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.     

MSE performance of the 2SHI estimator in a regression model with multivariate t error terms

t error terms and derive the explicit formula of the mean squared error (MSE) of the two-stage hierarchial information (2SHI) estimator. It is shown by numerical evaluations that the 2SHI estimator has smaller MSE than the positive-part Stein-rule (PSR) estimator over a wide region of the parameter space. Received: November 6, 1998; revised version: October 15, 1999     

MSE performance of the weighted average estimators consisting of shrinkage estimators

In this paper, we consider a regression model and propose estimators which are the weighted averages of two estimators among three estimators; the Stein-rule (SR), the minimum mean squared error (MMSE), and the adjusted minimum mean-squared error (AMMSE) estimators. It is shown that one of the proposed estimators has smaller mean-squared error (MSE) than the positive-part Stein-rule (PSR) estimator over a moderate region of parameter space when the number of the regression coefficients is small (i.e., 3), and its MSE performance is comparable to the PSR estimator even when the number of the regression coefficients is not so small.     

On the use of the Stein variance estimator in the double <Emphasis Type="Italic">k</Emphasis>-class estimator when each individual regression coefficient is estimated

In this paper we consider the double k-class estimator which incorporates the Stein variance estimator. This estimator is called the SVKK estimator. We derive the explicit formula for the mean squared error (MSE) of the SVKK estimator for each individual regression coefficient. It is shown analytically that the MSE performance of the Stein-rule estimator for each individual regression coefficient can be improved by utilizing the Stein variance estimator. Also, MSE’s of several estimators included in a family of the SVKK estimators are compared by numerical evaluations.     

Minimum mean squared error estimation of each individual coefficient in a linear regression model

In this paper, we derive the exact mean squared error (MSE) of the minimum MSE estimator for each individual coefficient in a linear regression model, and show a sufficient condition for the minimum MSE estimator for each individual coefficient to dominate the OLS estimator. Numerical results show that when the number of independent variables is 2 and 3, the minimum MSE estimator for each individual coefficient can be a good alternative to the OLS and Stein-rule estimators.     

State space modeling of multiple time series: a comment

This paper provides guidance in choosing k₁ andk₂ of the double k-class (KK) estimator such that it will improve upon both the ordinary least squares (OLS) and Stein-rule (SR) estimators in predictive mean squared error (PMSE). Asymptotic bias and mean squared error (MSE) results are derived for nonnormal and other cases. A simulation compares the KK estimator with the OLS and SR estimators.     

Using improved estimation strategies to combat multicollinearity

In this approach, some generalized ridge estimators are defined based on shrinkage foundation. Completely under the suspicion that some sub-space restrictions may occur, we present the estimators of the regression coefficients combining the idea of preliminary test estimator and Stein-rule estimator with the ridge regression methodology for normal models. Their exact risk expressions in addition to biases are derived and the regions of optimality of the estimators are exactly determined along with some numerical analysis. In this regard, the ridge parameter is determined in different disciplines.     

Unbiased estimation of the MSE matrices of improved estimators in linear regression

Stein-rule and other improved estimators have scarcely been used in empirical work. One major reason is that it is not easy to obtain precision measures for these estimators. In this paper, we derive unbiased estimators for both the mean squared error (MSE) and the scaled MSE matrices of a class of Stein-type estimators. Our derivation provides the basis for measuring the estimators' precision and constructing confidence bands. Comparisons are made between these MSE estimators and the least squares covariance estimator. For illustration, the methodology is applied to data on energy consumption.     

SEPARATE VERSUS SYSTEM METHODS OF STEIN-RULE ESTIMATION IN SEEMINGLY UNRELATED REGRESSION MODELS

ABSTRACT

Despite the sizeable literature associated with the seemingly unrelated regression models, not much is known about the use of Stein-rule estimators in these models. This gap is remedied in this paper, in which two families of Stein-rule estimators in seemingly unrelated regression equations are presented and their large sample asymptotic properties explored and evaluated. One family of estimators uses a shrinkage factor obtained solely from the equation under study while the other has a shrinkage factor based on all the equations of the model. Using a quadratic loss measure and Monte-Carlo sampling experiments, the finite sample risk performance of these estimators is also evaluated and compared with the traditional feasible generalized least squares estimator.     

A synthesis of stein-rule and mixed regression procedures inlinear regression models under non-normality

Stein-rule philosophy and mixed regression technique are combined to develop two families of improved estimators of regression coefficients in the linear regression model under incomplete prior information. The properties of these estimators are studied when disturbances are small and non-normal. Conditions for their dominance over mixed regression estimator are derived taking risk as the criterion for performance.     

Bootstrapping estimators for the seemingly unrelated regressions model

In this paper we consider the possibility of using the bootstrap to estimate the finite sample variability of feasible generalized least squares and improved estimators applied to the seemingly unrelated regressions model. The improved estimators we employ include members of the Stein-rule family and a hierarchical Bayes estimator proposed by Blattberg and George (1991). Simulation experiments are carried out using several SUR examples as well as a very large example based on the price-promotion model, and data, from marketing research.     

On an adjustment of degrees of freedom in the minimim mean squared error ertimator

In this paper, we consider an adjustment of degrees of freedom in the minimum mean squared error (MMSE) estimator, We derive the exact MSE of the adjusted MMSE (AMMSE) estimator, and compare the MSE of the AMMSE estimator with those of the Stein-(SR), positive-part Stein-rule (PSR) and MMSE estimators by numerical evaluations. It is shown that the adjustment of degrees of freedom is effective when the noncentrality parameter is close to zero, and the MSE performance of the MMSE estimator can be improved in the wide region of the noncentrality parameter by the adjustment, ft is also shown that the AMMSE estimator can have the smaller MSE than the PSR estimator in the wide region of the noncentrality parameter     

On risk comparisons of some estimators in a linear regression model under inagaki’S loss function and nonnormal error terms

In this paper we consider the risk performances of some estimators for both location and scale parameters in a linear regression model under Inagaki’s loss function We prove that the pre-test estimator for location parameter is dominated by the Stein-rule estimator under Inagaki’s loss function when the distribution of error terms is expressed by the scale mixture of normal distribution and the variance of error terms is unknown.. It is an extension of the results in Nagata (1983) to our situation Also we perform numerical calculations to draw the shapes of the risks.     

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     

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.     

Small sample performance of jackknife confidence intervals for the james-stein estimator

The primary goal of this paper is to examine the small sample coverage probability and size of jackknife confidence intervals centered at a Stein-rule estimator. A Monte Carlo experiment is used to explore the coverage probabilities and lengths of nominal 90% and 95% delete-one and infinitesimal jackknife confidence intervals centered at the Stein-rule estimator; these are compared to those obtained using a bootstrap procedure.     

Pooling multivariate data

In case it is doubtful whether two sets of data have the same mean vector, four estimation strategies have been developed for the target mean vector. In this situation, the estimates based on a preliminary test as well as on Stein-rule are advantageous. Two measures of relative efficiency are considered; one based on thequadratic loss function, and the other on the determinant of the mean square error matrix. A max-min rule for the size of the preliminary test of significance is presented. It is demonstrated that the shrinkage estimator dominates the classical estimator, whereas none of the shrinkage estimator and the preliminary test estimator dominate each other. The range in the parameter space where preliminary test estimator dominates shrinkage is investigated analytically and computationally. It is found that the shrinkage estimator outperform the preliminary test estimator except in a region around the null hypothesis. Moreover, for large values of a, the level of statistical significance, shrinkage estimator dominates the preliminary test estimator uniformly. The relative dominance of the estimators is presented.     

Stein-rule estimation in models with a lagged-dependemt variable

There is an extensive literature on the Stein-rule estimation of the parameters in a regression model. An important result in this literature is that the Stein-rule does not dominate the least squares estimator in the lower mean squared error sense when there are two or less regressors in the model. However, we note that not much is known about the Stein-rule estimation in dynamic models. This paper is a modest attempt in this direction and it shows that Stein-rule estimation of the lagged coefficient does not dominate the least square indicating that the result of regresion model goes through in the dynamic case.     

Comparison of the Iterative Stein-Rule and the Usual Estimators of the Disturbance Variance Under the Pitman Nearness Criterion in a Linear Regression Model with Proxy Variables

In a linear regression model with proxy variables, the iterative Stein-rule estimator and the usual estimator of the disturbance variance is compared under the Pitman Nearness Criterion. The exact expression of Pitman Nearness probability is derived and numerically evaluated.     

Properties of the ordinary least squares and stein-rule predictions in linear regression models with proxy variables

This article considers a linear regression model in which misspecification relates to the use of a stochastic proxy variable. The analysis indicates the decline in efficiency of the predictions arising from the ordinary least squares and the Stein-rule estimation procedures when a proxy variable is used in the place of an unobservable variable. However, the performance of the Stein-rule predictions is still found to be better than the ordinary least squares predictions over a broad range of k, the characterizing scalar of the Stein-rule estimator.