On equality of ordinary least squares estimator,best linear unbiased estimator and best linear unbiased predictor in the general linear model

The equality of ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE) and best linear unbiased predictor (BLUP) in the general linear model with new observations is investigated through matrix rank method, some new necessary and sufficient conditions are given.     

Estimation of linear models for field experiments

General mixed linear models for experiments conducted over a series of sltes and/or years are described. The ordinary least squares (OLS) estlmator is simple to compute, but is not the best unbiased estimator. Also, the usuaL formula for the varlance of the OLS estimator is not correct and seriously underestimates the true variance. The best linear unbiased estimator is the generalized least squares (GLS) estimator. However, t requires an inversion of the variance-covariance matrix V, whlch is usually of large dimension. Also, in practice, V is unknown.

We presented an estlmator [Vcirc] of the matrix V using the estimators of variance components [for sites, blocks (sites), etc.]. We also presented a simple transformation of the data, such that an ordinary least squares regression of the transformed data gives the estimated generalized least squares (EGLS) estimator. The standard errors obtained from the transformed regression serve as asymptotic standard errors of the EGLS estimators. We also established that the EGLS estlmator is unbiased.

An example of fitting a linear model to data for 18 sites (environments) located in Brazil is given. One of the site variables (soil test phosphorus) was measured by plot rather than by site and this established the need for a covariance model such as the one used rather than the usual analysis of variance model. It is for this variable that the resulting parameter estimates did not correspond well between the OLS and EGLS estimators. Regression statistics and the analysis of variance for the example are presented and summarized.     

Two Kinds of Restricted Estimators in Singular Linear Regression

In this article, the parameter estimators in singular linear model with linear equality restrictions are considered. The restricted root estimator and the generalized restricted root estimator are proposed and some properties of the estimators are also studied. Furthermore, we compare them with the restricted unified least squares estimator and show their sufficient conditions under which their superior over the restricted unified least squares estimator in terms of mean squares error, and discuss the choice of the unknown parameters of the generalized restricted root estimator.     

Simple least squares estimation versus best linear unbiased prediction

Necessary and sufficient conditions are developed for the simple least squares estimator to coincide with the best linear unbiased predictor. The conditions obtained are valid for a general linear model and are generalizations of the condition given by Watson (1972). Also, as a preliminary result, a new representation of the best linear unbiased predictor is established.     

Some overall properties of seemingly unrelated regression models

Seemingly unrelated regression models are extensions of linear regression models which allow correlated errors between equations. Estimations and inferences of singular seemingly unrelated regression models involve some complicated operations of the given matrices in the models and their generalized inverses. In this study, we characterize the consistency, natural restrictions, estimability of parametric functions under a singular seemingly unrelated regression model using the matrix rank method. We also derive necessary and sufficient conditions for the ordinary least squares estimators and the best linear unbiased estimators of parametric functions to be equal under seemingly unrelated regression models.     

Moment properties of estimators for a type 1 extreme-value regression model

A regression model is considered in which the response variable has a type 1 extreme-value distribution for smallest values. Bias approximations for the maximum likelihood estimators are pivm and a bias reduction estimator for the scale parameter is proposed. The small sample moment properties of the maximum likelihood estimators are compared with the properties of the ordinary least squares estimators and the best linear unbiased estimators based on order statistics for grouped data.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

Good ridge estimators based on prior information

Ridge regression is re-examined and ridge estimators based on prior information are introduced. A necessary and sufficient condition is given for such ridge estimators to yield estimators of every nonnull linear combination of the regression coefficients with smaller mean square error than that of the Gauss-Markov best linear unbiased estimator.     

Use of Quasi-Ranges in the Estimation of Scale Parameter of a Symmetric Distribution

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the scale parameter of a symmetric distribution using quasi-ranges is considered. We also prove a sufficient condition for the non negativity of the scale estimator obtained by the above method. Further, we obtain necessary and sufficient conditions for the derived estimators to be constant multiple of the sample range.     

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.      

Some conditions for optimality of the almost unbiased estimators of regression coefficients

The purpose of this paper is to examine the asymptotic properties of the operational almost unbiased estimator of regression coefficients which includes almost unbiased ordinary ridge estimator a s a special case. The small distrubance approximations for the bias and mean square error matrix of the estimator are derived. As a consequence, it is proved that, under certain conditions, the estimator is more efficient than a general class of estimators given by Vinod and Ullah (1981). Also it is shown that, if the ordinary ridge estimator (ORE) dominates the ordinary least squares estimator then the almost unbiased ordinary ridge estimator does not dominate ORE under the mean square error criterion.     

BETWEEN OLSE AND BLUE

Several estimators of X β under the general Gauss–Markov model are considered. Particular attention is paid to those estimators whose efficiency lies between that of the ordinary least squares estimator and that of the best linear unbiased estimator.     

Criteria for the equality between ordinary least squares and best linear unbiased estimators under certain linear models

A new necessary and sufficient condition is derived for the equality between the ordinary least-squares estimator and the best linear unbiased estimator of the expectation vector in linear models with certain specific design matrices. This condition is then applied to special cases involving one-way and two-way classification models.     

On the weighted least-squares,the ordinary least-squares and the best linear unbiased estimators under a restricted growth curve model

Necessary and sufficient conditions are given for a restricted growth curve model to be consistent. The general expressions of the weighted least-squares estimators (WLSEs), the ordinary least-squares estimators (OLSEs) and the best linear unbiased estimator (BLUE) under this model are also derived. Moreover, some algebraic and statistical properties of these estimators are presented by rank method.     

The Equality of the Ordinary Least Squares Estimator and the Best Linear Unbiased Estimator

It is well known that the ordinary least squares estimator of Xβ in the general linear model E _y = Xβ, cov y = σ² V, can be the best linear unbiased estimator even if V is not a multiple of the identity matrix. This article presents, in a historical perspective, the development of the several conditions for the ordinary least squares estimator to be best linear unbiased. Various characterizations of these conditions, using generalized inverses and orthogonal projectors, along with several examples, are also given. In addition, a complete set of references is provided.     

Superiority of the r–k Class Estimator Over Some Estimators In A Linear Model

In regression analysis, to overcome the problem of multicollinearity, the r ? k class estimator is proposed as an alternative to the ordinary least squares estimator which is a general estimator including the ordinary ridge regression estimator, the principal components regression estimator and the ordinary least squares estimator. In this article, we derive the necessary and sufficient conditions for the superiority of the r ? k class estimator over each of these estimators under the Mahalanobis loss function by the average loss criterion. Then, we compare these estimators with each other using the same criterion. Also, we suggest to test to verify if these conditions are indeed satisfied. Finally, a numerical example and a Monte Carlo simulation are done to illustrate the theoretical results.     

A Note on the Use of Some Functions of Spacings in the Estimation of Common Scale Parameter of Several Symmetric Distributions

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the common scale parameter of several symmetric distributions using some functions of spacings of all observations taken from individual samples. We also proved a sufficient condition for the non negativity of the common scale estimator obtained by the above method. Furthermore, we obtained necessary and sufficient conditions for the derived estimators to be constant multiple of the sum of first and last spacings of the pooled sample.     

Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.     

The relative efficiency of the restricted estimators in linear regression models

This paper deals with the problem of multicollinearity in a multiple linear regression model with linear equality restrictions. The restricted two parameter estimator which was proposed in case of multicollinearity satisfies the restrictions. The performance of the restricted two parameter estimator over the restricted least squares (RLS) estimator and the ordinary least squares (OLS) estimator is examined under the mean square error (MSE) matrix criterion when the restrictions are correct and not correct. The necessary and sufficient conditions for the restricted ridge regression, restricted Liu and restricted shrunken estimators, which are the special cases of the restricted two parameter estimator, to have a smaller MSE matrix than the RLS and the OLS estimators are derived when the restrictions hold true and do not hold true. Theoretical results are illustrated with numerical examples based on Webster, Gunst and Mason data and Gorman and Toman data. We conduct a final demonstration of the performance of the estimators by running a Monte Carlo simulation which shows that when the variance of the error term and the correlation between the explanatory variables are large, the restricted two parameter estimator performs better than the RLS estimator and the OLS estimator under the configurations examined.     

Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.