On the relationship between the matrix operators,vech and vecd

We introduce a matrix operator, which we call “vecd” operator. This operator stacks up “diagonals” of a symmetric matrix. This operator is more convenient for some statistical analyses than the commonly used “vech” operator. We show an explicit relationship between the vecd and vech operators. Using this relationship, various properties of the vecd operator are derived. As applications of the vecd operator, we derive concise and explicit expressions of the Wald and score tests for equal variances of a multivariate normal distribution and for the diagonality of variance coefficient matrices in a multivariate generalized autoregressive conditional heteroscedastic (GARCH) model, respectively.     

Some results on commutation matrices,with statistical applications

The commutation matrix P _mn changes the order of multiplication of a Kronecker matrix product. The vec operator stacks columns of a matrix one under another in a single column. It is possible to express the vec of a Kronecker matrix product in terms of a Kronecker product of vecs of matrices. The commutation matrix plays an important role here. “Super-vec-operators” like vec A ? vec A vec ( A ? A ), and vec{( A ? A ) P _nn} are very convenient. Several of their properties are being studied. Both the traditional commutation matrix and vec operator and the newer concepts developed from these are applied to multivariate statistical and related problems.     

New HEAVY Models for Fat-Tailed Realized Covariances and Returns

ABSTRACT

We develop a new score-driven model for the joint dynamics of fat-tailed realized covariance matrix observations and daily returns. The score dynamics for the unobserved true covariance matrix are robust to outliers and incidental large observations in both types of data by assuming a matrix-F distribution for the realized covariance measures and a multivariate Student's t distribution for the daily returns. The filter for the unknown covariance matrix has a computationally efficient matrix formulation, which proves beneficial for estimation and simulation purposes. We formulate parameter restrictions for stationarity and positive definiteness. Our simulation study shows that the new model is able to deal with high-dimensional settings (50 or more) and captures unobserved volatility dynamics even if the model is misspecified. We provide an empirical application to daily equity returns and realized covariance matrices up to 30 dimensions. The model statistically and economically outperforms competing multivariate volatility models out-of-sample. Supplementary materials for this article are available online.     

Marriage and Family Periodical Indexes

Abstract

Auburn University Libraries wanted to reduce the amount of work devoted to current print issues in a manner acceptable to public service librarians and patrons. The Current Periodicals collection in the Ralph Brown Draughon (RBD) Library, the main library, has been reduced to only those periodical issues for which reliable online access is not available. Current issues that have online access are kept in the Acquisitions Department in a “closed stacks” arrangement. “Closed stacks” issues are available to patrons upon request. The experience at Auburn University demonstrates that reconfiguring processes for Current Periodicals involves collaboration between technical and public services with minimal difficulties and introduces new efficiencies. The librarians hope to save time and resources and reduce missing issues and claims. They are monitoring patron reaction to the change.     

Bayesian Inference on Multivariate Normal Covariance and Precision Matrices in a Star-Shaped Model with Missing Data

In this article, we study Bayesian estimation for the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in the star-shaped model with missing data. Based on a Cholesky-type decomposition of the precision matrix Ω = Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements, we develop the Jeffreys prior and a reference prior for Ψ. We then introduce a class of priors for Ψ, which includes the invariant Haar measures, Jeffreys prior, and reference prior. The posterior properties are discussed and the closed-form expressions for Bayesian estimators for the covariance matrix Σ and the precision matrix Ω are derived under the Stein loss, entropy loss, and symmetric loss. Some simulation results are given for illustration.     

Order-preserving Estimators and an Inequality on the Integration of Zonal Polynomial

ABSTRACT

Suppose X , p × p p.d. random matrix, has the distribution which depends on a p × p p.d. parameter matrix Σ and this distribution is orthogonally invariant. The orthogonally invariant estimator of Σ which has the eigenvalues of the same order as the eigenvalues of X is called order-preserving. We conjecture that a non-order-preserving estimator is dominated by modified order-preserving estimators with respect to the entropy (Stein's) loss function. We show that an inequality on the integration of zonal polynomial is sufficient for this conjecture. We also prove this inequality for the case p = 2.     

Mixing and moments properties of a non-stationary copula-based Markov process

Abstract

We provide conditions under which a non-stationary copula-based Markov process is geometric β-mixing and geometric ρ-mixing. Our results generalize some results of Beare who considers the stationary case. As a particular case we introduce a stochastic process, that we call convolution-based Markov process, whose construction is obtained by using the C-convolution operator which allows the increments to be dependent. Within this subclass of processes we characterize a modified version of the standard random walk where copulas and marginal distributions involved are in the same elliptical family. We study mixing and moments properties to identify the differences compared to the standard case.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

A note on joint mix random vectors

Abstract

This note studies the dependence of joint mix random vectors from the perspective of covariance matrix. We first provide two useful methods in simulations to construct joint mix for Normal distribution. Then, we propose to characterize joint mix by covariance matrix for general marginal distribution. We present some examples showing that our methodology could provide supplementary results to relevant studies in literature.     

SUFFICIENT CONDITIONS FOR A GEOMETRIC TAIL IN A QBD PROCESS WITH MANY COUNTABLE LEVELS AND PHASES

Abstract

In this paper, we present sufficient conditions, under which the stationary probability vector of a QBD process with both infinite levels and phases decays geometrically, characterized by the convergence norm η and the 1/η-left-invariant vector x of the rate matrix R. We also present a method to compute η and x based on spectral properties of the censored matrix of a matrix function constructed with the repeating blocks of the transition matrix of the QBD process. What makes this method attractive is its simplicity; finding η reduces to determining the zeros of a polynomial. We demonstrate the application of our method through a few interesting examples.     

A note on efficient simulation of multidimensional spatial autoregressive processes

In applications of spatial statistics, it is necessary to compute the product of some matrix W of spatial weights and a vector y of observations. The weighting matrix often needs to be adapted to the specific problems, such that the computation of Wy cannot necessarily be done with available R-packages. Hence, this article suggests one possibility treating such issues. The proposed technique avoids the computation of the matrix product by calculating each entry of Wy separately. Initially, a specific spatial autoregressive process is introduced. The performance of the proposed program is briefly compared to a basic program using the matrix multiplication.     

Regression model for surrogate data in high dimensional statistics

Abstract

This paper deals with the problem of estimating the regression of a surrogated scalar response variable given a functional random one. We construct an estimator of the regression operator by using, in addition to the available (true) response data, a surrogate data. We then establish some asymptotic properties of the constructed estimator in terms of the almost-complete and the quadratic mean convergences. Notice that the obtained results generalize a part of the results obtained in the finite dimensional framework. Finally, an illustration on the applicability of our results on both simulated data and a real dataset was realized. We have thus shown the superiority of our estimator on classical estimators when we are lacking complete data.     

Connecting Print Patrons to Electronic Holdings with Shelf Dummies

Abstract

After a two-year renovation, the Kent State University Music Library reopened as the Performing Arts Library. Before the closure, some serials were sent to the depository or reclassified within the library. In order to assist patrons browsing print journals or searching for articles in the stacks, library staff created shelf dummies detailing print and electronic holdings information as well as indexing for each title. Patron feedback has been positive, and the project has helped both library staff and patrons better understand accessibility and location of journals.     

A note on Whittaker–Henderson graduation: Bisymmetry of the smoother matrix

Abstract

Whittaker–Henderson (WH) graduation is a popular smoothing method that has been used for mortality table construction in the actuarial sciences and for the trend-cycle decomposition in time series econometrics. This paper proves that the smoother matrix of WH graduation is bisymmetric (i.e., symmetric centrosymmetric). This result implies, for example, that the first row of the smoother matrix is equivalent to the last row of it in reverse order. We also provide some related results.     

An extension of the multivariate Chebyshev's inequality to a random vector with a singular covariance matrix

ABSTRACT

We extend Chebyshev's inequality to a random vector with a singular covariance matrix. Then we consider the case of a multivariate normal distribution for this generalization.     

Adaptive LASSO estimation for ARDL models with GARCH innovations

ABSTRACT

In this paper, we show the validity of the adaptive least absolute shrinkage and selection operator (LASSO) procedure in estimating stationary autoregressive distributed lag(p,q) models with innovations in a broad class of conditionally heteroskedastic models. We show that the adaptive LASSO selects the relevant variables with probability converging to one and that the estimator is oracle efficient, meaning that its distribution converges to the same distribution of the oracle-assisted least squares, i.e., the least square estimator calculated as if we knew the set of relevant variables beforehand. Finally, we show that the LASSO estimator can be used to construct the initial weights. The performance of the method in finite samples is illustrated using Monte Carlo simulation.     

Matrix variate Macdonald distribution

Abstract

In this article, we generalize the univariate Macdonald distribution to the matrix case and give its derivation using matrix variate gamma distribution. We study several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of several functions of the Macdonald matrix. Some of these results are expressed in terms of special functions of matrix arguments and zonal polynomials.     

Perturbation analysis of Markov modulated fluid models

ABSTRACT

We consider perturbations of positive recurrent Markov modulated fluid models. In addition to the infinitesimal generator of the phases, we also perturb the rate matrix, and analyze the effect of those perturbations on the matrix of first return probabilities to the initial level. Our main contribution is the construction of a substitute for the matrix of first return probabilities, which enables us to analyze the effect of the perturbation under consideration.