Bounds for the largest Mahalanobis distance

Upper and lower bounds for the magnitude of the largest Mahalanobis distance, calculated from n multivariate observations of length p, are derived. These bounds are multivariate extensions of corresponding bounds that arise for the most deviant Z-score calculated from a univariate sample of size n. The approach taken is to pose optimization problems in a mathematical context and to employ variational methods to obtain solutions. The attainability of the bounds obtained is demonstrated. Bounds for related quantities (elements of the “hat matrix”) are also derived.     

Wishart ratios with dependent structure: New members of the bimatrix beta type IV

In multivariate statistics under normality, the problems of interest are random covariance matrices (known as Wishart matrices) and “ratios” of Wishart matrices that arise in multivariate analysis of variance (MANOVA) (see 24). The bimatrix variate beta type IV distribution (also known in the literature as bimatrix variate generalised beta; matrix variate generalization of a bivariate beta type I) arises from “ratios” of Wishart matrices. In this paper, we add a further independent Wishart random variate to the “denominator” of one of the ratios; this results in deriving the exact expression for the density function of the bimatrix variate extended beta type IV distribution. The latter leads to the proposal of the bimatrix variate extended F distribution. Some interesting characteristics of these newly introduced bimatrix distributions are explored. Lastly, we focus on the bivariate extended beta type IV distribution (that is an extension of bivariate Jones’ beta) with emphasis on P(X₁<X₂) where X₁ is the random stress variate and X₂ is the random strength variate.     

Moving zeros among matrices

We investigate the zero-patterns that can be created by unitary similarity in a given matrix, and the zero-patterns that can be created by simultaneous unitary similarity in a given sequence of matrices. The latter framework allows a “simultaneous Hessenberg” formulation of Pati’s tridiagonal result for 4 × 4 matrices. This formulation appears to be a strengthening of Pati’s theorem. Our work depends at several points on the simplified proof of Pati’s result by Davidson and Djokovi?. The Hessenberg approach allows us to work with ordinary similarity and suggests an extension from the complex to arbitrary algebraically closed fields. This extension is achieved and related results for 5 × 5 and larger matrices are formulated and proved.     

A method for proving polynomial enumeration formulas

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by “explaining” their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to prove a new refinement of the Bender-Knuth (ex-)Conjecture, which easily implies the Bender-Knuth (ex-)Conjecture itself. This is probably the most elementary way to prove this result currently known. Furthermore we adapt our method to q-polynomials, which allows us to derive generating function results as well. Finally we use this method to give another proof for the enumeration of semistandard tableaux of a fixed shape which differs from our proof of the Bender-Knuth (ex-)Conjecture in that it is a multivariate application of our method.     

Preserving multivariate dispersion: An application to the Wishart distribution

Univariate dispersive ordering has been extensively characterized by many authors over the last two decades. However, the multivariate version lacks extensive analysis. In this paper, sufficient and necessary conditions are given to preserve the strong multivariate dispersion order through properties of the corresponding transformation. Finally, these results are applied to the Wishart distribution which can be viewed as “the spread of the dispersion”.     

Finite sample tail behavior of multivariate location estimators

A finite sample performance measure of multivariate location estimators is introduced based on “tail behavior”. The tail performance of multivariate “monotone” location estimators and the halfspace depth based “non-monotone” location estimators including the Tukey halfspace median and multivariate L-estimators is investigated. The connections among the finite sample performance measure, the finite sample breakdown point, and the halfspace depth are revealed. It turns out that estimators with high breakdown point or halfspace depth have “appealing” tail performance. The tail performance of the halfspace median is very appealing and also robust against underlying population distributions, while the tail performance of the sample mean is very sensitive to underlying population distributions. These findings provide new insights into the notions of the halfspace depth and breakdown point and identify the important role of tail behavior as a quantitative measure of robustness in the multivariate location setting.     

Simultaneous modelling of the Cholesky decomposition of several covariance matrices

A method for simultaneous modelling of the Cholesky decomposition of several covariance matrices is presented. We highlight the conceptual and computational advantages of the unconstrained parameterization of the Cholesky decomposition and compare the results with those obtained using the classical spectral (eigenvalue) and variance-correlation decompositions. All these methods amount to decomposing complicated covariance matrices into “dependence” and “variance” components, and then modelling them virtually separately using regression techniques. The entries of the “dependence” component of the Cholesky decomposition have the unique advantage of being unconstrained so that further reduction of the dimension of its parameter space is fairly simple. Normal theory maximum likelihood estimates for complete and incomplete data are presented using iterative methods such as the EM (Expectation-Maximization) algorithm and their improvements. These procedures are illustrated using a dataset from a growth hormone longitudinal clinical trial.     

The rational canonical form via the splitting field

It is clear that a given rational canonical form can be further resolved to a Jordan canonical form with entries from the splitting field of its minimal polynomial. Conversely, with an a priori knowledge of the existence and uniqueness of the rational canonical form of a matrix with entries from a general field, one can modify its Jordan canonical form in the splitting field of its minimal polynomial to construct its rational canonical form in the original field. No author has tried this converse with the a priori existence–uniqueness condition removed. It is feared that “in many occasions when, after a result has been established for a matrix with entries in a given field, considered as a matrix with entries in a finite extension of that field, we cannot go back from the extension field to get the desired information in the original field” [I.N. Herstein, Topics in Algebra, Ginn and Company, Waltham, MA, 1964 (pp. 262–263)]. The present paper removes this a priori condition and uses a “symmetrization” to “integrate” back the Jordan canonical form of a matrix to its rational canonical form in the original field.     

Shrinkage estimation in the frequency domain of multivariate time series

In this paper on developing shrinkage for spectral analysis of multivariate time series of high dimensionality, we propose a new nonparametric estimator of the spectral matrix with two appealing properties. First, compared to the traditional smoothed periodogram our shrinkage estimator has a smaller L₂ risk. Second, the proposed shrinkage estimator is numerically more stable due to a smaller condition number. We use the concept of “Kolmogorov” asymptotics where simultaneously the sample size and the dimensionality tend to infinity, to show that the smoothed periodogram is not consistent and to derive the asymptotic properties of our regularized estimator. This estimator is shown to have asymptotically minimal risk among all linear combinations of the identity and the averaged periodogram matrix. Compared to existing work on shrinkage in the time domain, our results show that in the frequency domain it is necessary to take the size of the smoothing span as “effective sample size” into account. Furthermore, we perform extensive Monte Carlo studies showing the overwhelming gain in terms of lower L₂ risk of our shrinkage estimator, even in situations of oversmoothing the periodogram by using a large smoothing span.     

Modified Whittle estimation of multilateral models on a lattice

In the estimation of parametric models for stationary spatial or spatio-temporal data on a d-dimensional lattice, for d?2, the achievement of asymptotic efficiency under Gaussianity, and asymptotic normality more generally, with standard convergence rate, faces two obstacles. One is the “edge effect”, which worsens with increasing d. The other is the possible difficulty of computing a continuous-frequency form of Whittle estimate or a time domain Gaussian maximum likelihood estimate, due mainly to the Jacobian term. This is especially a problem in “multilateral” models, which are naturally expressed in terms of lagged values in both directions for one or more of the d dimensions. An extension of the discrete-frequency Whittle estimate from the time series literature deals conveniently with the computational problem, but when subjected to a standard device for avoiding the edge effect has disastrous asymptotic performance, along with finite sample numerical drawbacks, the objective function lacking a minimum-distance interpretation and losing any global convexity properties. We overcome these problems by first optimizing a standard, guaranteed non-negative, discrete-frequency, Whittle function, without edge-effect correction, providing an estimate with a slow convergence rate, then improving this by a sequence of computationally convenient approximate Newton iterations using a modified, almost-unbiased periodogram, the desired asymptotic properties being achieved after finitely many steps. The asymptotic regime allows increase in both directions of all d dimensions, with the central limit theorem established after re-ordering as a triangular array. However our work offers something new for “unilateral” models also. When the data are non-Gaussian, asymptotic variances of all parameter estimates may be affected, and we propose consistent, non-negative definite estimates of the asymptotic variance matrix.     

The positive definite completion problem revisited

In [R. Grone, C.R. Johnson, E. Sa, H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984) 109-124] the positive definite (semi-) completion problem in which the underlying graph is chordal was solved. For the positive definite case, the process was constructive and the completion was obtained by completing the partial matrix an entry at a time. For the positive semidefinite case, they obtained completions of a particular sequence of partial positive definite matrices with the same underlying graph and noted that there is a convergent subsequence of these completions that converges to the desired completion. Here, in the chordal case, we provide a constructive solution, based entirely on matrix/graph theoretic methods, to the positive (semi-)definite completion problem. Our solution associates a specific tree (called the “clique tree” [C.R. Johnson, M. Lundquist, Matrices with chordal inverse zero-patterns, Linear and Multilinear Algebra 36 (1993) 1-17]) with the (chordal) graph of the given partial positive (semi-)definite matrix. This tree structure allows us to complete the matrix a “block at a time” as opposed to an “entry at a time” (as in Grone et al. (1984) for the positive definite case). In Grone et al. (1984), using complex analytic techniques, the completion for the positive definite case was shown to be the unique determinant maximizing completion and was shown to be the unique completion that has zeros in its inverse in the positions corresponding to the unspecified entries of the partial matrix. Here, we show the same using only matrix/graph theoretic tools.     

New affine measures of symmetry for convex bodies

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”.To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are “far apart”. In particular, we give an example of a convex body whose centroid and Santaló point are “far apart”.     

On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3], Cressie and Read (1988) [4], Menéndez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a “restricted” estimator or an “unrestricted” estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.     

Mixed effects regression trees for clustered data

This paper presents an extension of the standard regression tree method to clustered data. Previous works extending tree methods to accommodate correlated data are mainly based on the multivariate repeated-measures approach. We propose a “mixed effects regression tree” method where the correlated observations are viewed as nested within clusters rather than as vectors of multivariate repeated responses. The proposed method can handle unbalanced clusters, allows observations within clusters to be split, and can incorporate random effects and observation-level covariates. We implemented the proposed method using a standard tree algorithm within the framework of the expectation-maximization (EM) algorithm. The simulation results show that the proposed regression tree method provides substantial improvements over standard trees when the random effects are non negligible. A real data example is used to illustrate the method.     

Who's afraid of reduced-rank parameterizations of multivariate models? Theory and example

Reduced-rank restrictions can add useful parsimony to coefficient matrices of multivariate models, but their use is limited by the daunting complexity of the methods and their theory. The present work takes the easy road, focusing on unifying themes and simplified methods. For Gaussian and non-Gaussian (GLM, GAM, mixed normal, etc.) multivariate models, the present work gives a unified, explicit theory for the general asymptotic (normal) distribution of maximum likelihood estimators (MLE). MLE can be complex and computationally hard, but we show a strong asymptotic equivalence between MLE and a relatively simple minimum (Mahalanobis) distance estimator. The latter method yields particularly simple tests of rank, and we describe its asymptotic behavior in detail. We also examine the method's performance in simulation and via analytical and empirical examples.     

Unitarily achievable zero patterns and traces of words in A and A

Our primary objective is to identify a natural and substantial problem about unitary similarity on arbitrary complex matrices: which 0-patterns may be achieved for any given n-by-n complex matrix via some unitary similarity of it. To this end, certain restrictions on “achievable” 0-patterns are mentioned, both positional and, more important, on the maximum number of achievable 0’s. Prior results fitting this general question are mentioned, as well as the “first” unresolved pattern (for 3-by-3 matrices!). In the process a recent question is answered.A closely related additional objective is to mention the best known bound for the maximum length of words necessary for the application of Specht’s theorem about which pairs of complex matrices are unitarily similar, which seems not widely known to matrix theorists. In the process, we mention the number of words necessary for small size matrices.     

Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors

We develop optimal rank-based procedures for testing affine-invariant linear hypotheses on the parameters of a multivariate general linear model with elliptical VARMA errors. We propose a class of optimal procedures that are based either on residual (pseudo-)Mahalanobis signs and ranks, or on absolute interdirections and lift-interdirection ranks, i.e., on hyperplane-based signs and ranks. The Mahalanobis versions of these procedures are strictly affine-invariant, while the hyperplane-based ones are asymptotically affine-invariant. Both versions generalize the univariate signed rank procedures proposed by Hallin and Puri (J. Multivar. Anal. 50 (1994) 175), and are locally asymptotically most stringent under correctly specified radial densities. Their AREs with respect to Gaussian procedures are shown to be convex linear combinations of the AREs obtained in Hallin and Paindaveine (Ann. Statist. 30 (2002) 1103; Bernoulli 8 (2002) 787) for the pure location and purely serial models, respectively. The resulting test statistics are provided under closed form for several important particular cases, including multivariate Durbin-Watson tests, VARMA order identification tests, etc. The key technical result is a multivariate asymptotic linearity result proved in Hallin and Paindaveine (Asymptotic linearity of serial and nonserial multivariate signed rank statistics, submitted).     

When the positivity of the leading principal minors implies the positivity of all principal minors of a matrix

An n-by-n real matrix A enjoys the “leading implies all” (LIA) property, if, whenever D   is a diagonal matrix such that A+D$A+D$ has positive leading principal minors (PMs), all PMs of A are positive. Symmetric and Z-matrices are known to have this property. We give a new class of matrices (“mixed matrices”) that both unifies and generalizes these two classes and their special diagonal equivalences by also having the LIA property. “Nested implies all” (NIA) is also enjoyed by this new class.     

Non-white Wishart ensembles

We consider non-white Wishart ensembles , where X is a p×N random matrix with i.i.d. complex standard Gaussian entries and Σ is a covariance matrix, with fixed eigenvalues, close to the identity matrix. We prove that the largest eigenvalue of such random matrix ensembles exhibits a universal behavior in the large-N limit, provided Σ is “close enough” to the identity matrix. If not, we identify the limiting distribution of the largest eigenvalues, focusing on the case where the largest eigenvalues almost surely exit the support of the limiting Marchenko-Pastur's distribution.