted analysis suggested by Boik (1988) can be recommended at all.     

Asymptotic expansion of the nonnull distribution of likelihood ratio statistic for testing multisample sphericity

Asymptotic expansion of the nonnull distribution of the likelihood ratio statistic for testing muitisample sphericity in q multinormal populations is derived for the alternatives close to the null hypothesis.     

Further asymptotic expansions of the distribution of the sphericity test

In this paper new asymptotic expansions of the distributions of the sphericity test criterion are obtained in the null and the non-null case when the alternatives are close to the hypothesis. These expansions are obtained for the first time in terms of beta distributions. These appear to be better than the ones available in the literature.     

Comparison of powers for the sphericity tests using both the asymptotic distribution and the bootstrap

The asymptotic distributions of two tests for sphericity:the locally most powerful invariant test and the likelihood ratio test are derived under the general alternaties ∑?σ² I. The powers of these two tests are then compared when the data are from a trivariate normal population. The bootstrap method is also used to obtain the powers and the powers obtained by this method agree with those from the asymptotic distributions.     

Tests for high-dimensional covariance matrices using the theory of U-statistics

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p?n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.     

Testing homogeneity of several covariance matrices and multi-sample sphericity for high-dimensional data under non-normality

A test for homogeneity of g ? 2 covariance matrices is presented when the dimension, p, may exceed the sample size, n_i, i = 1, …, g, and the populations may not be normal. Under some mild assumptions on covariance matrices, the asymptotic distribution of the test is shown to be normal when n_i, p → ∞. Under the null hypothesis, the test is extended for common covariance matrix to be of a specified structure, including sphericity. Theory of U-statistics is employed in constructing the tests and deriving their limits. Simulations are used to show the accuracy of tests.     

Testing sphericity using small samples

It is shown here that with small sample sizes, the null distribution of the ellipticity, or sphericity, likelihood criterion W(n, p) can be obtained very accurately, either by computation using the Meijer function, or by Monte Carlo simulation. Testing in repeated measures design can now be carried out with much more accuracy.     

Testing for sphericity in a two-way error components panel data model

This article is concerned with sphericity test for the two-way error components panel data model. It is found that the John statistic and the bias-corrected LM statistic recently developed by Baltagi et al. (2011 Baltagi, B. H., Feng, Q., Kao, C. (2011). Testing for sphericity in a fixed effects panel data model. Econometrics Journal 14:25–47.[Crossref], [Web of Science ®] , [Google Scholar])Baltagi et al. (2012 Baltagi, B. H., Feng, Q., Kao, C. (2012). A Lagrange multiplier test for cross-sectional dependence in a fixed effects panel data model. Journal of Econometrics 170:164–177.[Crossref], [Web of Science ®] , [Google Scholar], which are based on the within residuals, are not helpful under the present circumstances even though they are in the one-way fixed effects model. However, we prove that when the within residuals are properly transformed, the resulting residuals can serve to construct useful statistics that are similar to those of Baltagi et al. (2011 Baltagi, B. H., Feng, Q., Kao, C. (2011). Testing for sphericity in a fixed effects panel data model. Econometrics Journal 14:25–47.[Crossref], [Web of Science ®] , [Google Scholar])Baltagi et al. (2012 Baltagi, B. H., Feng, Q., Kao, C. (2012). A Lagrange multiplier test for cross-sectional dependence in a fixed effects panel data model. Journal of Econometrics 170:164–177.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results show that the newly proposed statistics perform well under the null hypothesis and several typical alternatives.     

Distribution of the Λ-Criterion for Sphericity Test and Its Connection to a Generalized Dirichlet Model

It is shown that the exact null distribution of the likelihood ratio criterion for sphericity test in the p-variate normal case and the marginal distribution of the first component of a (p ? 1)-variate generalized Dirichlet model with a given set of parameters are identical. The exact distribution of the likelihood ratio criterion so obtained has a general format for every p. A novel idea is introduced here through which the complicated exact null distribution of the sphericity test criterion in multivariate statistical analysis is converted into an easily tractable marginal density in a generalized Dirichlet model. It provides a direct and easiest method of computation of p-values. The computation of p-values and a table of critical points corresponding to p = 3 and 4 are also presented.     

On projection-based tests for directional and compositional data

A new class of nonparametric tests, based on random projections, is proposed. They can be used for several null hypotheses of practical interest, including uniformity for spherical (directional) and compositional data, sphericity of the underlying distribution and homogeneity in two-sample problems on the sphere or the simplex. The proposed procedures have a number of advantages, mostly associated with their flexibility (for example, they also work to test “partial uniformity” in a subset of the sphere), computational simplicity and ease of application even in high-dimensional cases.     

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

For two or more multivariate distributions with common covariance matrix, test statistics for certain special structures of the common covariance matrix are presented when the dimension of the multivariate vectors may exceed the number of such vectors. The test statistics are constructed as functions of location‐invariant estimators defined as U‐statistics, and the corresponding asymptotic theory is used to derive the limiting distributions of the proposed tests. The properties of the test statistics are established under mild and practical assumptions, and the same are numerically demonstrated using simulation results with small or moderate sample sizes and large dimensions.     

Comparison of combinatoric and likelihood ratio procedures for classifying samples

A random sample is to be classified as coming from one of two normally distributed populations with known parameters. Combinatoric procedures which classify the sample based upon the sample mean(s) and variance(s) are described for the univariate and multivariate problems. Comparisons of misclassification probabilities are made between the combinatoric and the likelihood ratio procedure in the univariate case and between two alternative combinatoric procedures in the bivariate case.     

Semi-Supervised Logistic Discrimination via Regularized Gaussian Basis Expansions

The problem of constructing classification methods based on both labeled and unlabeled data sets is considered for analyzing data with complex structures. We introduce a semi-supervised logistic discriminant model with Gaussian basis expansions. Unknown parameters included in the logistic model are estimated by regularization method along with the technique of EM algorithm. For selection of adjusted parameters, we derive a model selection criterion from Bayesian viewpoints. Numerical studies are conducted to investigate the effectiveness of our proposed modeling procedures.     

Parameter estimation of the generalized Pareto distribution—Part II

This is the second part of a paper which focuses on reviewing methods for estimating the parameters of the generalized Pareto distribution (GPD). The GPD is a very important distribution in the extreme value context. It is commonly used for modeling the observations that exceed very high thresholds. The ultimate success of the GPD in applications evidently depends on the parameter estimation process. Quite a few methods exist in the literature for estimating the GPD parameters. Estimation procedures, such as the maximum likelihood (ML), the method of moments (MOM) and the probability weighted moments (PWM) method were described in Part I of the paper. We shall continue to review methods for estimating the GPD parameters, in particular methods that are robust and procedures that use the Bayesian methodology. As in Part I, we shall focus on those that are relatively simple and straightforward to be applied to real world data.     

Repeated confidence intervals and prediction intervals using stochastic curtailment

Jennlson and Turnbull (1984,1989) proposed procedures for repeated confidence intervals for parameters of interest In a clinical trial monitored with group sequential methods. These methods are extended for use with stochastic curtailment procedures for two samples in the estimation of differences of means, differences of proportions, odds ratios, and hazard ratios. Methods are described for constructing 1) confidence intervals for these estimates at repeated times In the course of a trial, and 2) prediction intervals for predicted estimates at the end of a trial. Specific examples from several clinical trials are presented.     

An EM algorithm for multivariate Poisson distribution and related models

Multivariate extensions of the Poisson distribution are plausible models for multivariate discrete data. The lack of estimation and inferential procedures reduces the applicability of such models. In this paper, an EM algorithm for Maximum Likelihood estimation of the parameters of the Multivariate Poisson distribution is described. The algorithm is based on the multivariate reduction technique that generates the Multivariate Poisson distribution. Illustrative examples are also provided. Extension to other models, generated via multivariate reduction, is discussed.     

Modeling statistical dependence of Markov chains via copula models

Conditional probability distributions have been commonly used in modeling Markov chains. In this paper we consider an alternative approach based on copulas to investigate Markov-type dependence structures. Based on the realization of a single Markov chain, we estimate the parameters using one- and two-stage estimation procedures. We derive asymptotic properties of the marginal and copula parameter estimators and compare performance of the estimation procedures based on Monte Carlo simulations. At low and moderate dependence structures the two-stage estimation has comparable performance as the maximum likelihood estimation. In addition we propose a parametric pseudo-likelihood ratio test for copula model selection under the two-stage procedure. We apply the proposed methods to an environmental data set.     

ON THE DISTRIBUTION OF SPHERICITY TEST CRITERION IN THE MULTIVARIATE GAUSSIAN DISTRIBUTION

In this paper, the exact distribution of the likelihood-ratio criterion for testing the hypothesis of sphericity in a multivariate normal distribution is derived as a mixture of incomplete beta functions. Some advantages of the present derivation are also indicated.     

Forms for the distribution of ss ilkptxcxtt statistic for bivariate sphericity

This paper examines the nonnuU distribution of the ellipticity statistic used for testing sphericity in a bivariate normal population. The distribution function is expressed as mixtures of F distribution functions, where the mixing probabilities are binomial and mgatiire binomial. A simple approximation to the distribution is suggested and examined