Nonparametric Inference for High Dimensional Data

Learning from data, especially ?Big Data?, is becoming increasingly popular under names such as Data Mining, Data Science, Machine Learning, Statistical Learning and High Dimensional Data Analysis. In this dissertation we propose a new related field, which we call ?United Nonparametric Data Science? - applied statistics with ?just in time? theory. It integrates the practice of traditional and novel statistical methods for nonparametric exploratory data modeling, and it is applicable to teaching introductory statistics courses that are closer to modern frontiers of scientific research. Our framework includes small data analysis (combining traditional and modern nonparametric statistical inference), big and high dimensional data analysis (by statistical modeling methods that extend our unified framework for small data analysis).

The first part of the dissertation (Chapters 2 and 3) has been oriented by the goal of developing a new theoretical foundation to unify many cultures of statistical science and statistical learning methods using mid-distribution function, custom made orthonormal score function, comparison density, copula density, LP moments and comoments. It is also examined how this elegant theory yields solution to many important applied problems. In the second part (Chapter 4) we extend the traditional empirical likelihood (EL), a versatile tool for nonparametric inference, in the high dimensional context. We introduce a modified version of the EL method that is computationally simpler and applicable to a large class of ?large p small n? problems, allowing p to grow faster than n. This is an important step in generalizing the EL in high dimensions beyond the p ? n threshold where the standard EL and its existing variants fail. We also present detailed theoretical study of the proposed method.