The density function and the MSE dominance of the pre-test estimator in a heteroscedastic linear regression model with omitted variables

In this paper, we consider a heteroscedastic linear regression model with omitted variables. We derive the density function of the pre-test estimator consisting of the two-stage Aitken estimator (2SAE) and the ordinary least squares estimator (OLSE) after the pre-test for homoscedasticity. We also derive the first two moments based on the density function and show the sufficient condition for the pre-test estimator to dominate the 2SAE in terms of the MSE. Our numerical evaluations show that when this sufficient condition does not hold and when the magnitude of the specification error is large, the pre-test estimator can be dominated by the 2SAE, and further, the 2SAE can be dominated by the OLSE.     

The exact distribution and density functions of the stein-type estimator for normal variance

In this paper, we derive the exact distribution and density functions of the Stein-type estimator for the normal variance. It is shown by numerical evaluation that the density function of the Stein-type estimator is unimodal and concentrates around the mode more than that of the usual estimator.     

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.     

The exact density and distribution functions of the inequality constrained and pre-test estimators

We consider a bivariate normal linear regression model with an inequality restriction imposed on one of the regression coefficients. The exact analytical expressions for the density and distribution functions of the inequality constrained and pre-test estimators are derived and numerically evaluated. The implications of using the inequality constrained and pre-test estimators in confidence interval construction are also discussed and explored.     

PMSE dominance of the positive-part shrinkage estimator in a regression model with proxy variables

Consider a linear regression model with some relevant regressors are unobservable. In such a situation, we estimate the model by using the proxy variables as regressors or by simply omitting the relevant regressors. In this paper, we derive the explicit formula of predictive mean squared error (PMSE) of a general family of shrinkage estimators of regression coefficients. It is shown analytically that the positive-part shrinkage estimator dominates the ordinary shrinkage estimator even when proxy variables are used in place of the unobserved variables. Also, as an example, our result is applied to the double k-class estimator proposed by Ullah and Ullah (Double k-class estimators of coefficients in linear regression. Econometrica. 1978;46:705–722). Our numerical results show that the positive-part double k-class estimator with proxy variables has preferable PMSE performance.     

A modified estimator of error variance in a partly linear model

We consider estimation of the variance of the error in a partly linear model Our resulting estimator has smaller asymptotic variance than the estimators given in literature.     

On the sampling performance of an inequality pre-test estimator of the regression error variance under LINEX loss

We consider the estimation of the error variance of a linear regression model where prior information is available in the form of an (uncertain) inequality constraint on the coefficients. Previous studies on this and other related problems use the squared error loss in comparing estimator’s performance. Here, by adopting the asymmetric LINEX loss function, we derive and numerically evaluate the exact risks of the inequality constrained estimator and the inequality pre-test estimator which results after a preliminary test for an inequality constraint on the coefficients. The risks based on squared error loss are special cases of our results, and we draw appropriate comparisons.     

Risk behavior of a pre-test estimator for normal variance with the Stein-type estimator

In this paper, we examine the risk behavior of a pre-test estimator for normal variance with the Stein-type estimator. The one-sided pre-test is conducted for the null hypothesis that the population variance is equal to a specific value, and the Stein-type estimator is used if the null hypothesis is rejected. A sufficient condition for the pre-test estimator to dominate the Stein-type estimator is shown.     

Adaptive ridge estimator in a linear regression model with spherically symmetric error under constraint

Abstract

We consider adaptive ridge regression estimators in the general linear model with homogeneous spherically symmetric errors. A restriction on the parameter of regression is considered. We assume that all components are non negative (i.e. on the positive orthant). For this setting, we produce under general quadratic loss such estimators whose risk function dominates that of the least squares provided the number of regressors in the least fore.     

Testing the disturbance variance after a pre-test for a linear hypothesis on coefficients in a linear regression

In this paper, we examine the sampling performance of a two-stage test which consists of a pre-test for a linear hypothesis on regression coeffiecients followed by a main-test for a disturbance variance in a linear regression. It is shown that the actual size of the two-stage test can be well-controlled around the normal size if the suggested sizes presented in this paper are used in the pre-test. It is also shown that the two-stage test when the suggested sizes are used in the preferable to the usual test for the disturbance variable which incorporates no pre-test in terms of the power.     

Prediction distribution for a linear regression model with multivariate student-t error distribution

The distribution(s) of future response(s) given a set of data from an informative experiment is known as prediction distribution. The paper derives the prediction distribution(s) from a linear regression model with a multivari-ate Student-t error distribution using the structural relations of the model. We observe that the prediction distribution(s) are multivariate t-variate(s) with degrees of freedom which do not depend on the degrees of freedom of the error distribution.     

Dropping variables versus use of proxy variables in linear regression

In this paper we investigate under which conditions it is preferable to use proxies or to omit variables from the linear regression model with respect to the matrix mean square error criterion. Furthermore, some attention is paid to the admissibility of the proxies-based least squares estimator.     

Improved Liu estimator in a linear regression model

In this paper, we mainly aim to introduce the notion of improved Liu estimator (ILE) in the linear regression model y=Xβ+e. The selection of the biasing parameters is investigated under the PRESS criterion and the optimal selection is successfully derived. We make a simulation study to show the performance of ILE compared to the ordinary least squares estimator and the Liu estimator. Finally, the main results are applied to the Hald data.     

Omnibus tests for the error distribution in the linear regression model

Test procedures are constructed for testing the goodness-of-fit of the error distribution in the regression context. The test statistic is based on an L ²-type distance between the characteristic function of the (assumed) error distribution and the empirical characteristic function of the residuals. The asymptotic null distribution as well as the behavior of the test statistic under contiguous alternatives is investigated, while the issue of the choice of suitable estimators has been particularly emphasized. Theoretical results are accompanied by a simulation study.     

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.     

The exact distribution of the maximum likelihood estimators for the linear regression negative exponential model

The exact distribution of the maximum likelihood estimators in an exponential regression model are derived. The approach involves finding the distribution of the score statistic, since the log likelihood is globally concave, and then using the one-to-one correspondence between this and the estimator. The distribution is a weighted sum of independent exponential random variables. The exact p.d.f. is found by inverting the characteristic function by a straightforward application of residue theory.     

Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).