Vec and vech operators for matrices,with some uses in jacobians and multivariate statistics

The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one-to-one transformation X → Y is then ∣∣?(vecX)/?(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ?(vechX)/?(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics.     

Best quadratic and positive semidefinite unbiased estimation of the variance matrix of the multivariate normal distribution

A BQPUE (best quadratic and positive semidefinite unbiased estimator) of the matrix V for the distribution vec X^′∽N_np(vec M^′, U?V) is being given. It is unique, although still depending on U and M. When U = I and M = (μ,…,μ), we get a well-known (unique) result not depending on M.     

A basic lemma and the analysis of block and Kronecker product designs

In this paper the analysis of the class of block designs whose C matrix can be expressed in terms of the Kronecker product of some elementary matrices is considered. The analysis utilizes a basic result concerning the spectral decomposition of the Kronecker product of symmetric matrices in terms of the spectral decomposition of the component matrices involved in the Kronecker product. The property (A) of Kurkjian and Zelen (1963) is generalised and the analysis of generalised property (A) designs is given. It is proved that a design is balanced factorially if and only if it is a generalised property (A) design. A method of analysis of Kronecker product block designs whose component designs are equi-replicate and proper is also suggested.     

Second-Order Covariance Matrix Formula for Heteroskedastic Generalized Linear Models

For the first time, we provide a matrix formula for second-order covariances of maximum likelihood estimates in heteroskedastic generalized linear models, thus generalizing the results of Cordeiro (2004 Cordeiro , G. M. ( 2004 ). Second-order covariance matrix of maximum likelihood estimates in generalized linear models . Statist. Probab. Lett. 66 : 153 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro et al. (2006 Cordeiro , G. M. , Barroso , L. P. , Botter , D. A. (2006). Covariance matrix formula for generalized linear models with unknown dispersion. Commun. Statist. Theor. Meth. 35:113–120.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) related to the generalized linear models with known and unknown dispersion parameter, respectively. The covariance matrix formula does not involve cumulants of log-likelihood derivatives and can be easily obtained using simple matrix operations. We apply our main result to a simple model. Some simulations show that the second-order covariances can be quite pronounced in small to moderate samples. The usual covariances of the maximum likelihood estimates can be corrected by these second-order covariances.     

More on the Kronecker Structured Covariance Matrix

In this article, the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly focused is the estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The suggested estimation generalizes the procedure proposed by Srivastava et al. (2008 Srivastava , M. , von Rosen , T. , von Rosen , D. ( 2008 ). Models with a Kronecker product covariance structure: Estimation and testing Mathemat. Meth. Statist. 17 : 357 – 370 .[Crossref] , [Google Scholar]) for a separable covariance matrix. The restrictions imposed by separability and double separability are also discussed.     

Linear model estimation errors with estimated design parameters

The problem of error estimation of parameters b in a linear model,Y = Xb+ e, is considered when the elements of the design matrix X are functions of an unknown ‘design’ parameter vector c. An estimated value c is substituted in X to obtain a derived design matrix [Xtilde]. Even though the usual linear model conditions are not satisfied with [Xtilde], there are situations in physical applications where the least squares solution to the parameters is used without concern for the magnitude of the resulting error. Such a solution can suffer from serious errors.

This paper examines bias and covariance errors of such estimators. Using a first-order Taylor series expansion, we derive approximations to the bias and covariance matrix of the estimated parameters. The bias approximation is a sum of two terms:One is due to the dependence between ? and Y; the other is due to the estimation errors of ? and is proportional to b, the parameter being estimated. The covariance matrix approximation, on the other hand, is composed of three omponents:One component is due to the dependence between ? and Y; the second is the covariance matrix ∑_b corresponding to the minimum variance unbiased b, as if the design parameters were known without error; and the third is an additional component due to the errors in the design parameters. It is shown that the third error component is directly proportional to bb'. Thus, estimation of large parameters with wrong design matrix [Xtilde] will have larger errors of estimation. The results are illustrated with a simple linear example.     

Shelf-life and its estimation in drug stability studies

One important property of any drug product is its stability over time. Drug stability studies are routinely carried out in the pharmaceutical industry in order to measure the degradation of an active pharmaceutical ingredient of a drug product. One important study objective is to estimate the shelf-life of the drug; the estimated shelf-life is required by the US Food and Drug Administration to be printed on the package label of the drug. This involves a suitable definition of the true shelf-life and the construction of an appropriate estimate of the true shelf-life. In this paper, the true shelf-life T_β is defined as the time point at which 100β% of all the individual dosage units (e.g. tablets) of the drug have the active ingredient content no less than the lowest acceptable limit L, where β and L are prespecified constants. The value of T_β depends on the parameters of the assumed degradation model of the active ingredient content and so is unknown. A lower confidence bound T?_β for T_β is then provided and used as the estimated shelf-life of the drug.     

Solving Survo puzzles using matrix combinatorial products

We investigate combinatorial matrix problems that are related to restricted integer partitions. They arise from Survo puzzles, where the basic task is to fill an m×n table by integers 1, 2,?…?, mn, so that each number appears only once, when the column sums and the row sums are fixed. We present a new computational method for solving Survo puzzles with binary matrices that are recoded and combined using the Hadamard, Kronecker, and Khatri–Rao products. The idea of our method is based on using the matrix interpreter and other data analytic tools of Survo R, which represents the newest generation of the Survo computing environment, recently implemented as a multiplatform, open source R package. We illustrate our method with detailed examples.     

Stable spectral factorization with applications to the estimation of time series models

A method for generating a miniphase and inveitible spectral factor from an unstable v × v full rank polynomial matrix is proposed. The zeros inside the unit circle are reflected through the boundary |z|=1 using closed form algebraic manipulations. Also included in the procedure is a technique foi determining the stability of a polynomial operator that does not require the explicit construction of the determinant al equation. Application of the technique is illustrated and the implementation of the method in the statistical context of system estimation is discussed.     

A-optimal designs for heteroscedastic multifactor regression models

Abstract

This paper searches for A-optimal designs for Kronecker product and additive regression models when the errors are heteroscedastic. Sufficient conditions are given so that A-optimal designs for the multifactor models can be built from A-optimal designs for their sub-models with a single factor. The results of an efficiency study carried out to check the adequacy of the products of optimal designs for uni-factor marginal models when these are used to estimate different multi-factor models are also reported.     

New difference-based estimator of parameters in semiparametric regression models

The paper introduces a new difference-based Liu estimator β?_Ldiff=([Xtilde]′[Xtilde]+I)^?1([Xtilde]′[ytilde]+η β?_diff) of the regression parameters β in the semiparametric regression model, y=Xβ+f+?. Difference-based estimator, β?_diff=([Xtilde]′[Xtilde])^?1[Xtilde]′[ytilde] and difference-based Liu estimator are analysed and compared with respect to mean-squared error (mse) criterion. Finally, the performance of the new estimator is evaluated for a real data set. Monte Carlo simulation is given to show the improvement in the scalar mse of the estimator.     

Covariance Matrix Formula for Exponential Family Nonlinear Models

This article gives a matrix formula for second-order covariances of maximum likelihood estimators in exponential family nonlinear models, thus generalizing the result of Cordeiro (2004 Cordeiro , G. M. ( 2004 ). Second-order covariance matrix of maximum likelihood estimates in generalized linear models . Statist. Probab. Lett. 66 : 153 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]) valid for generalized linear models with known dispersion parameter. Some simulations show that the second-order covariances for exponential family nonlinear models can be quite pronounced in small to moderate sample sizes.     

Projector operators in the multivariate Zyskind-Martin model

Summary Two quadratic formsS _H andS _E for a testable hypothesis and for an error in the multivariate Zyskind-Martin model with singular covariance matrix are expressed by means of projector operators. Thus the results for the multivariate standard model with identity covariance matrix given by Humak (1977) and Christensen (1987, 1991) are generalized for the case of Zyskind-Martin model. Special cases of our results are formulae forS _H andS _E in Aitken's (1935) model. In the case of general Gauss-Markoff modelS _H andS _E can also be expressed by means of projector operators for some subclasses of testable hypotheses. For these hypotheses, testing in Gauss-Markoff model is equivalent to testing in a Zyskind-Martin model.     

Classification of Higher-order Data with Separable Covariance and Structured Multiplicative or Additive Mean Models

Although devised in 1936 by Fisher, discriminant analysis is still rapidly evolving, as the complexity of contemporary data sets grows exponentially. Our classification rules explore these complexities by modeling various correlations in higher-order data. Moreover, our classification rules are suitable to data sets where the number of response variables is comparable or larger than the number of observations. We assume that the higher-order observations have a separable variance-covariance matrix and two different Kronecker product structures on the mean vector. In this article, we develop quadratic classification rules among g different populations where each individual has κth order (κ ≥2) measurements. We also provide the computational algorithms to compute the maximum likelihood estimates for the model parameters and eventually the sample classification rules.     

Berry–Esseen type bounds in heteroscedastic errors-in-variables model

ABSTRACT

Consider the heteroscedastic partially linear errors-in-variables (EV) model y_i = x_iβ + g(t_i) + ε_i, ξ_i = x_i + μ_i (1 ? i ? n), where ε_i = σ_ie_i are random errors with mean zero, σ²_i = f(u_i), (x_i, t_i, u_i) are non random design points, x_i are observed with measurement errors μ_i. When f( · ) is known, we derive the Berry–Esseen type bounds for estimators of β and g( · ) under {e_i,?1 ? i ? n} is a sequence of stationary α-mixing random variables, when f( · ) is unknown, the Berry–Esseen type bounds for estimators of β, g( · ), and f( · ) are discussed under independent errors.     

Pseudo confidence regions for the solution of a multivariate linear functional relationship

The problem of finding confidence regions (CR) for a q-variate vector γ given as the solution of a linear functional relationship (LFR) Λγ = μ is investigated. Here an m-variate vector μ and an m × q matrix Λ = (Λ₁, Λ₂,…, Λ_q) are unknown population means of an m(q+1)-variate normal distribution $\text{N}{}_{\mathrm{m(q+1)}}\text{(ζΩ?Σ)}$, where ζ′ = (μ′, Λ₁′, Λ₂′,…, Λ_q′Σ is an unknown, symmetric and positive definite m × m matrix and Ω is a known, symmetric and positive definite (q+1) × (q+1) matrix and ? denotes the Kronecker product. This problem is a generalization of the univariate special case for the ratio of normal means.A CR for γ with level of confidence 1 ? α, is given by a quadratic inequality, which yields the so-called ‘pseudo’ confidence regions (PCR) valid conditionally in subsets of the parameter space. Our discussion is focused on the ‘bounded pseudo’ confidence region (BPCR) given by the interior of a hyperellipsoid. The two conditions necessary for a BPCR to exist are shown to be the consistency conditions concerning the multivariate LFR. The probability that these conditions hold approaches one under ‘reasonable circumstances’ in many practical situations. Hence, we may have a BPCR with confidence approximately 1 ? α. Some simulation results are presented.     

PROPERTIES OF THE PATTERNED VECH OPERATOR

The vec operator arranges the columns of a matrix one below the other. When the matrix is symmetric such elements are not distinct but an extraction of only the distinct elements on or below the diagonal forms the operation denoted by vech. For other types of patterned matrices a ‘patterned vech’ operator is defined. The transformations from vech to vec are not uniquely defined. Here we examine properties of linear transformations which overcome the lack of uniqueness and develop properties of such linear transformations.     

Selection of Variables in Multivariate Regression Models for Large Dimensions

The Akaike information criterion, AIC, and Mallows’ C _p statistic have been proposed for selecting a smaller number of regressors in the multivariate regression models with fully unknown covariance matrix. All of these criteria are, however, based on the implicit assumption that the sample size is substantially larger than the dimension of the covariance matrix. To obtain a stable estimator of the covariance matrix, it is required that the dimension of the covariance matrix is much smaller than the sample size. When the dimension is close to the sample size, it is necessary to use ridge-type estimators for the covariance matrix. In this article, we use a ridge-type estimators for the covariance matrix and obtain the modified AIC and modified C _p statistic under the asymptotic theory that both the sample size and the dimension go to infinity. It is numerically shown that these modified procedures perform very well in the sense of selecting the true model in large dimensional cases.     

Effect of outliers on the GFI quality adjustment index in structural equation model and proposal of alternative indices

This work is intended to suggest modifications in the construction of the GFI index using robust methods for estimating the unrestricted sample covariance matrix, leading to new indices called GFI_(MCD) and GFI_(MVE). The validation of this proposal was made using Monte Carlo simulation methods, considering differences between the unrestricted sample covariance matrix and those imposed by the structural model, and different numbers of outliers generated by distributions with deviations from symmetry and excess kurtosis. It was concluded that for larger samples size (n ? 100), given that the outliers are from distributions that are symmetrical, the GFI_(MCD) and GFI_(MVE) present similar results, including samples with high percentages of outliers.     

Using a Truncated C p Statistic for Variable Selection in Multiple Linear Regression

In multiple linear regression analysis each lower-dimensional subspace L of a known linear subspace M of ? ⁿ corresponds to a non empty subset of the columns of the regressor matrix. For a fixed subspace L, the C _p statistic is an unbiased estimator of the mean square error if the projection of the response vector onto L is used to estimate the expected response. In this article, we consider two truncated versions of the C _p statistic that can also be used to estimate this mean square error. The C _p statistic and its truncated versions are compared in two example data sets, illustrating that use of the truncated versions may result in models different from those selected by standard C _p .