Semi-parametric and Nonparametric Inference

半参数和非参数推理

• 批准号: RGPIN-2014-04621
• 负责人: Huang, MeiLing
• 金额: $ 0.8万
• 依托单位: Brock University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587917
• 关键词: Semi; parametric; Nonparametric; Inference

The objective of this research is to develop novel inference methods in two related areas: 1) Inference for heavy tailed distributions; 2) nonparametric inference (for general distributions). 1. Inference for Heavy Tailed Distributions Extreme events occur in financial markets, natural disasters, disease control and industrial quality control, which affect human life. It is important but difficult to study these extreme events by using suitable mathematical models. Heavy tailed distributions comprise one class of the extreme value distributions which has been applied widely to risk analysis with applications in economics, industrial engineering, actuarial science, medical research and networks. There are theoretical difficulties in the inference of heavy tailed distributions. The proposed program will explore innovative methods to overcome these difficulties. (1) Heavy tailed data is often complicated. A single distribution may not fit data well. Two new cluster and sieve methods are proposed, which are found to e generally better fitting compared with existing methods. (2) Estimation of a high quantile (Value-at-Risk) of a heavy tailed distribution is an important but difficult problem. Existing inference methods have bias and deficiency problems. A geometric mean method is proposed. Preliminary work shows that the new method yields improvements. Also, I will study estimation for high conditional quantiles (quantile regression). A weighted loss function to produce more accurate predictions is proposed. (3) I will explore innovative approximation methods for heavy tailed distributions, e.g., truncation, level crossing, generalized hyperexponential algorithm, etc. 2. Nonparametric Inference (for general distributions) Nonparametric methods are superior to parametric methods on data sets having complex distributions, e.g. multiple modes. However, compared to parametric methods, nonparametric methods may lack efficiency on tails of the distribution or have technique difficulties. The proposed program will develop novel methods to overcome these problems. (4) A family of weighted empirical distribution functions will be studied. I will explore the optimal weights that minimize the estimation errors to improve tail estimation, by utilizing criteria for measurement of the errors, e.g., Lp-norm, exceedance measure and Hellinger distance. (5) The classical kernel methods have selection problems of bandwidth and kernel. I will develop some non-kernel methods by using Hermite orthogonal series, wavelet and L-statistics to avoid these difficulties and keep good properties. Measurements of the proposed methods (1) to (5) will be: a) Comparing the proposed methods with existing methods on theoretical properties: efficiency, consistency, rate of convergence and robustness; b) Studying Monte Carlo simulations to search for good methods and confirm theoretical results; c) Applying the proposed methods to real problems, to find a best model fitting the data; estimating value-at-risk, survival functions, waiting time in networks and performing goodness of fit tests. Scientific Approach: a) The theoretical approach includes probability, statistical theory, order statistics, stochastic processes, harmonic analysis and approximation theory. b) The computational approach includes: Monte Carlo, bootstrapping, computer programming. Expected Impact: The work will provide an alternative approach for statistical inference. The results are expected to overcome some difficulties in the field, to explore and suggest new ideas for future research, and to benefit Canadian research on extreme values field. This research will contribute to solving real world problems and obtaining accurate solutions.

本研究的目的是在两个相关领域开发新的推理方法：1）重尾分布的推理; 2）非参数推理（一般分布）。 1.重尾分布的推论 极端事件发生在金融市场、自然灾害、疾病控制和工业质量控制等领域，影响着人类的生活。利用合适的数学模型对这些极端事件进行研究是一个重要而又困难的问题。重尾分布是一类在经济、工业工程、精算、医学研究和网络等领域有着广泛应用的极值分布。重尾分布的推断存在理论上的困难。该计划将探索克服这些困难的创新方法。 (1)重尾数据通常很复杂。单一分布可能无法很好地拟合数据。提出了两种新的聚类和筛选方法，与现有方法相比，它们具有更好的拟合效果。 (2)重尾分布的高分位数（风险值）的估计是一个重要而困难的问题。现有的推理方法存在偏差和不足。提出了一种几何平均法。初步工作表明，新方法产生了改进。此外，我将研究高条件分位数的估计（分位数回归）。提出了一种加权损失函数，以产生更准确的预测。 (3)我将探索重尾分布的创新近似方法，例如，截断、水平交叉、广义超指数算法等。 2.非参数推断（一般分布） 在具有复杂分布的数据集上，非参数方法上级参数方法，例如多模式。然而，与参数方法相比，非参数方法可能在分布的尾部缺乏效率或具有技术困难。该计划将开发新的方法来克服这些问题。 (4)将研究一族加权经验分布函数。我将探索最佳权重，最大限度地减少估计误差，以改善尾部估计，通过利用标准的测量误差，例如，Lp-范数、线性测度和Hellinger距离 (5)经典的核方法存在带宽和核的选择问题。为了避免这些困难，并保持良好的性质，我将利用Hermite正交级数、小波和L-统计量来发展一些非核方法。 所提出的方法（1）至（5）的测量将是： a）将所提出的方法与现有方法在理论性质上进行比较：效率、一致性、收敛速度和鲁棒性; B）研究蒙特卡罗模拟，寻找好的方法，验证理论结果; c）将所提出的方法应用于真实的问题，以找到拟合数据的最佳模型;估计风险值、生存函数、网络中的等待时间并执行拟合优度检验。 科学方法： a）理论方法包括概率论、统计理论、顺序统计、随机过程、调和分析和近似理论。 B）计算方法包括：蒙特卡罗法、自举法、计算机编程。 预期影响： 这项工作将为统计推断提供另一种方法。本文的研究结果有望克服该领域的一些困难，为今后的研究探索和提出新的思路，并对加拿大在极值领域的研究有所裨益。这一研究将有助于解决真实的世界问题并获得精确解。