Multivariate analysis of variance with fewer observations than the dimension

In this article, we consider the problem of testing a linear hypothesis in a multivariate linear regression model which includes the case of testing the equality of mean vectors of several multivariate normal populations with common covariance matrix Σ, the so-called multivariate analysis of variance or MANOVA problem. However, we have fewer observations than the dimension of the random vectors. Two tests are proposed and their asymptotic distributions under the hypothesis as well as under the alternatives are given under some mild conditions. A theoretical comparison of these powers is made.     

On the consistency properties of linear and quadratic discriminant analyses

The limit behavior of the conditional probability of error of linear and quadratic discriminant analyses is studied under wide assumptions on the class conditional distributions. Results obtained may help to explain analytically the behavior in applications of linear and quadratic discrimination techniques.     

Inference for the invariance of canonical analysis under linear transformations

Using a recent result about the invariance problem in linear canonical analysis (LCA), we introduce a criterion by means of which one can see if this invariance holds when the related random vectors are transformed by linear maps. Then, the estimation of this criterion is considered as well as the problem of testing for invariance of LCA. Particularly, a new test for additional information in canonical analysis is proposed and simulations are used to gain understanding of the finite sample performance of this test and to compare it with the likelihood ratio test.     

Exact probabilities of correct classifications for uncorrelated repeated measurements from elliptically contoured distributions

Euclidean distance-based classification rules are derived within a certain nonclassical linear model approach and applied to elliptically contoured samples having a density generating function g. Then a geometric measure theoretical method to evaluate exact probabilities of correct classification for multivariate uncorrelated feature vectors is developed. When doing this one has to measure suitably defined sets with certain standardized measures. The geometric key point is that the intersection percentage functions of the areas under investigation coincide with those of certain parabolic cylinder type sets. The intersection percentage functions of the latter sets can be described as threefold integrals. It turns out that these intersection percentage functions yield simultaneously geometric representation formulae for the doubly noncentral g-generalized F-distributions. Hence, we get beyond new formulae for evaluating probabilities of correct classification new geometric representation formulae for the doubly noncentral g-generalized F-distributions. A numerical study concerning several aspects of evaluating both probabilities of correct classification and values of the doubly noncentral g-generalized F-distributions demonstrates the advantageous computational properties of the present new approach. This impression will be supported by comparison with the literature.It is shown that probabilities of correct classification depend on the parameters of the underlying sample distribution through a certain well-defined set of secondary parameters. If the underlying parameters are unknown, we propose to estimate probabilities of correct classification.     

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.     

Kronecker product permutation matrices and their application to moment matrices of the normal distribution

In this paper, we consider the matrix which transforms a Kronecker product of vectors into the average of all vectors obtained by permuting the vectors involved in the Kronecker product. An explicit expression is given for this matrix, and some of its properties are derived. It is shown that this matrix is particularly useful in obtaining compact expressions for the moment matrices of the normal distribution. The utility of these expressions is illustrated through some examples.     

Conditional information criteria for selecting variables in linear mixed models

In this paper, we consider the problem of selecting the variables of the fixed effects in the linear mixed models where the random effects are present and the observation vectors have been obtained from many clusters. As the variable selection procedure, here we use the Akaike Information Criterion, AIC. In the context of the mixed linear models, two kinds of AIC have been proposed: marginal AIC and conditional AIC. In this paper, we derive three versions of conditional AIC depending upon different estimators of the regression coefficients and the random effects. Through the simulation studies, it is shown that the proposed conditional AIC’s are superior to the marginal and conditional AIC’s proposed in the literature in the sense of selecting the true model. Finally, the results are extended to the case when the random effects in all the clusters are of the same dimension but have a common unknown covariance matrix.     

Multivariate Behrens-Fisher problem with missing data

Inference about the difference between two normal mean vectors when the covariance matrices are unknown and arbitrary is considered. Assuming that the incomplete data are of monotone pattern, a pivotal quantity, similar to the Hotelling T² statistic, is proposed. A satisfactory moment approximation to the distribution of the pivotal quantity is derived. Hypothesis testing and confidence estimation based on the approximate distribution are outlined. The accuracy of the approximation is investigated using Monte Carlo simulation. Monte Carlo studies indicate that the approximate method is very satisfactory even for moderately small samples. The proposed methods are illustrated using an example.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

A unified approach to testing for and against a set of linear inequality constraints in the product multinomial setting

A problem that is frequently encountered in statistics concerns testing for equality of multiple probability vectors corresponding to independent multinomials against an alternative they are not equal. In applications where an assumption of some type of stochastic ordering is reasonable, it is desirable to test for equality against this more restrictive alternative. Similar problems have been considered heretofore using the likelihood ratio approach. This paper aims to generalize the existing results and provide a unified technique for testing for and against a set of linear inequality constraints placed upon on any probability vectors corresponding to r independent multinomials. The paper shows how to compute the maximum likelihood estimates under all hypotheses of interest and obtains the limiting distributions of the likelihood ratio test statistics. These limiting distributions are of chi bar square type and the expression of the weighting values is given. To illustrate our theoretical results, we use a real life data set to test against second-order stochastic ordering.     

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.     

Minimum distance classification rules for high dimensional data

In this article, the problem of classifying a new observation vector into one of the two known groups Π_i,i=1,2, distributed as multivariate normal with common covariance matrix is considered. The total number of observation vectors from the two groups is, however, less than the dimension of the observation vectors. A sample-squared distance between the two groups, using Moore-Penrose inverse, is introduced. A classification rule based on the minimum distance is proposed to classify an observation vector into two or several groups. An expression for the error of misclassification when there are only two groups is derived for large p and n=O(p^δ),0<δ<1.     

Asymptotic distributions of robust shape matrices and scales

It has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix Σ of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for Σ or another matrix Γ∝Σ. In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the estimators presented are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the scale function presented leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine [D. Paindaveine, A canonical definition of shape, Statistics and Probability Letters 78 (2008) 2240-2247] in the context of local asymptotic normality theory.     

Optimal training sample allocation and asymptotic expansions for error rates in discriminant analysis

The sample-based rule obtained from Bayes classification rule by replacing the unknown parameters by ML estimates from a stratified training sample is used for the classification of a random observationX into one ofL populations. The asymptotic expansions in terms of the inverses of the training sample sizes for cross-validation, apparent and plug-in error rates are found. These are used to compare estimation methods of the error rate for a wide range of regular distributions as probability models for considered populations. The optimal training sample allocation minimizing the asymptotic expected error regret is found in the cases of widely applicable, positively skewed distributions (Rayleigh and Maxwell distributions). These probability models for populations are often met in ecology and biology. The results indicate that equal training sample sizes for each populations sometimes are not optimal, even when prior probabilities of populations are equal.     

Likelihood ratio tests of correlated multivariate samples

We develop methods to compare multiple multivariate normally distributed samples which may be correlated. The methods are new in the context that no assumption is made about the correlations among the samples. Three types of null hypotheses are considered: equality of mean vectors, homogeneity of covariance matrices, and equality of both mean vectors and covariance matrices. We demonstrate that the likelihood ratio test statistics have finite-sample distributions that are functions of two independent Wishart variables and dependent on the covariance matrix of the combined multiple populations. Asymptotic calculations show that the likelihood ratio test statistics converge in distribution to central Chi-squared distributions under the null hypotheses regardless of how the populations are correlated. Following these theoretical findings, we propose a resampling procedure for the implementation of the likelihood ratio tests in which no restrictive assumption is imposed on the structures of the covariance matrices. The empirical size and power of the test procedure are investigated for various sample sizes via simulations. Two examples are provided for illustration. The results show good performance of the methods in terms of test validity and power.     

Sensitivity of GLS estimators in random effects models

This paper studies the sensitivity of random effects estimators in the one-way error component regression model. Maddala and Mount (1973) [6] give simulation evidence that in random effects models the properties of the feasible GLS estimator are not affected by the choice of the first-step estimator used for the covariance matrix. Taylor (1980) [8] gives a theoretical example of this effect. This paper provides a reason for this in terms of sensitivity. The properties of are transferred via an uncorrelated (and independent under normality) link, called sensitivity. The sensitivity statistic counteracts the improvement in . A Monte Carlo experiment illustrates the theoretical findings.     

Parameter estimation of selfsimilarity exponents

The characteristic feature of operator selfsimilar stochastic processes is that a linear rescaling in time is equal in the sense of distributions to a linear operator rescaling in space, which in turn is characterized by the selfsimilarity exponent. The growth behaviour of such processes in any radial direction is determined by the real parts of the eigenvalues of the selfsimilarity exponent. We extend an estimation method of Meerschaert and Scheffler [M.M. Meerschaert, H.-P. Scheffler, Moment estimator for random vectors with heavy tails, J. Multivariate Anal. 71 (1999) 145-159, M.M. Meerschaert, H.-P. Scheffler, Portfolio modeling with heavy tailed random vectors, in: S.T. Rachev (Ed.), Handbook of Heavy Tailed Distributions in Finance, Elsevier Science B.V., Amsterdam, 2003, pp. 595-640] to be applicable for estimating the real parts of the eigenvalues of the selfsimilarity exponent and corresponding spectral directions given by the eigenvectors. More generally, the results are applied to operator semi-selfsimilar processes, which obey a weaker scaling property, and to certain Ornstein-Uhlenbeck type processes connected to operator semi-selfsimilar processes via Lamperti's transformation.     

Noncentral matrix quadratic forms of the skew elliptical variables

In this paper, the noncentral matrix quadratic forms of the skew elliptical variables are studied. A family of the matrix variate noncentral generalized Dirichlet distributions is introduced as the extension of the noncentral Wishart distributions, the Dirichlet distributions and the noncentral generalized Dirichlet distributions. Main distributional properties are investigated. These include probability density and closure property under linear transformation and marginalization, the joint distribution of the sub-matrices of the matrix quadratic forms in the skew elliptical variables and the moment generating functions and Bartlett's decomposition of the matrix quadratic forms in the skew normal variables. Two versions of the noncentral Cochran's Theorem for the matrix variate skew normal distributions are obtained, providing sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the matrix variate noncentral generalized Dirichlet distributions. Applications include the properties of the least squares estimation in multivariate linear model and the robustness property of the Wilk's likelihood ratio statistic in the family of the matrix variate skew elliptical distributions.     

Direct variable selection for discrimination among several groups

We propose a criterion for variable selection in discriminant analysis. This criterion permits to arrange the variables in decreasing order of adequacy for discrimination, so that the variable selection problem reduces to that of the estimation of suitable permutation and dimensionality. Then, estimators for these parameters are proposed and the resulting method for selecting variables is shown to be consistent. In a simulation study, we compute proportions of correct classification after variable selection in order to gain understanding of the performance of our proposal and to compare it to existing methods.     

Asymptotic expansions for the pivots using log-likelihood derivatives with an application in item response theory

Asymptotic expansions of the distributions of the pivotal statistics involving log-likelihood derivatives under possible model misspecification are derived using the asymptotic cumulants up to the fourth-order and the higher-order asymptotic variance. The pivots dealt with are the studentized ones by the estimated expected information, the negative Hessian matrix, the sum of products of gradient vectors, and the so-called sandwich estimator. It is shown that the first three asymptotic cumulants are the same over the pivots under correct model specification with a general condition of the equalities. An application is given in item response theory, where the observed information is usually used rather than the estimated expected one.