Efficiency of the generalized difference-based Liu estimators in semiparametric regression models with correlated errors

In this paper, a generalized difference-based estimator is introduced for the vector parameter β in the semiparametric regression model when the errors are correlated. A generalized difference-based Liu estimator is defined for the vector parameter β in the semiparametric regression model. Under the linear nonstochastic constraint Rβ=r, the generalized restricted difference-based Liu estimator is given. The risk function for the β?_GRD(η) associated with weighted balanced loss function is presented. The performance of the proposed estimators is evaluated by a simulated data set.     

Ridge regression methodology in partial linear models with correlated errors

In this article, a generalized restricted difference-based ridge estimator is defined for the vector parameter in a partial linear model when the errors are dependent. It is suspected that some additional linear constraints may hold on to the whole parameter space. The estimator is a generalization of the well-known restricted least-squares estimator and is confined to the (affine) subspace which is generated by the restrictions. The risk functions of the proposed estimators are derived under balanced loss function. Finally, the performance of the new estimators is evaluated by a simulated data set.     

On Bayes Linear Unbiased Estimator Under the Balanced Loss Function

In this paper, the Bayes linear unbiased estimator (Bayes LUE) is derived under the balanced loss function. Moreover, the superiority of Bayes LUE over ordinary least square estimator is studied under the mean square error matrix criterion and Pitman closeness criterion. Furthermore, we compare Bayes LUE under the balanced loss function with Bayes LUE under the quadratic loss function.     

Stein-rule estimation under an extended balanced loss function

This paper extends the balanced loss function to a more general setup. The ordinary least squares estimator (OLSE) and Stein-rule estimator (SRE) are exposed to this general loss function with quadratic loss structure in a linear regression model. Their risks are derived when the disturbances in the linear regression model are not necessarily normally distributed. The dominance of OLSE and SRE over each other and the effect of departure from normality assumption of disturbances on the risk property are studied.     

A note on the admissibility of relative to the linex loss function

The purpose of this note is to give a correct proof of a result in Rojo (1987). Let 2 be the mean of a random sample of size n from a normal 2 distribution with unknown mean 0 and known variance o . Following earlier work by Zellner (1986), Rojo (1987) considered the admissibility of the linear estimator c; + d relative to Variants (1975) asymmetric LINEX loss function     

Risk performance of a pre-test estimator for normal variance with the Stein-variance estimator under the LINEX loss function

In this paper, we derive the exact formula of the risk function of a pre-test estimator for normal variance with the Stein-variance (PTSV) estimator when the asymmetric LINEX loss function is used. Fixing the critical value of the pre-test to unity which is a suggested critical value in some sense, we examine numerically the risk performance of the PTSV estimator based on the risk function derived. Our numerical results show that although the PTSV estimator does not dominate the usual variance estimator when under-estimation is more severe than over-estimation, the PTSV estimator dominates the usual variance estimator when over-estimation is more severe. It is also shown that the dominance of the PTSV estimator over the original Stein-variance estimator is robust to the extension from the quadratic loss function to the LINEX loss function.     

Bayesian Estimation of Regression Coefficients Under Extended Balanced Loss Function

Appreciating the desirability of simultaneously using both the criteria of goodness of fitted model and clustering of estimates around true parameter values, an extended version of the balanced loss function is presented and the Bayesian estimation of regression coefficients is discussed. The thus obtained optimal estimator is then compared with the least squares estimator and posterior mean vector with respect to the criteria like posterior expected loss, Bayes risk, bias vector, mean squared error matrix and risk function.     

New biased estimators under the LINEX loss function

In this paper, using the asymmetric LINEX loss function we derive the risk function of the generalized Liu estimator and almost unbiased generalized Liu estimator. We also examine the risk performance of the feasible generalized Liu estimator and feasible almost unbiased generalized Liu estimator when the LINEX loss function is used.     

Comparison of estimators for heteroskedastic models

Ordinary least squares (OLS) yield inefficient parameter estimates and inconsistent estimates of the covariance matrix in case of heteroskedastic errors. Robinson's adaptive estimator and the Cragg estimator avoid any explicit parameterization of heteroskedasticity, and reduce the danger of misspecification. A small Monte Carlo experiment is performed to compare the behavior of the adaptive estimator with the performance of the Cragg estimator. The Monte Carlo experiment includes simulations of the Generalized Least Squares (GLS) estimator. Indeed, an interesting question is how more sophisticated techniques, like the adaptive estimator, compare with GLS when the latter relies on an incorrect specification of the heteroskedastic process. It turns out that the regression parameters, when estimated adaptively, display small mean squared errors and great efficiency in case of medium or high heteroskedasticity. The covariance matrix, instead, is better estimated by the Cragg estimator or by GLS based on a misspecified error term, since the adaptive estimator overpredicts the standard errors of the regression parameters.     

The admissible minimax estimator in Gauss–Markov model under a balanced loss function

In this paper, we investigate the admissible minimax estimator (AME) of regression coefficient in Gauss–Markov model under a balanced loss function. In the class of homogeneous linear estimators, we obtain the AME under two occasions, respectively. We also prove that the AME is a shrinkage estimator of the best linear unbiased estimator (BLUE). Furthermore, we prove that the AME dominates the BLUE under certain conditions.     

An Appreciation of Balanced Loss Functions Via Regret Loss

We examine balanced loss functions, which account for both estimation error and goodness of fit (or proximity to a “target” estimator), in terms of their regret losses, providing new insight and interpretations. This also shows a connection between quadratic balanced loss and usual quadratic loss, which easily converts frequentist and Bayesian results for quadratic loss to related results for quadratic balanced loss and vice versa. Some implications of these results for Stein-rule estimators under linear regression are discussed. We also examine regret losses corresponding to several non-quadratic balanced loss functions.     

Efficient Estimation and Robust Inference of Linear Regression Models in the Presence of Heteroscedastic Errors and High Leverage Points

It is common for linear regression models that the error variances are not the same for all observations and there are some high leverage data points. In such situations, the available literature advocates the use of heteroscedasticity consistent covariance matrix estimators (HCCME) for the testing of regression coefficients. Primarily, such estimators are based on the residuals derived from the ordinary least squares (OLS) estimator that itself can be seriously inefficient in the presence of heteroscedasticity. To get efficient estimation, many efficient estimators, namely the adaptive estimators are available but their performance has not been evaluated yet when the problem of heteroscedasticity is accompanied with the presence of high leverage data. In this article, the presence of high leverage data is taken into account to evaluate the performance of the adaptive estimator in terms of efficiency. Furthermore, our numerical work also evaluates the performance of the robust standard errors based on this efficient estimator in terms of interval estimation and null rejection rate (NRR).     

Linearized Ridge Regression Estimator in Linear Regression

In this article, we aim to study the linearized ridge regression (LRR) estimator in a linear regression model motivated by the work of Liu (1993). The LRR estimator and the two types of generalized Liu estimators are investigated under the PRESS criterion. The method of obtaining the optimal generalized ridge regression (GRR) estimator is derived from the optimal LRR estimator. We apply the Hald data as a numerical example and then make a simulation study to show the main results. It is concluded that the idea of transforming the GRR estimator as a complicated function of the biasing parameters to a linearized version should be paid more attention in the future.     

Shrinkage estimator of regression model under asymmetric loss

This article investigates the performance of the shrinkage estimator (SE) of the parameters of a simple linear regression model under the LINEX loss criterion. The risk function of the estimator under the asymmetric LINEX loss is derived and analyzed. The moment-generating functions and the first two moments of the estimators are also obtained. The risks of the SE have been compared numerically with that of pre-test and least-square estimators (LSEs) under the LINEX loss criterion. The numerical comparison reveals that under certain conditions the LSE is inadmissible, and the SE is the best among the three estimators.     

A study of quadratic inference functions with alternative weighting matrices

Quadratic inference function (QIF) is an alternative methodology to the popular generalized estimating equations (GEE) approach, it does not involve direct estimation of the correlation parameter, and thus remains optimal even if the working correlation structure is misspecified. The idea is to represent the inverse of the working correlation matrix by a linear combination of some basis matrices. In this article, we present a modification of QIF with a robust variance estimator of the extended score function. Theoretical and numerical results show that the modified QIF attains better efficiency and achieves better small sample performance than the original QIF method.     

On the minimum mean squared error estimators in a regression model

In this paper we analyze the properties of two estimators oroposed by Farebrother (1975) for linear regression models.     

Preliminary test and Bayes Estimation of a Location Parameter Under Blinex Loss

In this article, the preliminary test estimator is considered under the BLINEX loss function. The problem under consideration is the estimation of the location parameter from a normal distribution. The risk under the null hypothesis for the preliminary test estimator, the exact risk function for restricted maximum likelihood and approximated risk function for the unrestricted maximum likelihood estimator, are derived under BLINEX loss and the different risk structures are compared to one another both analytically and computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary test estimator under LINEX loss and it is shown that the LINEX expected loss is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined and compared to the classical preliminary test estimator.     

Simultaneous estimation of the multivariate normal mean under balanced loss function

This paper considers simultaneous estimation of multivariate normal mean vector using Zellner's(1994) balanced loss function which is defined as follows:

where 0 < w < 1 and for i = 1,…,p and j = 1,…,n, X_ij is distributed as normal with mean θi and variance 1. It is shown that the sample mean, X, is admissible when p <3. For p ≥3, we obtain that James-Stein type estimator which has uniformly smaller risk than that of sample mean X.     

A multiparameter extension of two stage modified shrinkage estimator

A multiparameter extension is made to modified two stage shrinkage estimator proposed by Handa andKambo (1990), For aparticular class of shrinkage estimator, the local optimality of the extended modified estimator is shown over the two stage shrinkage estimator defined by Bhattacharya and Prakasa Rao(1990) in terms of quadratic loss.     

New types of shrinkage estimators of Poisson means under the normalized squared error loss

In estimating p( ? 2) independent Poisson means, Clevenson and Zidek (1975) have proposed a class of estimators that shrink the unbiased estimator to the origin and dominate the unbiased one under the normalized squared error loss. This class of estimators was subsequently enlarged in several directions. This article deals with the problem and proposes new classes of dominating estimators using prior information pertinently. Dominance is shown by partitioning the sample space into disjoint subsets and averaging the loss difference over each subset. Estimation of several Poisson mean vectors is also discussed. Further, simultaneous estimation of Poisson means under order restriction is treated and estimators which dominate the isotonic regression estimator are proposed for some types of order restrictions.