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.     

Linearly admissible estimators of the common mean parameter under balanced loss function

Admissibility of linear estimators of the common mean parameter is investigated in the context of a linear model under balanced loss function. Sufficient and necessary conditions for linear estimators to be admissible in classes of homogeneous and non homogeneous linear estimators are obtained, respectively.     

Linearly admissible estimators of mean vector with respect to balanced loss function in multivariate statistics

The authors discuss the bias of the estimate of the variance of the overall effect synthesized from individual studies by using the variance weighted method. This bias is proven to be negative. Furthermore, the conditions, the likelihood of underestimation and the bias from this conventional estimate are studied based on the assumption that the estimates of the effect are subject to normal distribution with common mean. The likelihood of underestimation is very high (e.g. it is greater than 85% when the sample sizes in two combined studies are less than 120). The alternative less biased estimates for the cases with and without the homogeneity of the variances are given in order to adjust for the sample size and the variation of the population variance. In addition, the sample size weight method is suggested if the consistence of the sample variances is violated Finally, a real example is presented to show the difference by using the above three estimate methods.     

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.     

The exact risks of some pre-test and stein-type regression estimators umder balanced loss

In regression analysis we are often interested in using an estimator which is “precise” and which simultaneously provides a model with “good fit”, In this paper we consider the risk properties of several estimators of the regression coefficient vector "trader “balanced” loss, This loss function (Zellner, 1994) reflects both of the described attributes. Under a particular form of balanced loss, we derive the predictive risk of the pre-test estimator which results after a test for exact linear restrictions on the coefficient vector. The corresponding risks of Stein-rule and positive-part Stein-rale estimators are also established. The risks based on loss functions which allow only for estimation precision, or only for goodness of fit, are special cases of our results, and we draw appropriate comparisons, In particular, we show that some of the well-known results under (quadratic) precision-only loss are not robust to our generalization of the loss function     

On geometry of the set of admissible invariant quadratic estimators in balanced two variance components model

Two variance components model for which each invariant quadratic admissible estimator of a linear function of variance components (under quadratic loss function) is a linear combination of two quadratic forms,Z ₁,Z ₂, say, is considered. A setD={(d ₁,d ₂)′:d ₁ Z ₁+d ₂ Z ₂ is admissible} is described by giving formulae on the boundary ofD. Different forms of the setD are presented on figures.     

Limiting admissible estimators for variance components

Simultaneous estimation of the vector of the variance components for mixed and random models under the quadratic loss function is considered. For a large class of such models there are identified classes of admissible biased invariant quadratic estimators that are better than some admissible unbiased estimators. Numerous numerical results presented in the paper show that for many of the commonly used balanced models the improvements in the quadratic risk may be considerable over a large set of the parameter space.     

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.     

The performance of the adaptive optimal estimator under the extended balanced loss function

The adaptive optimal estimator of Farebrother (1975) is discussed by many authors, but the goodness of fitted model criterion that is used to investigate the performance of estimators is quite often ignored. Shalabh, Toutenburg, and Heumann (2009) proposed the extended balanced loss function in which the mean squared error and the Zellner's balanced loss function are just special cases of it. In this paper, we discuss the performance of the adaptive optimal estimator of Farebrother (1975) under the extended balanced loss function. Moreover, a Monte Carlo simulation experiment is conducted to examine the performance of the estimator in finite samples.     

A simulation study of ridge and other regression estimators

We consider a number of estimators of regression coefficients, all of generalized ridge, or 'shrinkage' type. Results of a simulation study indicate that with respect to two commonly used mean square error criteria, two ordinary ridge estimators, one proposed by Hoerl, Kennard and Baldwin, and the other introduced here, perform substantially better than both least squares and the other estimators discussed here     

The exact risk performance of a pre-test estimator in a heteroskedastic linear regression model under the balanced loss function

We examine the risk of a pre-test estimator for regression coefficients after a pre-test for homoskedasticity under the Balanced Loss Function (BLF). We show analytically that the two stage Aitken estimator is dominated by the pre-test estimator with the critical value of unity, even if the BLF is used. We also show numerically that both the two stage Aitken estimator and the pre-test estimator can be dominated by the ordinary least squares estimator when “goodness of fit” is regarded as more important than precision of estimation.     

On the admissibility of ordered poisson parameter under the entropy loss function

Let X ₁, X ₂be two independent Poisson random variables with means θ ₁and θ ₂respectively. Assume 0 ≤ θ ₁ ≤ θ ₂ ≤ ∞. The problem is estimation of the ordered parameters which has received considerable attention in statistical literature during the last two decades, In this paper the main portion of the study is devoted to the problem of estimating the smallest of the two ordered Poisson means, when it is known which population corresponds to each mean under the entropy loss function. An extension of the problem to the k-ordered Poisson means is also discussed.     

A note on Pitman's measure of closeness with balanced loss function

In this note, we consider the problem of estimating an unknown parameter θ in the sense of the Pitman's measure of closeness (PMC) using the balanced loss function (BLF). We show that the PMC comparison of estimators under the BLF can be reduced to the PMC comparison under the usual absolute error loss. The Pitman-closest estimators of the location and scale parameters under BLF are also characterized. Illustrative examples are given to show the broad range applications of the obtained results.     

Admissibility of estimators in the non-regular family under entropy loss function

In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities. The admissibility of generalized Bayes and Pitman estimates of functions of parameters have been established under entropy loss function.     

Further results on the two-stage hierarchial information (2shi) estimators in the linear regression models

The paper considers a class of 2SHI estimators for the linear regression models and provides some results regarding the dominance in quadratic loss of this class over the OLS and usual Stein-rule estimators.     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

Robustness of theil's mixed regression estimators

An efficient method for incorporating incomplete prior information in regression analysis was developed by Theil [1963]. In this paper we take up the estimator of coefficients given by this procedure and study its robustness to departures from normality of prior estimators of coefficients. The use of incomplete or biased prior information in regression analysis is also considered and a new estimator for the regression coefficient is suggested.     

Sufficient conditions for the admissibility under the LINEX loss function in non-regular case

Consider an estimation problem under the LINEX loss function in one-parameter non-regular distributions where the endpoint of the support depends on an unknown parameter. The purpose of this paper is to give sufficient conditions for a generalized Bayes estimator of a parametric function to be admissible. Also, it is shown that the main result in this paper is an extension of the quadratic loss case. Some examples are given.     

A simulation comparison of estimators for a regression coefficient under differential non-response

We consider the estimation of a regression coefficient in a linear regression when observations are missing due to nonresponse. Response is assumed to be determined by a nonobservable variable which is linearly related to an observable variable. The values of the observable variable are assumed to be available for the whole sample but the variable is not includsd in the regression relationship of interest . Several alternative estimators have been proposed for this situation under various simplifying assumptions. A sampling theory approach provides three alternative estimatrs by considering the observatins as obtained from a sub-sample, selected on the basis of the fully observable variable , as formulated by Nathan and Holt (1980). Under an econometric approach, Heckman (1979) proposed a two-stage (probit and OLS) estimator which is consistent under specificconditions. A simulation comparison of the four estimators and the ordinary least squares estimator , under multivariate normality of all the variables involved, indicates that the econometric approach estimator is not robust to departures from the conditions underlying its derivation, while two of the other estimators exhibit a similar degree of stable performance over a wide range of conditions. Simulations for a non-normal distribution show that gains in performance can be obtained if observations on the independent variable are available for the whole population.