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

对于结构复杂的数据，如基因组数据或金融数据，我们必须修改或改进有序统计方法。本课题的目的是提出新方法，研究新方法在非参数设定下的基本性质。我们得到以下结果。1. 在不假设底层分布的情况下，我们得到了Cornish-Fisher近似和归一化变换的反演的渐近表示。使用这些表示，我们从理论上比较了反转的均方误差。我们还提出了新的置信区间来改进有序方法。2.基于贝叶斯方法，我们得到了新的信息准则。将新准则应用于复杂的统计模型，得到了新的统计方法，提高了统计推断的准确性。我们还提出了新的正则基展开式，并得到了它们的理论性质。3.提出了新的二元正态分布均值向量差置信域。该置信区域基于序列方法，并利用数学规划方法证明了该置信区域优于有序区域。4.在不确定性条件下，引入马尔可夫模型，研究了基于非加性随机动态规划的最优性。利用嵌入方法，我们还证明了准则为非线性时的最优性，并表明这些结果适用于统计推断。5.对于离散观测的扩散过程，我们得到了基于近似鞅随机方程的新的统计推理方法。我们还提出了一个小变化扩散过程漂移参数的新估计量，并证明了新估计量的相合性和渐近正态性。

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.