Effects of the generalized Box–Cox transformation on Type I error rate and power of Hotelling's T 2

Most multivariate statistical techniques rely on the assumption of multivariate normality. The effects of nonnormality on multivariate tests are assumed to be negligible when variance–covariance matrices and sample sizes are equal. Therefore, in practice, investigators usually do not attempt to assess multivariate normality. In this simulation study, the effects of skewed and leptokurtic multivariate data on the Type I error and power of Hotelling's T ² were examined by manipulating distribution, sample size, and variance–covariance matrix. The empirical Type I error rate and power of Hotelling's T ² were calculated before and after the application of generalized Box–Cox transformation. The findings demonstrated that even when variance–covariance matrices and sample sizes are equal, small to moderate changes in power still can be observed.     

CLASSIFICATION AND PROPORTION ESTIMATION

Assume that a number of individuals are to be classified into one of two populations and that, at the same time, the proportion of members of each population needs to be estimated. The allocated proportions given by the Bayes classification rule are not consistent estimates of the true proportions, so a different classification rule is proposed; this rule yields consistent estimates with only a small increase in the probability of misclassification. As an illustration, the case of two normal distributions with equal covariance matrices is dealt with in detail.     

A parametric bootstrap solution to the MANOVA under heteroscedasticity

In this article, we consider the problem of comparing several multivariate normal mean vectors when the covariance matrices are unknown and arbitrary positive definite matrices. We propose a parametric bootstrap (PB) approach and develop an approximation to the distribution of the PB pivotal quantity for comparing two mean vectors. This approximate test is shown to be the same as the invariant test given in [Krishnamoorthy and Yu, Modified Nel and Van der Merwe test for the multivariate Behrens–Fisher problem, Stat. Probab. Lett. 66 (2004), pp. 161–169] for the multivariate Behrens–Fisher problem. Furthermore, we compare the PB test with two existing invariant tests via Monte Carlo simulation. Our simulation studies show that the PB test controls Type I error rates very satisfactorily, whereas other tests are liberal especially when the number of means to be compared is moderate and/or sample sizes are small. The tests are illustrated using an example.     

On discrimination and classification with multivariate repeated measures data

We study the problem of classification with multiple q-variate observations with and without time effect on each individual. We develop new classification rules for populations with certain structured and unstructured mean vectors and under certain covariance structures. The new classification rules are effective when the number of observations is not large enough to estimate the variance–covariance matrix. Computational schemes for maximum likelihood estimates of required population parameters are given. We apply our findings to two real data sets as well as to a simulated data set.     

Sequential discrimination

An analysis of the 1-stage classification decision with two candidate populations is provided in this paper. When the successive posterior probabilities follow a first order markov process it it shown that the optimal classification rules are greatly simplified. A detailed analysis and example are provided for the important case of multivariate normality with equal covariance matrices.     

Classification Rules under Autoregressive and General Circulant Covariance

We develop classification rules for data that have an autoregressive circulant covariance structure under the assumption of multivariate normality. We also develop classification rules assuming a general circulant covariance structure. The new classification rules are efficient in reducing the misclassification error rates when the number of observations is not large enough to estimate the unknown variance–covariance matrix. The proposed classification rules are demonstrated by simulation study for their validity and illustrated by a real data analysis for their use. Analyses of both simulated data and real data show the effectiveness of our new classification rules.     

Distances between normal populations when covariance matrices are unequal

The definition of distance between two populations of equal covariance matrices is extended to two and more than two populations with unequal covariance matrices and Rao’s U test for testing the conditional contribution of a subset of variables to the distance is extended to this situation, even when sample sizes are not necessarily the same.     

Performance evaluation of likelihood-ratio tests for assessing similarity of the covariance matrices of two multivariate normal populations

Abstract

In analyzing two multivariate normal data sets, the assumption about equality of covariance matrices is usually used as a default for doing subsequence inferences. If this equality doesn’t hold, later inferences will be more complex and usually approximate. If one detects some identical components between two decomposed non equal covariance matrices and uses this extra information, one expects that subsequence inferences can be more accurately performed. For this purpose, in this article we consider some statistical tests about the equality of components of decomposed covariance matrices of two multivariate normal populations. Our emphasis is on the spectral decomposition of these matrices. Hypotheses about the equalities of sizes, shapes, and set of directions as components of these two covariance matrices are tested by the likelihood ratio test (LRT). Some simulation studies are carried out to investigate the accuracy and power of the LRT. Finally, analyses of two real data sets are illustrated.     

Alternative Methods of Estimating the Likelihood Ratio in Classification of Multivariate Normal Observations

An observation is to be classified into one of two multivariate normal distributions with equal covariance matrices. When the parameters are unknown, four methods of estimating the likelihood ratio, that is, the plug-in method, the test procedure, the Bayesian approach, and the best invariant estimate method, are reviewed. The assumptions, interpretations, and consequences of the four approaches are given. It is shown that the last three methods yield the same classification procedure.     

The Covariance Inflation Criterion for Adaptive Model Selection

We propose a new criterion for model selection in prediction problems. The covariance inflation criterion adjusts the training error by the average covariance of the predictions and responses, when the prediction rule is applied to permuted versions of the data set. This criterion can be applied to general prediction problems (e.g. regression or classification) and to general prediction rules (e.g. stepwise regression, tree-based models and neural nets). As a by-product we obtain a measure of the effective number of parameters used by an adaptive procedure. We relate the covariance inflation criterion to other model selection procedures and illustrate its use in some regression and classification problems. We also revisit the conditional bootstrap approach to model selection.     

A note on the linear discriminant function when group means are equal

The performance of the sample linear discriminant function with known, proportional, covariance matrices and equal but unknown mean vectors is considered. Unconditional misclassification rates are obtained from the Student-t distribution. These results can be used as an aid in verifying simulation programs incorporating the linear discriminant function when Gaussian densities with unequal covariance matrices are used.     

The Asymptotic Covariance Matrix of the Least Squares Estimator in the Stochastic Linear Regression Model: The Case of Elliptically Symmetric Distribution

This article considers the unconditional asymptotic covariance matrix of the least squares estimator in the linear regression model with stochastic explanatory variables. The asymptotic covariance matrix of the least squares estimator of regression parameters is evaluated relative to the standard asymptotic covariance matrix when the joint distribution of the dependent and explanatory variables is in the class of elliptically symmetric distributions. An empirical example using financial data is presented. Numerical examples and simulation experiments are given to illustrate the difference of the two asymptotic covariance matrices.     

Classification rules for triply multivariate data with an AR(1) correlation structure on the repeated measures over time

In this article we study the problem of classification of three-level multivariate data, where multiple q$q$-variate observations are measured on u$u$-sites and over p$p$-time points, under the assumption of multivariate normality. The new classification rules with certain structured and unstructured mean vectors and covariance structures are very efficient in small sample scenario, when the number of observations is not adequate to estimate the unknown variance–covariance matrix. These classification rules successfully model the correlation structure on successive repeated measurements over time. Computation algorithms for maximum likelihood estimates of the unknown population parameters are presented. Simulation results show that the introduction of sites in the classification rules improves their performance over the existing classification rules without the sites.     

ESTIMATION OF FAMILIAL CORRELATIONS UNDER AUTOREGRESSIVE CIRCULAR COVARIANCE

Circular covariance matrices play an important role in modeling phenomena in numerous epidemiological, communications and physical contexts. In this article, we propose a parsimonious, autoregressive type of circular covariance structure for modeling correlations between the “siblings” of a “family”. This structure, similar to AR(1) structure used in time series models, involves only two parameters. We derive the maximum likelihood estimators of these parameters, and discuss testing of hypotheses about the autoregressive parameter. Estimation of “parent-sib” correlation, namely, the interclass correlation, is also considered. Estimation of the parameters when there are unequal numbers of siblings in different families is also discussed.     

A Parametric Bootstrap Solution to Two-way MANOVA without Interaction under Heteroscedasticity: Fixed and Mixed Models

In this article, we propose a parametric bootstrap (PB) test for heteroscedastic two-way multivariate analysis of variance without Interaction. For the problem of testing equal main effects of factors, we obtain a PB approach and compare it with existing modified Brown–Forsythe (MBF) test and approximate Hotelling T² (AHT) test by an extensive simulation study. The PB test is a symmetric function in samples, and does not depend on the chosen weights used to define the parameters uniquely. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations when the homogeneity assumption is seriously violated, and tends to outperform the AHT test for moderate or larger samples in terms of power and controlling size. The MBF test, the AHT test, and the PB test have similar robustness to violations of underlying assumptions. It is also noted that the same PB test can be used to test the significance of random effect vector in a two-way multivariate mixed effects model with unequal cell covariance matrices.     

A hierarchical eigenmodel for pooled covariance estimation

Summary.  Although the covariance matrices corresponding to different populations are unlikely to be exactly equal they can still exhibit a high degree of similarity. For example, some pairs of variables may be positively correlated across most groups, whereas the correlation between other pairs may be consistently negative. In such cases much of the similarity across covariance matrices can be described by similarities in their principal axes, which are the axes that are defined by the eigenvectors of the covariance matrices. Estimating the degree of across-population eigenvector heterogeneity can be helpful for a variety of estimation tasks. For example, eigenvector matrices can be pooled to form a central set of principal axes and, to the extent that the axes are similar, covariance estimates for populations having small sample sizes can be stabilized by shrinking their principal axes towards the across-population centre. To this end, the paper develops a hierarchical model and estimation procedure for pooling principal axes across several populations. The model for the across-group heterogeneity is based on a matrix-valued antipodally symmetric Bingham distribution that can flexibly describe notions of 'centre' and 'spread' for a population of orthogonal matrices.     

Bayesian Estimation Using Warner's Randomized Response Model through Simple and Mixture Prior Distributions

The linear discriminant function (LDF) is known to be optimal in the sense of achieving an optimal error rate when sampling from multivariate normal populations with equal covariance matrices. Use of the LDF in nonnormal situations is known to lead to some strange results. This paper will focus on an evaluation of misclassification probabilities when the power transformation could have been used to achieve at least approximate normality and equal covariance matrices in the sampled populations for the distribution of the observed random variables. Attention is restricted to the two-population case with bivariate distributions.     

Distribution of multivariate quadratic forms under certain covariance structures

Necessary and sufficient conditions are given for the covariance structure of all the observations in a multivariate factorial experiment under which certain multivariate quadratic forms are independent and distributed as a constant times a Wishart. It is also shown that exact multivariate test statistics can be formed for certain covariance structures of the observations when the assumption of equal covariance matrices for each normal population is relaxed. A characterization is given for the dependency structure between random vectors in which the sample mean and sample covariance matrix have certain properties.     

Linear Discriminant Analysis of Multivariate Spatial–Temporal Regressions

Abstract.  We consider classification of the realization of a multivariate spatial–temporal Gaussian random field into one of two populations with different regression mean models and factorized covariance matrices. Unknown means and common feature vector covariance matrix are estimated from training samples with observations correlated in space and time, assuming spatial–temporal correlations to be known. We present the first-order asymptotic expansion of the expected error rate associated with a linear plug-in discriminant function. Our results are applied to ecological data collected from the Lithuanian Economic Zone in the Baltic Sea.     

Discriminant analysis of multivariate repeated measures data with a Kronecker product structured covariance matrices

This paper proposes new classifiers under the assumption of multivariate normality for multivariate repeated measures data with Kronecker product covariance structures. These classifiers are especially effective when the number of observations is not large enough to estimate the covariance matrices, and thus the traditional classifiers fail. Computational scheme for maximum likelihood estimates of required class parameters are also given. The quality of these new classifiers are examined on some real data.