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.     

A projector oriented approach to the best linear unbiased estimator

The paper provides a projector based approach to the best linear unbiased estimator (BLUE). By revisiting the so called generalized projection operator, introduced in Rao (J R Stat Soc Ser B Stat Methodol 36:442–448, 1974), a number of new formulae for BLUE is established. Furthermore, some attention is paid to the coincidence of the BLUE and the ordinary least squares estimator.     

Improving the best linear unbiased estimator for the scale parameter of symmetric distributions by using the absolute value of ranked set samples

Ranked set sampling is a cost efficient sampling technique when actually measuring sampling units is difficult but ranking them is relatively easy. For a family of symmetric location-scale distributions with known location parameter, we consider a best linear unbiased estimator for the scale parameter. Instead of using original ranked set samples, we propose to use the absolute deviations of the ranked set samples from the location parameter. We demonstrate that this new estimator has smaller variance than the best linear unbiased estimator using original ranked set samples. Optimal allocation in the absolute value of ranked set samples is also discussed for the estimation of the scale parameter when the location parameter is known. Finally, we perform some sensitivity analyses for this new estimator when the location parameter is unknown but estimated using ranked set samples and when the ranking of sampling units is imperfect.     

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.     

An extension of a rank criterion for the least squares estimator to be the best linear unbiased estimator

Among criteria for the least squares estimator in a linear model (y, Xβ, V) to be simultaneously the best linear unbiased estimator, one convenient for applications is that of Anderson (1971, 1972). His result, however, has been developed under assumptions of full column rank for X and nonsingularity for V. Subsequently, this result has been extended by Styan (1973) to the case when the restriction on X is removed. In this note, it is shown that also the restriction on V can be relaxed and, consequently, that Anderson's criterion is applicable to the general linear model without any rank assumptions at all.     

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.     

Efficiency of the generalized-difference-based weighted mixed almost unbiased two-parameter estimator in partially linear model

In this paper, a generalized difference-based estimator is introduced for the vector parameter β in partially linear model when the errors are correlated. A generalized-difference-based almost unbiased two-parameter estimator is defined for the vector parameter β. Under the linear stochastic constraint r = Rβ + e, we introduce a new generalized-difference-based weighted mixed almost unbiased two-parameter estimator. The performance of this new estimator over the generalized-difference-based estimator and generalized- difference-based almost unbiased two-parameter estimator in terms of the MSEM criterion is investigated. The efficiency properties of the new estimator is illustrated by a simulation study. Finally, the performance of the new estimator is evaluated for a real dataset.     

A pseudo‐empirical best linear unbiased prediction approach to small area estimation using survey weights

The authors develop a small area estimation method using a nested error linear regression model and survey weights. In particular, they propose a pseudo‐empirical best linear unbiased prediction (pseudo‐EBLUP) estimator to estimate small area means. This estimator borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. The proposed estimator also has a nice self‐benchmarking property. The authors also obtain an approximation to the model mean squared error (MSE) of the proposed estimator and a nearly unbiased estimator of MSE. Finally, they compare the proposed estimator with the EBLUP estimator and the pseudo‐EBLUP estimator proposed by Prasad & Rao (1999), using data analyzed earlier by Battese, Harter & Fuller (1988).     

Efficiency of an almost unbiased two-parameter estimator in linear regression model

This paper deals with parameter estimation in the linear regression model and an almost unbiased two-parameter estimator is introduced. The performance of this new estimator over the ordinary least-squares estimator and the two-parameter estimator [M.R. Özkale and S. Kaçiranlar, The restricted and unrestricted two-parameter estimator, Comm. Statist. Theory Methods 36 (2007), pp. 2707–2725] in terms of scalar mean-squared error criterion is investigated and a simulation study is done.     

Optimum linear unbiased estimation of scale parameter by absolute values of order statistics in symmetric distributions

A new estimator of the scale parameter by the optimum linear combination of absolute values of order statistics in symmetric location-scale families with known location parameter (without loss of generality assumed to be zero) from complete and Type II censored samples is introduced and is termed as optimum unbiased absolute estimator of the scale parameter. The new estimator of the scale parameter is compared with the corresponding best linear unbiased estimator (BLUE) in the rectangular and normal distributions. Generally it is found that the new estimator is more efficient than the BLUE.     

On the restricted almost unbiased two-parameter estimator in linear regression model

Özkale and Kaçiranlar introduced the restricted two-parameter estimator (RTPE) to deal with the well-known multicollinearity problem in linear regression model. In this paper, the restricted almost unbiased two-parameter estimator (RAUTPE) based on the RTPE is presented. The quadratic bias and mean-squared error of the proposed estimator is discussed and compared with the corresponding competitors in literatures. Furthermore, a numerical example and a Monte Carlo simulation study are given to explain some of the theoretical results.     

The BLUE's covariance matrix revisited: A review

In this paper we comment on and review some unexpected but interesting features of the BLUE (best linear unbiased estimator) of the expectation vector in the general linear model and in particular, the BLUE's covariance matrix. Most of these features appear in the literature but are rather scattered or hidden.     

Positive-rule stein-type almost unbiased ridge estimator in linear regression model

Abstract

In this article, when it is suspected that regression coefficients may be restricted to a subspace, we discuss the parameter estimation of regression coefficients in a multiple regression model. Then, in order to improve the preliminary test almost ridge estimator, we study the positive-rule Stein-type almost unbiased ridge estimator based on the positive-rule stein-type shrinkage estimator and almost unbiased ridge estimator. After that, quadratic bias and quadratic risk values of the new estimator are derived and compared with some relative estimators. And we also discuss the option of parameter k. Finally, we perform a real data example and a Monte Carlo study to illustrate theoretical results.     

On the best linear unbiased estimator and the linear sufficiency of a general growth curve model

Growth curve models are used to analyze repeated measures data (longitudinal data), which are functions of time. In this paper, some necessary and sufficient conditions for linear function B₁YB₂ to be the best linear unbiased estimator (BLUE) of estimable functions X₁ΘX₂ (or K₁ΘK₂) under the general growth curve model were established. In addition, the representations of BLUE(K₁ΘK₂) (or BLUE(X₁ΘX₂)) were derived when the conditions are satisfied. Two special notions of linear sufficiency with respect to the general growth curve model are given in the end. The findings of this paper enrich some known results in the literature.     

Nonlinear unbiased estimation in the linear regression model with nonnormal disturbances

In the application of the linear regression model there continues to be wide-spread use of the Least Squares Estimator (LSE) due to its theoretical optimality. For example, it is well known that the LSE is the best unbiased estimator under normality while it remains best linear unbiased estimator (BLUE) when the normality assumption is dropped. In this paper we extend an approach given in Knautz (1993) that allows improvement of the LSE in the context of nonnormal and nonsymmetric error distributions. It will be shown that there exist linear plus quadratic (LPQ) estimators, consisting of linear and quadratic terms in the dependent variable, which dominate the LS estimator, depending on second, third and fourth moments of the error distribution. A simulation study illustrates that this remains true if the moments have to be estimated from the data. Computation of confidence intervals using bootstrap methods reveal significant improvement compared with inference based on the LS especially for nonsymmetric distributions of the error term.     

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.     

Optimum linear unbiased estimation of the scale parameter by absolute values of order statistics in the double exponential and the double weibull distributions

The first two moments and product moments of absolute values of order statistics are obtained for the double exponential and the double Weibull distributions. In both of the distributions an optimum linear unbiased estimator of the scale parameter, by absolute values of the order statistics, is obtained from complete and censored samples of size n=3(1)10. It is found that the new estimator is generally more efficient than the best linear unbiased estimator (BLUE) of the scale parameter by order statistcs in both of the distributions.     

Calibrated linear unbiased estimators in finite population sampling

Sugden and Smith [2002. Exact linear unbiased estimation in survey sampling. J. Stat. Plann. Inf. 102, 25–38] and Rao [2002. Discussion of “Exact linear unbiased estimation in survey sampling”. J. Stat. Plann. Inf. 102, 39–40] suggested some useful techniques of deriving a linear unbiased estimator of a finite population total by modifying a given linear estimator. In this paper we suggest various generalizations of their results. In particular, we search for estimators satisfying the calibration property with respect to a related auxiliary variable and obtain some new calibrated unbiased ratio-type estimators for arbitrary sampling designs. We also explore a few properties of one of the estimators suggested in Sugden and Smith [2002. Exact linear unbiased estimation in survey sampling. J. Stat. Plann. Inf. 102, 25–38].     

On the almost unbiased generalized liu estimator and unbiased estimation of the bias and mse

In this paper, we derive the almost unbiased generalized Liu estimator and examine an exact unbiased estimator of the bias and mean squared error of the feasible generalized Liu estimator . We compare the almost unbiased generalized Liu estimator (AUGLE) with the generalized Liu estimator (GLE) and with the ordinary least squares estimator (OLSE).     

On the almost unbiased ridge regression estimator

The purpose of this paper is two-fold. One is to compare the almost unbiased generalized ridge regression (AUGRR) estimator proposed by Singh, Chaubey and Dwivedi (1986) with the generalized ridge regression (GRR) estimator and with the ordinary least squares (OLS) estimator in terms of the mean squared error criterion. Second is to examine small sample properties of the operational almost unbiased ordinary ridge regression (AUORR) estimator by Monte Carlo experiments.