Regularized covariance matrix estimation under the common principal components model

The common principal components (CPC) model provides a way to model the population covariance matrices of several groups by assuming a common eigenvector structure. When appropriate, this model can provide covariance matrix estimators of which the elements have smaller standard errors than when using either the pooled covariance matrix or the per group unbiased sample covariance matrix estimators. In this article, a regularized CPC estimator under the assumption of a common (or partially common) eigenvector structure in the populations is proposed. After estimation of the common eigenvectors using the Flury–Gautschi (or other) algorithm, the off-diagonal elements of the nearly diagonalized covariance matrices are shrunk towards zero and multiplied with the orthogonal common eigenvector matrix to obtain the regularized CPC covariance matrix estimates. The optimal shrinkage intensity per group can be estimated using cross-validation. The efficiency of these estimators compared to the pooled and unbiased estimators is investigated in a Monte Carlo simulation study, and the regularized CPC estimator is applied to a real dataset to demonstrate the utility of the method.     

Covariance Matrix Formula for Generalized Linear Models with Unknown Dispersion

In this paper, the existence of the Uniformly Minimum Risk Equivariant (UMRE) estimator of parameters in SURE model under some quadratic losses and matrix losses is studied. The necessary and sufficient conditions for existence of the UMRE estimator of linearly estimable function vectors of regression coefficients under an affine group of transformations are obtained. It is proved that no UMRE estimator of the covariance matrix under any one of two affine groups of transformations exists.     

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.     

Covariance Matrix Estimation via Network Structure

In this article, we employ a regression formulation to estimate the high-dimensional covariance matrix for a given network structure. Using prior information contained in the network relationships, we model the covariance as a polynomial function of the symmetric adjacency matrix. Accordingly, the problem of estimating a high-dimensional covariance matrix is converted to one of estimating low dimensional coefficients of the polynomial regression function, which we can accomplish using ordinary least squares or maximum likelihood. The resulting covariance matrix estimator based on the maximum likelihood approach is guaranteed to be positive definite even in finite samples. Under mild conditions, we obtain the theoretical properties of the resulting estimators. A Bayesian information criterion is also developed to select the order of the polynomial function. Simulation studies and empirical examples illustrate the usefulness of the proposed methods.     

Two Graphical Techniques Useful in Detecting Correlation Structure in Repeated Measures Data

Analysis of repeated measures data using a mixed model includes specifying a form for the covariance matrix of the within-subject observations. This reduction in the number of estimated parameters from the unspecified structure may improve the efficiency of inferences made. An implementation of this technique has been incorporated in the MIXED procedure of the SAS® statistical package, and includes a wide range of options for the structure of the covariance matrix. It is demonstrated that draftman's display plots and/or plots in a coordinate system with parallel axes can aid in visualizing the dispersion structure.     

OPTIMAL QUADRATIC UNBIASED ESTIMATION FOR MODELS WITH LINEAR TOEPLITZ COVARIANCE STRUCTURE

This paper deals with the problem of quadratic unbiased estimation for models with linear Toeplitz covariance structure. These serial covariance models are very useful to modelize time or spatial correlations by means of linear models. Optimality and local optimality is examined in different ways. For the nested Toeplitz models, it is shown that there does not exist a Uniformly Minimum Variance Quadratic Unbiased Estimator for at least one linear combination of covariance parameters. Moreover, empirical unbiased estimators are identified as Locally Minimum Variance Quadratic Unbiased Estimators for a particular choice on covariance parameters corresponding to the case where the covariance matrix of the observed random vector is proportional to the identity matrix. The complete Toeplitz-circulant model is also studied. For this model, the existence of a Uniformly Minimum Variance Quadratic Unbiased Estimator for each covariance parameter is proved.     

Testing a block exchangeable covariance matrix

Testing hypotheses about the structure of a covariance matrix for doubly multivariate data is often considered in the literature. In this paper the Rao's score test (RST) is derived to test the block exchangeable covariance matrix or block compound symmetry (BCS) covariance structure under the assumption of multivariate normality. It is shown that the empirical distribution of the RST statistic under the null hypothesis is independent of the true values of the mean and the matrix components of a BCS structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Simulation studies are performed for the sample size consideration, and for the estimation of the empirical quantiles of the null distribution of the test statistic. The RST procedure is illustrated on a real data set from the medical studies.     

Bayesian analysis of joint mean and covariance models for longitudinal data

Efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure. If misspecification occurs, it may lead to inefficient or biased estimators of parameters in the mean. One of the most commonly used methods for handling the covariance matrix is based on simultaneous modeling of the Cholesky decomposition. Therefore, in this paper, we reparameterize covariance structures in longitudinal data analysis through the modified Cholesky decomposition of itself. Based on this modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit lower triangular matrix involving moving average coefficients and a diagonal matrix involving innovation variances, which are modeled as linear functions of covariates. Then, we propose a fully Bayesian inference for joint mean and covariance models based on this decomposition. A computational efficient Markov chain Monte Carlo method which combines the Gibbs sampler and Metropolis–Hastings algorithm is implemented to simultaneously obtain the Bayesian estimates of unknown parameters, as well as their standard deviation estimates. Finally, several simulation studies and a real example are presented to illustrate the proposed methodology.     

Supervised classifiers for high-dimensional higher-order data with locally doubly exchangeable covariance structure

We explore the performance accuracy of the linear and quadratic classifiers for high-dimensional higher-order data, assuming that the class conditional distributions are multivariate normal with locally doubly exchangeable covariance structure. We derive a two-stage procedure for estimating the covariance matrix: at the first stage, the Lasso-based structure learning is applied to sparsifying the block components within the covariance matrix. At the second stage, the maximum-likelihood estimators of all block-wise parameters are derived assuming the doubly exchangeable within block covariance structure and a Kronecker product structured mean vector. We also study the effect of the block size on the classification performance in the high-dimensional setting and derive a class of asymptotically equivalent block structure approximations, in a sense that the choice of the block size is asymptotically negligible.     

Large sample test criteria on Toeplitz covariance structure

In the analysis of stationary stochastic process, one has to deal with covariance matrix of Toeplitz (or Laurent) structure. Such structure has a feature that not only the elements on the principal diagonal but also those lying on each of the parallel sub-diagonals are equal as well. The present investigation is on the problem of large sample testing of the Toeplitz pattern of the population covariance matrix. Apart from usual application of likelihood ratio and Rao’s efficient score criteria, some heuristic two-stage tests are suggested. The results of Monte Carlo experiment are reported for the size of the proposed tests.     

ON UMP INVARIANT F-TEST PROCEDURES IN A GENERAL LINEAR MODEL

A simple method of setting linear hypotheses for a split mean vector testable by F-tests in a general linear model, when the covariance matrix has a general form and is completely unknown, is provided by extending the method discussed in Ukita et al. The critical functions in these F-tests are constructed as UMP invariants, when the covariance matrix has a known structure. Further critical functions in F-tests of linear hypotheses for the other split mean vector in the model are shown to be UMP invariant if the same known structure of the covariance matrix is assumed.     

Criterion for the Selection of a Working Correlation Structure in the Generalized Estimating Equation Approach for Longitudinal Balanced Data

The generalized estimating equation is a popular method for analyzing correlated response data. It is important to determine a proper working correlation matrix at the time of applying the generalized estimating equation since an improper selection sometimes results in inefficient parameter estimates. We propose a criterion for the selection of an appropriate working correlation structure. The proposed criterion is based on a statistic to test the hypothesis that the covariance matrix equals a given matrix, and also measures the discrepancy between the covariance matrix estimator and the specified working covariance matrix. We evaluated the performance of the proposed criterion through simulation studies assuming that for each subject, the number of observations remains the same. The results revealed that when the proposed criterion was adopted, the proportion of selecting a true correlation structure was generally higher than that when other competing approaches were adopted. The proposed criterion was applied to longitudinal wheeze data, and it was suggested that the resultant correlation structure was the most accurate.     

Bayesian tests on components of the compound symmetry covariance matrix

Complex dependency structures are often conditionally modeled, where random effects parameters are used to specify the natural heterogeneity in the population. When interest is focused on the dependency structure, inferences can be made from a complex covariance matrix using a marginal modeling approach. In this marginal modeling framework, testing covariance parameters is not a boundary problem. Bayesian tests on covariance parameter(s) of the compound symmetry structure are proposed assuming multivariate normally distributed observations. Innovative proper prior distributions are introduced for the covariance components such that the positive definiteness of the (compound symmetry) covariance matrix is ensured. Furthermore, it is shown that the proposed priors on the covariance parameters lead to a balanced Bayes factor, in case of testing an inequality constrained hypothesis. As an illustration, the proposed Bayes factor is used for testing (non-)invariant intra-class correlations across different group types (public and Catholic schools), using the 1982 High School and Beyond survey data.     

Influence in covariance structure analysis: with an application to confirmatory factor analysis

Influence functions are derived for the parameters in covariance structure analysis, where the parameters are estimated by minimizing a discrepancy function between the assumed covariance matrix and the sample covariance matrix. The case of confirmatory factor analysis is studied precisely with a numerical example. Comparing with a general procedure called one-step estimation, the proposed procedure has two advantages:1) computing cost is cheaper, 2) the property that arbitrary influence can be decomposed into a fi-nite number of components discussed by Tanaka and Castano-Tostado(1990) can be used for efficient computing and the characterization of a covariance structure model from the sensitivity perspective. A numerical comparison is made among the confirmatory factor analysis and some procedures of ex-ploratory factor analysis by using the decomposition mentioned above.     

Effects of the estimation of covariance matrix parameters in the generalized multivariate linear model

We Consider the generalized multivariate linear model and assume the covariance matrix of the p x 1 vector of responses on a given individual can be represented in the general linear structure form described by Anderson (1973). The effects of the use of estimates of the parameters of the covariance matrix on the generalized least squares estimator of the regression coefficients and on the prediction of a portion of a future vector, when only the first portion of the vector has been observed, are investigated. Approximations are derived for the covariance matrix of the generalized least squares estimator and for the mean square error matrix of the usual predictor, for the practical case where estimated parameters are used.     

Adaptive robust estimation in joint mean–covariance regression model for bivariate longitudinal data

The estimation of the covariance matrix is important in the analysis of bivariate longitudinal data. A good estimator for the covariance matrix can improve the efficiency of the estimators of the mean regression coefficients. Furthermore, the covariance estimation itself is also of interest, but it is a challenging job to model the covariance matrix of bivariate longitudinal data due to the complex structure and positive definite constraint. In addition, most of existing approaches are based on the maximum likelihood, which is very sensitive to outliers or heavy-tail error distributions. In this article, an adaptive robust estimation method is proposed for bivariate longitudinal data. Unlike the existing likelihood-based methods, the proposed method can adapt to different error distributions. Specifically, at first, we utilize the modified Cholesky block decomposition to parameterize the covariance matrices. Secondly, we apply the bounded Huber's score function to develop a set of robust generalized estimating equations to estimate the parameters both in the mean and the covariance models simultaneously. A data-driven approach is presented to select the parameter c in the Huber's score function, which can ensure that the proposed method is robust and efficient. A simulation study and a real data analysis are conducted to illustrate the robustness and efficiency of the proposed approach.     

HIGH DIMENSIONAL COVARIANCE MATRIX ESTIMATION IN APPROXIMATE FACTOR MODELS

The variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics. Popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems. Classical methods of estimating the covariance matrices are based on the strict factor models, assuming independent idiosyncratic components. This assumption, however, is restrictive in practical applications. By assuming sparse error covariance matrix, we allow the presence of the cross-sectional correlation even after taking out common factors, and it enables us to combine the merits of both methods. We estimate the sparse covariance using the adaptive thresholding technique as in Cai and Liu (2011), taking into account the fact that direct observations of the idiosyncratic components are unavailable. The impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied.     

Maximum likelihood mean and covariance matrix estimation constrained to general positive semi-definiteness

Maximum likelihood estimation of a mean and a covariance matrix whose structure is constrained only to general positive semi-definiteness is treated in this paper. Necessary and sufficient conditions for the local optimality of mean and covariance matrix estimates are given. Observations are assumed to be independent. When the observations are also assumed to be identically distributed, the optimality conditions are used to obtain the mean and covariance matrix solutions in closed form. For the nonidentically distributed observation case, a general numerical technique which integrates scoring and Newton's iterations to solve the optimality condition equations is presented, and convergence performance is examined.     

Sensitivity analysis in covariance structure analysis with equality constraints

Influence functions are derived for covariance structure analysis with equality constraints, where the parameters are estimated by minimizing a discrepancy function between the assumed covariance matrix and the sample covariance matrix. As a special case maximum likelihood exploratory factor analysis is studied precisely with a numerical example. Comparison is made with the the results of Tanaka and Odaka (1989), who have proposed a sensitivity analysis procedure in maximum likelihood exploratory factor analysis using the perturbation expansion of a certain function of eigenvalues and eigenvectors of a real symmetric matrix. Also the present paper gives a generalization of Tanaka, Watadani and Moon (1991) to the case with equality constraints.