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

1.决定系数通常采用普通最小二乘(OLS)估计。在这项研究中，我们处理了用Stein-Rule(SR)估计定义的决定系数。虽然我们基于随机共振估计导出了确定系数矩的精确公式，但矩的计算公式非常复杂，而且依赖于未知参数。当估计器的矩公式非常复杂且依赖于未知参数时，很难对估计的精度进行评估。然而，如果我们使用Efron(1979)提出的Bootstrap方法，估计的精度是可能的。因此，我们考虑了如何应用Bootstrap方法来估计估计的精度和基于SR估计的决定系数的可信区间。我们还通过蒙特卡罗实验和COMP…得到了估计精度的经验估计更多的是用精确的公式估计的精度。蒙特卡罗结果表明，Bootstrap方法是有效的。在这项研究中，我们检验了决定系数的小样本性质，当通过对回归系数的线性限制进行预测检验时，所选择的模型。我们推导了决定系数的检验前估计矩的精确公式，并比较了决定系数的检验前估计的偏差和均方根误差与通常估计的偏差和均方误差。我们的数值结果表明，尽管决定系数的预检验估计量的偏差比通常的决定系数的偏差要小，但均方误差性能取决于预试的大小。现在，我们正在开发Bootstrap方法的程序。给出了回归误差方差的预检验估计量的精确分布，并考察了预检验估计量的小样本性质。现在，我们正在开发Bootstrap方法的程序。较少

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