Research for practical use and new application of resampling method

在本研究项目中，我们从实际应用的角度出发，对重采样方法的问题进行了研究，试图获得应用重采样方法的有效途径。我们取得了以下成果：1.在非参数统计推断中，我们研究了提高正态分布收敛速度的高阶正规化变换，并在此基础上得到了新的可信区间。我们还将这些区间与Bootstrap迭代进行了比较。此外，我们还提出了一种新的Bootstrap方法，它是Bootstrap迭代的一种近似，并且减少了计算时间。2.针对强非线性的时间序列数据，提出了嵌入维数、延迟时间和Lyapunov指数的新估计。为了估计这些估计量的方差，我们应用了Bootstrap方法并试图改进这些估计量。3.提出了评价预测方法的新的模型准则，并研究了该准则的理论性质。为了分析协方差结构，对基于Bootstrap方法的推论和基于渐近展开的推论进行了理论研究。4.将Bootstrap方法应用于序贯统计推断，提出了一种新的条件均值置信域，该置信域基于两阶段过程，满足一个大小条件。5.在考察重抽样方法在运筹学中的应用时，我们提出了两个新的准则，即取分数表达式和评价受控马氏链的期望。利用递归方法，我们还得到了确定性或动态系统的最优和非最优问题的新解。我们还开发了计算机程序，使我们能够将上述结果用于实际。

In this research project, we have studied problems of resampling method from, a viewpoint of practical use and tried to obtain efficient way of applying the method. We have obtained the following results.1. In non-parametric statistical inference, we have studied the higher order normalizing transformations which improve convergence rates to normal distribution and obtained new confidence intervals based on the transformations. We have also compared those intervals with bootstrap iteration. Further we propose new bootstrap method which is an approximation of the bootstrap iteration and reduces computation time. 2. For analyzing the time series data, which are strongly non-linear, we propose new estimators of embedded dimension, delay time and Lyapunov exponent. In order to evaluate the variances of these estimators, we apply the bootstrap method and try to improve the estimators. 3. We propose new model criteria, which evaluate the prediction method, and study theoretical properties of the criteria. For analyzing covariance structure, theoretical studies of inferences based on bootstrap method and inferences based on asymptotic expansions have done. 4. Applying the bootstrap method to sequential statistical inferences, we propose a new confidence region of conditional mean, which is based on two-stage procedure and satisfies a size condition. 5. While examining applications of the resampling method to operation research, we propose two new criteria which takes fractional expressions and evaluate expectations of controlled Markov chain. Using recursive method, we also obtain new solutions of optimal and non-optimal problems for deterministic or dynamic systems.We also develop computer programs which allow us to use the above results practically.