Estimation in Singular Linear Models with Stochastic Linear Restrictions

This article generalizes the ordinary mixed estimator (OME) in theory, and obtains the estimator of the unknown regression parameters in singular linear models with stochastic linear restrictions: singular mixed estimator (SME). We also give some properties of SME obtained in this article, and prove that it is superior to unrestricted least squared estimator (LSE) in singular linear models in the sense of the covariance matrix and generalized mean square error (GMSE). After that, we also have a discussion about the two-stage estimator of SME. The result we give in this article could be regarded as generalizations of both OME and unrestricted LSE at the same time.     

On singular elliptical models

The probability density function (pdf) ofsingular elliptical distributions is represented as an integralseries of singular normal distributions. Explicit formulas for the pdf and the cdf of the generalized Chi-square distribution are derived under singular elliptical assumptions extending the result of Díaz-García [(2002). Singular elliptical distribution: density and applications. Commun. Stat.—Theory Methods 31:665–681]. Applications are given of the proposed result for singular mixedmodels.     

Admissible linear estimation in singular linear models

The admissibility results of Rao (1976), proved in the context of a nonsingular covariance matrix, are exteneded to the situation where the covariance matrix is singular. Admi.s s Lb Le linear estimators in the Gauss-Markoff model are characterized and admis-sibility of the best linear unbiased estimator is investigated.     

Admissible linear estimation in singular linear models with respect to a restricted parameter set

The admissibility results of Hoffmann (1977), proved in the context of a nonsingular covariance matrix are extended to the situation where the covariance matrix is singular. Admissible linear estimators in the Gauss-Markoff model are characterised and admissibility of the Best Linear Unbiased Estimator is investigated.     

An identity for reflexive g-inverses in the context with BLU estimators

In sampling from finite populations conditional minimax estimators are equivalent to BLU estimators. This has been shown by Gabler (1988). If the covariance matrix is singular the BLU estimator can be represented in two different ways.     

On the natural restrictions in the singular Gauss–Markov model

A Gauss–Markov model is said to be singular if the covariance matrix of the observable random vector in the model is singular. In such a case, there exist some natural restrictions associated with the observable random vector and the unknown parameter vector in the model. In this paper, we derive through the matrix rank method a necessary and sufficient condition for a vector of parametric functions to be estimable, and necessary and sufficient conditions for a linear estimator to be unbiased in the singular Gauss–Markov model. In addition, we give some necessary and sufficient conditions for the ordinary least-square estimator (OLSE) and the best linear unbiased estimator (BLUE) under the model to satisfy the natural restrictions.      

The General Multivariate Gauss—Markov Model of the Incomplete Block Design

The aim of the paper is to generalize testing and estimation for the multivariate standard incomplete block model (Rao & Mitra, 1971a) to the general multivariate Gauss—Markov incomplete block model with singular covariance matrix. The results of this paper can be applied to particular cases of the multivariate Gauss—Markov incomplete block model, including the Zyskind—Martin model.     

Least squares theory for possibly singular models

In a recent paper, Scobey (1975) observed that the usual least squares theory can be applied even when the covariance matrix σ²V of Y in the linear model Y = Xβ + e is singular by choosing the Moore-Penrose inverse (V+XX′)⁺ instead of V^-1 when V is nonsingular. This result appears to be wrong. The appropriate treatment of the problem in the singular case is described.     

The Density Functions of the Singular Quaternion Normal Matrix and the Singular Quaternion Wishart Matrix

In this article, we give the density functions of the singular quaternion normal matrix and the singular quaternion Wishart matrix. Furthermore, we also give the density functions of certain singular quaternion β-matrix and the singular quaternion F-matrix in terms of the density function of the singular quaternion Wishart matrix and hypergeometric functions of quaternion matrix argument.     

Linearly admissible estimators on linear functions of regression coefficient under balanced loss function

Linearly admissible estimators on linear functions of regression coefficient are studied in a singular linear model and balanced loss when the design matrix has not full column rank. The sufficient and necessary conditions for linear estimators to be admissible are obtained respectively in homogeneous and inhomogeneous classes.     

Numerical treatment of restricted gauss-markov model 1

The singular value decomposition (SVD) has been widely used in the ordinary linear model and other statistical problems. In this paper, we shall introduce the generalized singular value decomposition (GSVD) of any two matrices X and H having the same number of columns to moti-vate the numerical treatment of large scale restricted Gauss-Markov model (y,Xβ\Hβ = r,σ²1), a situation to reveal the relationship (or restriction) existing among the parameters of the model. Many approaches to restricted linear model are already available. Those approaches apply the generalized inverse of matrices and emphasize the the-oretical solution of the problem rather than the development of efficient and numerical stable algorithm for the computation of estimators. The possible merit of the method present here might lie in the facts that they directly lead to an efficient, numerically stable and easily programmed algorithm for     

Singular Ridge Regression With Stochastic Constraints

This article considers the estimation of the restricted ridge regression parameter in singular models. The problem is commenced with considering elliptically contoured equality constrained and then followed by proposing the preliminary test estimator. Along with proposing some important properties of this estimator, a real example satisfying the elliptical assumption is also given to bring the problem into a noticeable issue.     

Recovery of inter-block information: extensions in a two variance component model

For a two variance component mixed linear model, it is shown that under suitable conditions there exists a nonlinear unbiased estimator that is better than a best linear unbiased estimator defined with respect to a given singular covariance matrix. It is also shown how this result applies to improving on intra-block estimators and on estimators like the unweighted means estimator in a random one-way model.     

Mean square error matrix comparisons of estimators in linear regression

The aim of this paper is to provide criteria which allow to compare two estimators of the parameter vector in the linear regression model with respect to their mean square error matrices, where the main interest is focussed on the case when the difference of the covariance matrices is singular. The results obtained are applied to equality restricted and pretest estimators.     

Estimation of parameters in the singular gauss-markoff model

Consider the Gauss-Markoff model (Y, Xβ, σ² V) in the usual notation (Rao, 1973a, p. 294). If V is singular, there exists a matrix N such that N'Y has zero covariance. The minimum variance unbiased estimator of an estimable parametric function p'β is obtained in the wider class of (non-linear) unbiased estimators of the form f(N'Y) + Y'g(N'Y) where f is a scalar and g is a vector function.     

Feasible algorithm for linear mixed model for massive data

This article studies computation problem in the context of estimating parameters of linear mixed model for massive data. Our algorithms combine the factored spectrally transformed linear mixed model method with a sequential singular value decomposition calculation algorithm. This combination solves the operation limitation of the method and also makes this algorithm feasible to big dataset, especially when the data has a tall and thin design matrix. Our simulation studies show that our algorithms make the calculation of linear mixed model feasible for massive data on ordinary desktop and have same estimating accuracy with the method based on the whole data.     

Categorical information and the singular linear model

C. R. Rao (1978) discusses estimation for the common linear model in the case that the variance matrix σ² Q has known singular form Q . In the more general context of inference, this model exhibits certain special features and illustrates how information concerning unknowns can separate into a categorical component and a statistical component. The categorical component establishes that certain parameters are known in value and thus are not part of the statistical inference.     

A note on the analysis of association in cross-classifications having ordered categories

The use of the singular value decomposition of a matrix in the analysis of cross-classifications having ordered categories la presented? Utilizing some matrix properties of a two-way contingency table, the singular value decomposition approach la applied on models such as the null association, uniform association and row-column effect models discussed recently in the literature. Some properties of estimates resulting from the singular value decomposition approach are discussed     

A solvable singular control problem driven by a jump diffusion process with applications

We consider a class of singular control problems driven by a double exponential jump diffusion process, which come from the reversible investment problem. In some interesting cases (e.g., the running cost function is given by the so-called Cobb-Douglas production function), we give the explicit solutions to the singular control problem by using the connection between singular control and optimal switching. We solve a collection of consistent optimal switching problems and yield the explicit solution for the singular control problem. We then give an application to a particular inventory control problem in a single random period.     

Wishart and Pseudo-Wishart distributions under elliptical laws and related distributions in the shape theory context

In this paper, we determine the density of a singular elliptically contoured matrix. From this, the study of Wishart and Pseudo-Wishart distributions, whether central or noncentral, whether singular or nonsingular, is extended to the case of elliptical models. Some elated distributions are studied in the context of shape theory. Particular attention is paid to singular size-and-shape and size-and-shape cone densities.