Semi-parametric and Nonparametric Inference

拟议研究计划的目标：我的研究目标是在两个领域发展新的推理方法：1)重尾分布和网络的推理方法。2)分位数回归的加权和非参数推断，培养高素质人才开展研究。1.重尾分布和网络的推断极端事件发生在金融市场、自然灾害、疾病控制和工业风险中。对于通常应用重尾分布的极端事件的分析，找到合适的数学模型是很重要的。重尾分布的推断存在理论上的困难。拟议的方案探索了三个创新的推理目标，以克服困难：目标(1)(Prop)。对分布和分位数的推断：探索新的方法来减少估计高分位数和分布的偏差和误差。从理论和计算上对新方法和现有方法进行了比较。目标(2)(主张)。重尾分布的聚类和近似：重尾数据通常很复杂，因此单一分布可能不能很好地适应数据。探索用于估计重尾分布的新的簇、Hermite级数、超指数近似方法。目标(3)(主张)。随机模型和随机网络的推理：从理论和计算上探索与重尾分布相关的极端更新过程和随机网络的创新推理方法。2.条件分布尾部分位数回归估计的加权和非参数推断是一个具有挑战性的目标。本文主要研究条件分位数的估计(分位数回归)。我将研究两个新的目标如下：目标(4)(主张)。分位数回归的加权方法：探索使估计误差最小化的最佳权重。利用几个标准来测量误差。目标(5)(道具)。分位数回归的直接非参数方法：发展具有更有效算法的非参数分位数回归。研究理论效率、一致性、收敛速度和稳健性。对拟议目标的评估(1)至(5)1)理论方法包括概率论、统计理论、随机过程、积分方程式、组合学、群和近似。2)计算方法包括蒙特卡罗模拟、Bootstrapping，以验证理论结果。3)在实际算例中的应用，找出最佳的模型拟合数据，得出合理的结论。预期意义：该程序为统计推断提供了一种新的替代方法。这些结果有望克服这一领域的理论和计算困难。该计划训练HQP带来新的想法和技能，以建立合适的数学模型来解决现实世界的问题。

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.