Improvement of the nonparametric statistical inference under complex statistical model and its application

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For the data which has complex structure, like genome or financial data, we have to modify or improve ordinal statistical methods. Our purpose of this project is to propose new methods and study basic properties of the new methods under nonparametric setting. We obtain the following results. 1. Without assuming the underlying distribution, we obtain asymptotic representations of inversions of the Cornish-Fisher approximation and normalizing transformation. Using these representations, we compare mean squared errors of the inversions, theoretically. We also propose new confidence intervals that improve the ordinal method. 2.Based on the Bayes approach, we obtain new information criteria. Applying the new criteria to complex statistical model, we obtain new statistical methods which improve accuracy of statistical inference. We also propose new regularized basis expansions, and obtain theoretical properties of them. 3.We propose new confidence region of difference between mean vectors of bivariate normal distributions. This confidence region is based on the sequential method, and using mathematical programming approach, we prove that the new region is superior to ordinal region. 4.Under the uncertainty, we introduce Markov model, and study optimality based on non-additive stochastic dynamic programming. Using embedded method, we also prove optimality when the criterion is non-linear, and show that those results are applicable to the statistical inference. 5.For discretely observed diffusion process, we obtain new statistical inference methods, based on approximate martingale stochastic equation. We also propose a new estimator of a drift parameter for diffusion process with small variation, and prove consistency and asymptotic normality of the new estimator.