Partitioning k multivariate normal populations according to equivalence with respect to a standard vector

We propose optimal procedures to achieve the goal of partitioning k multivariate normal populations into two disjoint subsets with respect to a given standard vector. Definition of good or bad multivariate normal populations is given according to their Mahalanobis distances to a known standard vector as being small or large. Partitioning k multivariate normal populations is reduced to partitioning k non-central Chi-square or non-central F distributions with respect to the corresponding non-centrality parameters depending on whether the covariance matrices are known or unknown. The minimum required sample size for each population is determined to ensure that the probability of correct decision attains a certain level. An example is given to illustrate our procedures.     

A Class of Convolution‐Based Models for Spatio‐Temporal Processes with Non‐Separable Covariance Structure

Abstract. In this article, we propose a new parametric family of models for real‐valued spatio‐temporal stochastic processes S ( x , t ) and show how low‐rank approximations can be used to overcome the computational problems that arise in fitting the proposed class of models to large datasets. Separable covariance models, in which the spatio‐temporal covariance function of S ( x , t ) factorizes into a product of purely spatial and purely temporal functions, are often used as a convenient working assumption but are too inflexible to cover the range of covariance structures encountered in applications. We define positive and negative non‐separability and show that in our proposed family we can capture positive, zero and negative non‐separability by varying the value of a single parameter.     

Consistent variable selection via the optimal discovery procedure in multiple testing

In this paper, we translate variable selection for linear regression into multiple testing, and select significant variables according to testing result. New variable selection procedures are proposed based on the optimal discovery procedure (ODP) in multiple testing. Due to ODP’s optimality, if we guarantee the number of significant variables included, it will include less non significant variables than marginal p-value based methods. Consistency of our procedures is obtained in theory and simulation. Simulation results suggest that procedures based on multiple testing have improvement over procedures based on selection criteria, and our new procedures have better performance than marginal p-value based procedures.     

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.     

A permutation-based Bayesian approach for inverse covariance estimation

Abstract

Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In particular, we utilize multiple maximum a posteriori estimates under the DAG-Wishart prior for each permutation, and subsequently construct the final estimate of the inverse covariance matrix. The proposed estimator has smaller variability and yields order-invariant property. We establish posterior convergence rates under mild assumptions and illustrate that our method outperforms existing approaches in estimating the inverse covariance matrices via simulation studies.     

Selection of the best with a preliminary test for two-parameter exponential distributions

This paper deals with the problem of selecting the best population from among k(≥ 2) two-parameter exponential populations. New selection procedures are proposed for selecting the unique best. The procedures include preliminary tests which allow the xperimenter to have an option to not select if the statistical evidence is not significant. Two probabilities, the probability to make a selection and the probability of a correct selection, are controlled by these selection procedures. Comparisons between the proposed selection procedures and certain earlier existing procedures are also made. The results show the superiority of the proposed selection procedures in terms of the required sample size.     

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.     

Surveillance of the covariance matrix of multivariate nonlinear time series

In this paper, sequential procedures for the surveillance of the covariance matrices of multivariate nonlinear time series are introduced. Two different types of control charts are proposed. The first type is based on the exponential smoothing of each component of a local measure for the covariances. The control statistic is equal to the Mahalanobis distance of this quantity with its in-control mean. In our second approach, the Mahalanobis distance is first determined and after that it is exponentially smoothed. We discuss three examples of local measures.

Several properties of the proposed schemes are discussed assuming the target process to be generated by a multivariate GARCH(1, 1) model. The generalization to the family of spherical distributions allows the modelling of frequently observed fat tails in financial data. Some results of an extensive Monte Carlo simulation study are provided in order to judge the performance of the presented control schemes. As a performance measure we use the average run length. An empirical example illustrates the importance of the fast detection of the changes in the covariance structure of the returns of financial assets.     

A selection procedure for estimating the number of signal components

This paper proposes a selection procedure to estimate the multiplicity of the smallest eigenvalue of the covariance matrix. The unknown number of signals present in a radar data can be formulated as the difference between the total number of components in the observed multivariate data vector and the multiplicity of the smallest eigenvalue. In the observed multivariate data, the smallest eigenvalues of the sample covariance matrix may in fact be grouped about some nominal value, as opposed to being identically equal. We propose a selection procedure to estimate the multiplicity of the common smallest eigenvalue, which is significantly smaller than the other eigenvalues. We derive the probability of a correct selection, P(CS), and the least favorable configuration (LFC) for our procedures. Under the LFC, the P(CS) attains its minimum over the preference zone of all eigenvalues. Therefore, a minimum sample size can be determined from the P(CS) under the LFC, P(CS|LFC), in order to implement our new procedure with a guaranteed probability requirement. Numerical examples are presented in order to illustrate our proposed procedure.     

MULTIVARLYTE ANALYSIS OF CLUSTERED SURVEY DATA: TESTING EQUALITY OF COVARLANCE MATRICES

The problem of testing the hypothesis of equality of covariance matrices in the presence of two-stage sampling is considered. Asymptotic test procedures based on linearization, grouping and jackknifing with or without transformation are proposed. The finite sample properties of these procedures are investigated in sampling experiments both from simulated known distributions and from a natural population.     

Selecting the best normal population better than a standard under unequal variances

In this article we consider a problem of selecting the best normal population that is better than a standard when the variances are unequal. Single-stage selection procedures are proposed when the variances are known. Wilcox (1984) and Taneja and Dudewicz (1992) proposed two-stage selection procedures when the variances are unknown. In addition to these procedures, we propose a two-stage selection procedure based on the method of Lam (1988). Comparisons are made between these selection procedures in terms of the sample sizes.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

A two-stage procedure for estimating a linear function of K multinormal mean vectors when covariance matrices are unknown

We consider the problem of constructing a fixed-size confidence region for a linear function of mean vectors of k multinormal populations, where all covariance matrices are completely unknown. A two-stage procedure is proposed to construct such a confidence region. It is shown that the proposed two-stage procedure is consistent and its asymptotic property for the expected sample size is also given. A Monte Carlo simulation study is given for an illustration.     

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).     

New aspects of Bregman divergence in regression and classification with parametric and nonparametric estimation

In statistical learning, regression and classification concern different types of the output variables, and the predictive accuracy is quantified by different loss functions. This article explores new aspects of Bregman divergence (BD), a notion which unifies nearly all of the commonly used loss functions in regression and classification. The authors investigate the duality between BD and its generating function. They further establish, under the framework of BD, asymptotic consistency and normality of parametric and nonparametric regression estimators, derive the lower bound of their asymptotic covariance matrices, and demonstrate the role that parametric and nonparametric regression estimation play in the performance of classification procedures and related machine learning techniques. These theoretical results and new numerical evidence show that the choice of loss function affects estimation procedures, whereas has an asymptotically relatively negligible impact on classification performance. Applications of BD to statistical model building and selection with non‐Gaussian responses are also illustrated. The Canadian Journal of Statistics 37: 119‐139; 2009 © 2009 Statistical Society of Canada     

Subset selection in two-factor experiments using randomization restricted designs

This paper studies subset selection procedures for screening in two-factor treatment designs that employ either a split-plot or strip-plot randomization restricted experimental design laid out in blocks. The goal is to select a subset of treatment combinations associated with the largest mean. In the split-plot design, it is assumed that the block effects, the confounding effects (whole-plot error) and the measurement errors are normally distributed. None of the selection procedures developed depend on the block variances. Subset selection procedures are given for both the case of additive and non-additive factors and for a variety of circumstances concerning the confounding effect and measurement error variances. In particular, procedures are given for (1) known confounding effect and measurement error variances (2) unknown measurement error variance but known confounding effect (3) unknown confounding effect and measurement error variances. The constants required to implement the procedures are shown to be obtainable from available FORTRAN programs and tables. Generalization to the case of strip-plot randomization restriction is considered.     

Bayesian model-averaged regularization for Gaussian graphical models

The graphical lasso has now become a useful tool to estimate high-dimensional Gaussian graphical models, but its practical applications suffer from the problem of choosing regularization parameters in a data-dependent way. In this article, we propose a model-averaged method for estimating sparse inverse covariance matrices for Gaussian graphical models. We consider the graphical lasso regularization path as the model space for Bayesian model averaging and use Markov chain Monte Carlo techniques for the regularization path point selection. Numerical performance of our method is investigated using both simulated and real datasets, in comparison with some state-of-art model selection procedures.     

Alternative estimators of the common regression matrix in two GMANOVA models under weighted quadratic losses

We consider the problem of estimating the common regression matrix of two GMANOVA models with different unknown covariance matrices under certain type of loss functions which include a weighted quadratic loss function as a special case. We consider a class of estimators, which contains the Graybill–Deal-type estimator proposed by Sugiura and Kubokawa (Ann. Inst. Statist. Math. 40 (1988) 119), and we give its risk representation via Kubokawa and Srivastava's (Ann. Statist. 27 (1999) 600; J. Multivariate Anal. 76 (2001) 138) identities when the error matrices follow the elliptically contoured distributions. Using the method similar to an approximate minimization of the unbiased risk estimate due to Stein (Studies in the Statistical Theory of Estimation, vol. 74, Nauka, Leningrad, 1977, p. 4), we obtain an alternative estimator to the Graybill–Deal-type estimator which was given under the normality assumption. However, it seems difficult to evaluate the risk of our proposed estimator analytically because of complex nature of its risk function. Instead, we conduct a Monte-Carlo simulation to evaluate the performance of our proposed estimator. The results indicate that our proposed estimator compares favorably with the Graybill–Deal-type estimator.     

Dynamic Equicorrelation

A new covariance matrix estimator is proposed under the assumption that at every time period all pairwise correlations are equal. This assumption, which is pragmatically applied in various areas of finance, makes it possible to estimate arbitrarily large covariance matrices with ease. The model, called DECO, involves first adjusting for individual volatilities and then estimating correlations. A quasi-maximum likelihood result shows that DECO provides consistent parameter estimates even when the equicorrelation assumption is violated. We demonstrate how to generalize DECO to block equicorrelation structures. DECO estimates for U.S. stock return data show that (block) equicorrelated models can provide a better fit of the data than DCC. Using out-of-sample forecasts, DECO and Block DECO are shown to improve portfolio selection compared to an unrestricted dynamic correlation structure.     

Analyzing multiple vector autoregressions through matrix-variate normal distribution with two covariance matrices

Abstract

This article proposes a new approach to analyze multiple vector autoregressive (VAR) models that render us a newly constructed matrix autoregressive (MtAR) model based on a matrix-variate normal distribution with two covariance matrices. The MtAR is a generalization of VAR models where the two covariance matrices allow the extension of MtAR to a structural MtAR analysis. The proposed MtAR can also incorporate different lag orders across VAR systems that provide more flexibility to the model. The estimation results from a simulation study and an empirical study on macroeconomic application show favorable performance of our proposed models and method.