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.

这项研究的目的是在两个相关领域开发新的推理方法：1）对重尾部分布的推断； 2）非参数推断（用于一般分布）。 1。尾部分布的推断 极端事件发生在金融市场，自然灾害，疾病控制和工业质量控制中，这会影响人类的生活。使用合适的数学模型研究这些极端事件很重要，但很难研究。重型尾部分布包括一类极值分布，这些分布已广泛应用于经济学，工业工程，精算科学，医学研究和网络中的应用中的风险分析。重大尾部分布的推断存在理论上的困难。拟议的计划将探讨克服这些困难的创新方法。 （1）重型尾部数据通常很复杂。单个分布可能不符合数据。提出了两种新的群集和筛子方法，与现有方法相比，通常发现它们的拟合通常更好。 （2）对重尾部分布的高分位数（价值）的估计是一个重要但困难的问题。现有的推理方法存在偏见和缺陷问题。提出了一种几何平均方法。初步工作表明，新方法可以改善。另外，我将研究高条件分位数（分位数回归）的估计。提出了加权损失函数以产生更准确的预测。 （3）我将探索用于重型尾部分布的创新近似方法，例如，截断，水平交叉，广义过度过度算法等。 2。非参数推断（用于一般分布） 非参数方法优于具有复杂分布的数据集的参数方法，例如多种模式。但是，与参数方法相比，非参数方法可能缺乏分布尾部或技术困难的效率。拟议的计划将开发新的方法来克服这些问题。 （4）将研究一个加权经验分布功能的家族。我将通过利用标准来测量误差，例如LP-NORM，超级测量和Hellinger距离，从而探索最大程度地减少估计误差以改善尾部估计的最佳权重。 （5）经典内核方法具有带宽和内核的选择问题。我将通过使用Hermite正交系列，小波和L统计量来开发一些非内核方法，以避免这些困难并保持良好的特性。 拟议方法（1）至（5）的测量将是： a）将提议的方法与现有方法进行理论特性进行比较：效率，一致性，收敛速度和鲁棒性； b）研究蒙特卡洛模拟以寻找良好的方法并确认理论结果； c）将提出的方法应用于实际问题，以找到适合数据的最佳模型；估计风险的价值，生存功能，网络中的等待时间以及进行拟合测试的优点。 科学方法： a）理论方法包括概率，统计理论，顺序统计，随机过程，谐波分析和近似理论。 b）计算方法包括：蒙特卡洛，自举，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。预计结果将克服该领域的一些困难，探索并提出新的想法以供未来的研究，并使加拿大关于极端价值领域的研究受益。这项研究将有助于解决现实世界中的问题并获得准确的解决方案。