Some Advances in Nonparametric Statistics

Nonparametric statistics is an important branch of statistics that utilize infinite dimensional modelsto achieve great flexibility. However, such flexibility often comes with difficulties in computations and convergent properties. One approach is to study the natural patterns for one type of datasets and summarize such patterns into mathematical assumptions that can potentially provide computational and theoretical benefits. I carried out the above idea on three different problems. The first problem is the classification trees for imbalanced datasets, where I formulate the regularity of shapes into surface-to-volumeratio and develop satisfactory theory and methodology using this ratio. The second problem is the approximation of Gaussian process, where I observe the critical role of spatial decaying covariance function in Gaussian process approximations and use such decaying properties to prove the approximation error for my proposed method. The last problem is the posterior contraction rates in Kullback-Leibler (KL) divergence, where I am motivated by the dismatch between KL divergence and Hellinger distance and develop a posterior contraction theory entirely based on KL divergence