Choice of best linear in the guass-markoff model with a singular dispersion matrix

In this paper we obtain the complete class of representations and useful subclasses of MV-UB-LE and MV-MB-LE (minimum variance unbiased and minimum bias linear estimators) of linear parametric functions in the Gauss-Markoff model (Y,Xβ, σ ²V) when V is possibly singular.     

Admissible linear estimators in the general Gauss-Markov model

This paper derives a complete characterization of estimators that are admissible for a given identifiable vector of parametric functions among the set of linear estimators under the general Gauss-Markov model with a dispersion matrix possibly singular. The characterization obtained implies some corollaries, which are then compared with the results known in the literature.     

On O.L.S. estimation in singular linear models with general positive definite dispersion matrix

A condition in Graybill [1976] for the O.L.S.estimator to be B.L.U.E. in a linear model with positive definite dispersion not necessarily proportional to the identity matrix is extended to cover the case of a singular linear model.     

On pre-test estimation of parametric functions in the general Gauss-Markov model with quadratic risk

The approach to preliminary test estimation based on comparing the weighted quadratic risk function of two competing estimators of β under the linear regression model {y,Xβ, σ² I} is extended to the case when a given vector of parametric functions κ=Kβ is to be estimated under the general Gauss-Markov 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.     

Some matrix results related to a partitioned singular linear model

Consider the linear model (y, Xβ V), where the model matrix X may not have a full column rank and V might be singular. In this paper we introduce a formula for the difference between the BLUES of Xβ under the full model and the model where one observation has been deleted. We also consider the partitioned linear regression model where the model matrix is (X₁: X₂) the corresponding vector of unknown parameters being (β′₁ : β′₂)′. We show that the BLUE of X₁ β₁ under a specific reduced model equals the corresponding BLUE under the original full model and consider some interesting consequences of this result.     

Recursive improvement of estimates in a Gauss-Markov model with linear restrictions

The minimum-dispersion linear unbiased estimator of a set of estimable functions in a general Gauss-Markov model with double linear restrictions is considered. The attention is focused on developing a recursive formula in which an initial estimator, obtained from the unrestricted model, is corrected with respect to the restrictions successively incorporated into the model. The established formula generalizes known results developed for the simple Gauss-Markov model.     

Estimation of the seemingly unrelated regression model when the error covariance matrix is singular

A two-step generalized least-squares (GLS) estimator proposed by Zellner for seemingly unrelated regression (SUR) models is implementable when the estimated covariance matrix of the errors in the SUR system is non-singular. Violating the premise of non-singularity is a common problem among many applications in economics, business and management. We present methods of resolving this problem and propose an efficient procedure. The simulation study shows that the estimator of Haff performs better for small-sized observations, whereas the estimator of Ullah and Racine performs better for larger sized observations. Furthermore, the Ullah-Racine estimate is simple to calculate and easy to use. The empirical analysis involves the study of the diffusion processes of videocassette recorders across different geographic regions in the US, which exhibits a singular covariance matrix. The empirical results show that the procedures efficiently deal with the problem and provide plausible estimation results.     

Optimality of blue's in a general linear model with incorrect design matrix

We consider the Gauss-Markoff model (Y,X₀β,σ²V) and provide solutions to the following problem: What is the class of all models (Y,Xβ,σ²V) such that a specific linear representation/some linear representation/every linear representation of the BLUE of every estimable parametric functional p'β under (Y,X₀β,σ²V) is (a) an unbiased estimator, (b) a BLUE, (c) a linear minimum bias estimator and (d) best linear minimum bias estimator of p'β under (Y,Xβ,σ²V)? We also analyse the above problems, when attention is restricted to a subclass of estimable parametric functionals.     

Influence diagnostic analysis in the possibly heteroskedastic linear model with exact restrictions

The local influence method has proven to be a useful and powerful tool for detecting influential observations on the estimation of model parameters. This method has been widely applied in different studies related to econometric and statistical modelling. We propose a methodology based on the Lagrange multiplier method with a linear penalty function to assess local influence in the possibly heteroskedastic linear regression model with exact restrictions. The restricted maximum likelihood estimators and information matrices are presented for the postulated model. Several perturbation schemes for the local influence method are investigated to identify potentially influential observations. Three real-world examples are included to illustrate and validate our methodology.     

Two group inspection-based control charts for dispersion matrix

In this article, we propose the two control charts, i.e. the ‘VMAX Group Runs’ (VMAX-GR) and ‘VMAX Modified Group Runs’ (VMAX-MGR) control charts based on the bivariate normal processes, for monitoring the covariance matrix. The proposed charts give the faster detection of a process change and have better diagnostic feature. It is verified that the VMAX-GR and the VMAX-MGR charts give a significant reduction in the out-of-control ‘Average Run Length’ (ARL) in the zero state, as well as in the steady state, as compared to the synthetic control chart based on the VMAX statistic and the generalized variance |S| chart.     

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.     

On some tests in manocova model with different dispersion matrices

In this article we have considered the problem of testing linear hypothesis in MANOCOVA model with different dispersion Matrices by two test procedures in the line of Anderson (1963) and fihargava CI971). Also efficiencies of the two tests have often compared.     

A replacement model with general age-dependent failure rates

In this paper we derive optimal replacement rules for a repairable system where both the operating and repair times are random variables having general age-dependent failure rates. First a finite horizon setting is studied. The optimal replacement problem for the infinite horizon case follows from the finite horizon setting by taking limits. Several submodels will also be considered.     

A doubly restricted exponential dispersion model

In this paper, we propose and develop a doubly restricted exponential dispersion model, i.e. a varying dispersion generalized linear model with two sets of restrictions, a set of linear restrictions for the mean response, and at the same time, for another set of linear restrictions for the dispersion of the distribution. This model would be useful to consider several situations where it is necessary to control/analyze drug-doses, active effects in factorial experiments, mean-variance relationships, among other situations. A penalized likelihood function is proposed and developed in order to achieve the restricted parameters and to develop the inferential results. Several special cases from the literature are commented on. A simply restricted varying dispersion beta regression model is exemplified by means of real and simulated data. Satisfactory and promising results are found.     

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.