Evaluation of precision of estimation of shrinkage estimators by the bootstrap method, and its applications to empirical researches

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1. The coefficient of determination is usually by using the ordinary least squares (OLS) estimator. In this research, we dealt with the coefficient of determination defined by using the Stein-Rule (SR) estimator. Although we derived the exact formula for the moments of the coefficient of determination based on the SR estimator, the formula for the moments is very complicated and it depends on unknown parameters. When the formula for the moments of estimators is very complicated and it depends on unknown parameter, it is very difficult to evaluate the precision of estimation. However, if we use the bootstrap method proposed by Efron (1979), it is possible to evaluate the precision of estimation. Thus, we considered how we apply the bootstrap method to estimating the precision of estimation and the confidence interval of the coefficient of determination based on the SR estimator. We also generated the empirical estimates of the precision of estimation by Monte Carlo experiments, and comp … More ared them with the precision of estimation evaluated by the exact formula. The Monte Carlo results showed that the bootstrap method worked effectively.2. In this research, we examined the small sample properties of the coefficient of determination when a model is selected by a pre-test for linear restrictions on regression coefficients. We derived the exact formula for the moments of the pre-test estimator for the coefficient of determination and compare the bias and MSE of the pre-test estimator for the coefficient of determination with those of the usual estimator. Our numerical results show that although the bias of the pre-test estimator for he coefficient of determination is smaller than that of the usual coefficient of determination, the MSE performance depends on the size of the pre-test. Now, we are developing the procedure of the bootstrap method.3. We derived the exact distribution of the pre-test estimator for the regression error variance, and examined the small sample properties of the pre-test estimator. Now, we are developing the procedure of the bootstrap method. Less