Positive-part Stein-rule Research Articles

MSE performance of the weighted average estimators consisting of shrinkage estimators

ABSTRACTIn 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.

Risk Comparison of Improved Estimators in a Linear Regression Model with MultivariatetErrors under Balanced Loss Function

Under a balanced loss function, we derive the explicit formulae of the risk of the Stein-rule (SR) estimator, the positive-part Stein-rule (PSR) estimator, the feasible minimum mean squared error (FMMSE) estimator, and the adjusted feasible minimum mean squared error (AFMMSE) estimator in a linear regression model with multivariateterrors. The results show that the PSR estimator dominates the SR estimator under the balanced loss and multivariateterrors. Also, our numerical results show that these estimators dominate the ordinary least squares (OLS) estimator when the weight of precision of estimation is larger than about half, and vice versa. Furthermore, the AFMMSE estimator dominates the PSR estimator in certain occasions.

A note on the properties of Stein-rule and inequality restricted estimators when the regression model is over-fitted

This paper considers the effect of an erroneous inclusion of regressors on the risk properties of the Stein-rule, positive-part Stein-rule and inequality restricted and pre-test estimators in a linear regression model. The two Stein-rule estimators are considered when extraneous information is available in the form of a set of multiple equality constraints on the coefficients, while the inequality estimators are considered under the case of a single inequality constraint. It is shown that the inclusion of wrong regressors has only minimal effect on the properties of the Stein-rule and positive-part Stein-rule estimators, and no effect at all on the inequality restricted and pre-test estimators when there is a single inequality constraint.

Robustness of Stein-type estimators under a non-scalar error covariance structure

The Stein-rule (SR) and positive-part Stein-rule (PSR) estimators are two popular shrinkage techniques used in linear regression, yet very little is known about the robustness of these estimators to the disturbances’ deviation from the white noise assumption. Recent studies have shown that the OLS estimator is quite robust, but whether this is so for the SR and PSR estimators is less clear as these estimators also depend on the F statistic which is highly susceptible to covariance misspecification. This study attempts to evaluate the effects of misspecifying the disturbances as white noise on the SR and PSR estimators by a sensitivity analysis. Sensitivity statistics of the SR and PSR estimators are derived and their properties are analyzed. We find that the sensitivity statistics of these estimators exhibit very similar properties and both estimators are extremely robust to MA(1) disturbances and reasonably robust to AR(1) disturbances except for the cases of severe autocorrelation. The results are useful in light of the rising interest of the SR and PSR techniques in the applied literature.

PMSE performance of the Stein-rule and positive-part Stein-rule estimators in a regression model with or without 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 the predictive mean squared error (PMSE) of the Stein-rule (SR) estimator and the positive-part Stein-rule (PSR) estimator for the regression coefficients when the proxy variables are used. We examine the effect of using the proxy variables on the risk performances of the SR and PSR estimators. It is shown analytically that the PSR estimator dominates the SR estimator even when the proxy variables are used. Also, our numerical results show that using the proxy variables is preferable to omitting the relevant regressors.

PMSE PERFORMANCE OF THE BIASED ESTIMATORS IN A LINEAR REGRESSION MODEL WHEN RELEVANT REGRESSORS ARE OMITTED

In this paper, we consider a linear regression model when relevant regressors are omitted. We derive the explicit formulae for the predictive mean squared errors (PMSEs) of the Stein-rule (SR) estimator, the positive-part Stein-rule (PSR) estimator, the minimum mean squared error (MMSE) estimator, and the adjusted minimum mean squared error (AMMSE) estimator. It is shown analytically that the PSR estimator dominates the SR estimator in terms of PMSE even when there are omitted relevant regressors. Also, our numerical results show that the PSR estimator and the AMMSE estimator have much smaller PMSEs than the ordinary least squares estimator even when the relevant regressors are omitted.

IMPROVING THE 'HKB' ORDINARY TYPE RIDGE ESTIMATOR

The HKB estimator of Hoerl, Kennard and Baldwin is known to be an ordinary ridge type shrinkage estimator and an adjustment of degrees of freedom in the ordinary ridge estimator of Farebrother. The HKB estimator has a smaller predictive mean squared error (MSE) than the positive-part Stein-rule (PP) estimator in the wide region of the noncentral parameter when the degrees of freedom q =3 ∼ 6, but does not satisfy the sufficient condition to dominate the OLS estimator of Baranchik or Efron and Morris when q = 3. In this paper, a sufficient condition to dominate the OLS estimator is derived, and the modified HKB estimator is constructed to dominate the OLS estimator and have a smaller MSE than the PP estimator in the wide region of the noncentral parameter.

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

In this paper, we consider a linear regression model with multivariate 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.

MSE dominance of the PT-2SHI estimator over the positive-part Stein-rule estimator in regression

Abstract In this paper, we consider a heterogeneous pre-test estimator which consists of the two-stage hierarchial information (2SHI) estimator and the Stein-rule (SR) estimator. This estimator is called the pre-test 2SHI (PT-2SHI) estimator. It is shown analytically that the PT-2SHI estimator dominates the SR estimator in terms of mean squared error (MSE) if the parameter values in the PT-2SHI estimator are chosen appropriately. Moreover, our numerical results show that the appropriate PT-2SHI estimator dominates the positive-part Stein-rule (PSR) estimator.

MSE performance of a heterogeneous pre-test estimator

In this paper, we consider a linear regression model and examine the MSE performance of a heterogeneous pre-test estimator whose components are the adjusted minimum mean-squared error (AMMSE) estimator and the Stein-rule (SR) estimator. It is shown that the heterogeneous pre-test estimator dominates the SR estimator if a critical value of the pre-test is chosen appropriately. By numerical evaluations it is shown that if the number of independent variables (say, k) is 3 and the critical value of the pre-test is chosen appropriately, then the heterogeneous pre-test estimator dominates the positive-part Stein-rule (PSR) estimator. Also, when k = 4 and 5, the MSE performances of the heterogeneous pre-test and PSR estimators are comparable.

Mes performance of the minimum mean squared error estimators in a linear regression model when relevant regressors are omitted

In this paper, we consider a linear regression model when relevant regressors are omitted. We derive the explicit formulae of the mean squared errors (MSE's) of the feasible minimum MSE (FMMSE) estimator and the adjusted FMMSE (AFMMSE) estimator. By numerical evaluations, we compare the MSE performances of the FMMSE and AFMMSE estimators with those of the Stein-rule (SR) and positive-part Stein-rule (PSR) estimators. It is shown that when there are omitted regressors, the MSE performances of the AFMMSE and PSR estimators are comparable when the number of regressors included in the specified model (sayk 1) is larger than or equal to 8, and the MSE performance of the AFMMSE estimator is better than that of the PSR estimator when k 1≤5 and the model misspecification is severe.

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

A Comparison of the Stein-Rule and Positive-Part Stein-Rule Estimators in a Misspecified Linear Regression Model

In this paper, we examine the performance of the predictive risk of the Steinrule (SR) and positive-part Stein-rule (PSR) estimators when relevant regressors are omitted in the specified model. The exact formula of the predictive risk of the PSR estimator is derived, and the sufficient condition for the PSR estimator to dominate the SR estimator under a specification error is given. It is shown by numerical computation that the PSR estimator seems to be the best choice among the OLS, SR, and PSR estimators even when there are omitted variables.