Semi-parametric and Nonparametric Inference

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Objectives of the Proposed Research Program: The objective of my research is to develop novel inference methodology in two areas: 1) Inference methods for heavy-tailed distributions and networks. 2) Weighted and nonparametric inference for quantile regression; Train high quality personnel (HQP) to carry out research. 1. Inference for Heavy-Tailed Distributions and Networks Extreme events occur in financial markets, natural disasters, disease control and industrial risk. It is important to find suitable mathematical models for analyzing of extreme events where heavy-tailed distributions are usually applied. There are theoretical difficulties in the inference of heavy-tailed distributions. The proposed program explores three innovative inference objectives to overcome difficulties: Objective (1)(Prop) . Inference for Distributions and Quantiles: Explore new methods to reduce bias and errors for estimation of high quantiles and distributions. Compare new methods with existing methods theoretically and computationally. Objective (2)(Prop). Cluster and Approximation for Heavy Tailed Distributions: Heavy tailed data is often complicated, such that a single distribution may not fit data well. Explore new cluster, Hermite series, hyperexponential approximation methods for estimating heavy tailed distributions. Objective (3)(Prop). Inference for stochastic Models and Random Networks: Explore innovative inference methods on extreme renewal process and random network related to heavy tailed distributions theoretically and computationally. 2. Weighted and Nonparametric Inference for Quantile Regression Estimation on the tail of a conditional distribution is a challenging objective. This work focuses on estimating the conditional quantiles (Quantile Regression). I will study two novel objectives as follows: Objective (4)(Prop). Weighted Methods for Quantile Regression: Explore the optimal weights that minimize the estimation errors. Utilize several criteria for measurement of the errors. Objective (5)(prop). Direct Nonparametric Methods for Quantile Regression: Develop nonparametric quantile regression with more effective algorithms. Study theoretical efficiency, consistency, rate of convergence, and robustness. Assessment of the Proposed Objectives (1) to (5) 1) The theoretical approach includes probability theory, statistical theory, stochastic processes, integral equations, combinatorics, groups, and approximation. 2) The computational approach includes Monte Carlo simulations, bootstrapping, to confirm the theoretic results. 3) Applications on real-word examples, find best model fitting data with reasonable conclusions. Expected Significance: The program provides a new alternative approach for Statistical inference. The results are expected to overcome theoretical and computational difficulties in this field. The program trains HQPs to bring new ideas and skills for building suitable Mathematical models to solve real-world problems.