Research for practical use and new application of resampling method

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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.