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.     

Convergence of Sample Eigenvectors of Spiked Population Model

In this paper, for the general non Gaussian spiked population model, where a few fixed eigenvalues of the population covariance matrix are separated from others, we investigate the convergence properties of the eigenvectors of sample covariance matrices corresponding to the spiked population eigenvalues and angle between the population eigenvectors and sample eigenvectors as both the sample size and population size are large.     

Testing the equality of two high‐dimensional spatial sign covariance matrices

This paper is concerned with testing the equality of two high‐dimensional spatial sign covariance matrices with applications to testing the proportionality of two high‐dimensional covariance matrices. It is interesting that these two testing problems are completely equivalent for the class of elliptically symmetric distributions. This paper develops a new test for testing the equality of two high‐dimensional spatial sign covariance matrices based on the Frobenius norm of the difference between two spatial sign covariance matrices. The asymptotic normality of the proposed testing statistic is derived under the null and alternative hypotheses when the dimension and sample sizes both tend to infinity. Moreover, the asymptotic power function is also presented. Simulation studies show that the proposed test performs very well in a wide range of settings and can be allowed for the case of large dimensions and small sample sizes.     

Further theoretical results and a comparison between two methods for approximating eigenvalues of perturbed covariance matrices

Covariance matrices, or in general matrices of sums of squares and cross-products, are used as input in many multivariate analyses techniques. The eigenvalues of these matrices play an important role in the statistical analysis of data including estimation and hypotheses testing. It has been recognized that one or few observations can exert an undue influence on the eigenvalues of a covariance matrix. The relationship between the eigenvalues of the covariance matrix computed from all data and the eigenvalues of the perturbed covariance matrix (a covariance matrix computed after a small subset of the observations has been deleted) cannot in general be written in closed-form. Two methods for approximating the eigenvalues of a perturbed covariance matrix have been suggested by Hadi (1988) and Wang and Nyquist (1991) for the case of a perturbation by a single observation. In this paper we improve on these two methods and give some additional theoretical results that may give further insight into the problem. We also compare the two improved approximations in terms of their accuracies.     

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.     

Accent on Teaching Materials

Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

Separability tests for high-dimensional,low-sample size multivariate repeated measures data

Longitudinal imaging studies have moved to the forefront of medical research due to their ability to characterize spatio-temporal features of biological structures across the lifespan. Valid inference in longitudinal imaging requires enough flexibility of the covariance model to allow reasonable fidelity to the true pattern. On the other hand, the existence of computable estimates demands a parsimonious parameterization of the covariance structure. Separable (Kronecker product) covariance models provide one such parameterization in which the spatial and temporal covariances are modeled separately. However, evaluating the validity of this parameterization in high dimensions remains a challenge. Here we provide a scientifically informed approach to assessing the adequacy of separable (Kronecker product) covariance models when the number of observations is large relative to the number of independent sampling units (sample size). We address both the general case, in which unstructured matrices are considered for each covariance model, and the structured case, which assumes a particular structure for each model. For the structured case, we focus on the situation where the within-subject correlation is believed to decrease exponentially in time and space as is common in longitudinal imaging studies. However, the provided framework equally applies to all covariance patterns used within the more general multivariate repeated measures context. Our approach provides useful guidance for high dimension, low-sample size data that preclude using standard likelihood-based tests. Longitudinal medical imaging data of caudate morphology in schizophrenia illustrate the approaches appeal.     

SOME EXPECTED VALUES FOR THE ERROR RATES OF THE SAMPLE QUADRATIC DISCRIMINANT FUNCTION1

The expected error rates of the sample quadratic discriminant function are studied in the context of two populations with different means and proportional covariance matrices. For the general case of all population parameters unknown the expected error rates are expressed in the form of asymptotic expansions which are evaluated numerically and tabulated for several combinations of the population parameters.     

ON ESTIMATION OF THE POPULATION SPECTRAL DISTRIBUTION FROM A HIGH‐DIMENSIONAL SAMPLE COVARIANCE MATRIX

Sample covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the sample size, the sample covariance matrix demonstrates significant deviations from the underlying population covariance matrix. There is an urgent need to develop new estimation tools in such cases with high‐dimensional data to recover the characteristics of the population covariance matrix from the observed sample covariance matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross‐validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.     

Maximum likelihood estimation of parameter structures for the Wishart distribution using constraints

Maximum likelihood estimation under constraints for estimation in the Wishart class of distributions, is considered. It provides a unified approach to estimation in a variety of problems concerning covariance matrices. Virtually all covariance structures can be translated to constraints on the covariances. This includes covariance matrices with given structure such as linearly patterned covariance matrices, covariance matrices with zeros, independent covariance matrices and structurally dependent covariance matrices. The methodology followed in this paper provides a useful and simple approach to directly obtain the exact maximum likelihood estimates. These maximum likelihood estimates are obtained via an estimation procedure for the exponential class using constraints.     

Parametric and permutation testing for multivariate monotonic alternatives

We are firstly interested in testing the homogeneity of k mean vectors against two-sided restricted alternatives separately in multivariate normal distributions. This problem is a multivariate extension of Bartholomew (in Biometrica 46:328–335, 1959b) and an extension of Sasabuchi et al. (in Biometrica 70:465–472, 1983) and Kulatunga and Sasabuchi (in Mem. Fac. Sci., Kyushu Univ. Ser. A: Mathematica 38:151–161, 1984) to two-sided ordered hypotheses. We examine the problem of testing under two separate cases. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but common. For the general case that covariance matrices are known the test statistic is obtained using the likelihood ratio method. When the known covariance matrices are common and diagonal, the null distribution of test statistic is derived and its critical values are computed at different significance levels. A Monte Carlo study is also presented to estimate the power of the test. A test statistic is proposed for the case when the common covariance matrices are unknown. Since it is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown, we first present a reformulation of the test statistic based on the orthogonal projections on the closed convex cones and then determine the upper bounds for its p-values. Also we provide a general nonparametric solution based on the permutation approach and nonparametric combination of dependent tests.     

INDEPENDENCE DISTRIBUTION-PRESERVING NONNEGATIVE-DEFINITE COVARIANCE STRUCTURES FOR THE SAMPLE VARIANCE

This paper explicitly characterizes the general nonnegative-definite covariance structure for a multivariate normal random vector such that the univariate sample variance is distributed exactly as if the sample observations are normal independent and identically distributed (i.i.d.). This work extends the results of Baldessari (1965) and Stadje (1984) who have characterized the general positive-definite covariance matrix such that the univariate sample variance is distributed exactly as if the sample observations are normal i.i.d.     

A monte carlo study on two methods of calculating the mle's covariance matrix in a seemingly unrelated nonlinear regression. *

Econometric techniques to estimate output supply systems, factor demand systems and consumer demand systems have often required estimating a nonlinear system of equations that have an additive error structure when written in reduced form. To calculate the ML estimate's covariance matrix of this nonlinear system one can either invert the Hessian of the concentrated log likelihood function, or invert the matrix calculated by pre-multiplying and post multiplying the inverted MLE of the disturbance covariance matrix by the Jacobian of the reduced form model. Malinvaud has shown that the latter of these methods is the actual limiting distribution's covariance matrix, while Barnett has shown that the former is only an approximation.

In this paper, we use a Monte Carlo simulation study to determine how these two covariance matrices differ with respect to the nonlinearity of the model, the number of observations in the dataet, and the residual process. We find that the covariance matrix calculated from the Hessian of the concentrated likelihood function produces Wald statistics that are distributed above those calculated with the other covariance matrix. This difference becomes insignificant as the sample size increases to one-hundred or more observations, suggesting that the asymptotics of the two covariance matrices are quickly reached.     

A monte carlo study on two methods of calculating the mle's covariance matrix in a seemingly unrelated nonlinear regression.

Econometric techniques to estimate output supply systems, factor demand systems and consumer demand systems have often required estimating a nonlinear system of equations that have an additive error structure when written in reduced form. To calculate the ML estimate's covariance matrix of this nonlinear system one can either invert the Hessian of the concentrated log likelihood function, or invert the matrix calculated by pre-multiplying and post multiplying the inverted MLE of the disturbance covariance matrix by the Jacobian of the reduced form model. Malinvaud has shown that the latter of these methods is the actual limiting distribution's covariance matrix, while Barnett has shown that the former is only an approximation.

In this paper, we use a Monte Carlo simulation study to determine how these two covariance matrices differ with respect to the nonlinearity of the model, the number of observations in the dataet, and the residual process. We find that the covariance matrix calculated from the Hessian of the concentrated likelihood function produces Wald statistics that are distributed above those calculated with the other covariance matrix. This difference becomes insignificant as the sample size increases to one-hundred or more observations, suggesting that the asymptotics of the two covariance matrices are quickly reached.     

A Selection Procedure for the Number of Signals in Presence of Colored Noise

Ranking and selection theory is used to estimate the number of signals present in colored noise. The data structure follows the well-known MUSIC (MUltiple SIgnal Classification) model. We deal with the eigenvalues of a covariance matrix, using the MUSIC model and colored noise. The data matrix can be written as the product of two matrices. The first matrix is the sample covariance matrix of the observed vectors. The second matrix is the inverse of the sample covariance matrix of reference vectors. We propose a multi-step selection procedure to construct a confidence interval on the number of signals present in a data set. Properties of this procedure will be stated and proved. Those properties will be used to compute the required parameters (procedure constants). Numerical examples are given to illustrate our theory.     

Averaging estimation for conditional covariance models

Estimating conditional covariance matrices is important in statistics and finance. In this paper, we propose an averaging estimator for the conditional covariance, which combines the estimates of marginal conditional covariance matrices by Model Averaging MArginal Regression of Li, Linton, and Lu. This estimator avoids the “curse of dimensionality” problem that the local constant estimator of Yin et al. suffered from. We establish the asymptotic properties of the averaging weights and that of the proposed conditional covariance estimator. The finite sample performances are augmented by simulation. An application to portfolio allocation illustrates the practical superiority of the averaging estimator.     

A Test for the Equality of Means of Two Groups with Different Variances When the Sample Size and the Dimension are Large

We consider a class of test statistics including the Dempster trace criterion in the case of two groups without assuming equal covariance matrices. The test statistics in the class are valid when the dimension is larger than the sample size. We obtain asymptotic distributions of the test statistics in the class and use these distributions to derive the limiting power in each case. We obtain the most powerful test in the class with respect to this limiting power.     

Two sample inference in functional linear models

We propose a method of comparing two functional linear models in which explanatory variables are functions (curves) and responses can be either scalars or functions. In such models, the role of parameter vectors (or matrices) is played by integral operators acting on a function space. We test the null hypothesis that these operators are the same in two independent samples. The complexity of the test statistics increases as we move from scalar to functional responses and relax assumptions on the covariance structure of the regressors. They all, however, have an asymptotic chi‐squared distribution with the number of degrees of freedom which depends on a specific setting. The test statistics are readily computable using the R package fda , and have good finite sample properties. The test is applied to egg‐laying curves of Mediterranean flies and to data from terrestrial magnetic observatories. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application

Let X _n = (x _{i j} ) be a k ×n data matrix with complex‐valued, independent and standardized entries satisfying a Lindeberg‐type moment condition. We consider simultaneously R sample covariance matrices , where the Q _r 's are non‐random real matrices with common dimensions p ×k (k ≥p ). Assuming that both the dimension p and the sample size n grow to infinity, the limiting distributions of the eigenvalues of the matrices { B _{n r} } are identified, and as the main result of the paper, we establish a joint central limit theorem (CLT) for linear spectral statistics of the R matrices { B _{n r} }. Next, this new CLT is applied to the problem of testing a high‐dimensional white noise in time series modelling. In experiments, the derived test has a controlled size and is significantly faster than the classical permutation test, although it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix (R = 1).