Semi-parametric and Nonparametric Inference

这项研究的目的是在两个相关领域发展新的推断方法：1)重尾分布的推断；2)非参数推断(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影响人类生活的领域。使用合适的数学模型来研究这些极端事件是重要的，但也是困难的。重尾分布是一类广泛应用于风险分析的极值分布，在经济学、工业工程、精算、医学和网络等领域有着广泛的应用。重尾分布的推断存在理论上的困难。拟议的计划将探索克服这些困难的创新方法。 (1)重尾数据往往比较复杂。单一的分布可能不能很好地拟合数据。提出了两种新的聚类法和筛选法，与已有的方法相比，这两种方法具有更好的拟合效果。 (2)重尾分布高分位数(在险价值)的估计是一个重要而又困难的问题。现有的推理方法存在偏差和不足的问题。提出了一种几何平均法。初步工作表明，新方法取得了改进。此外，我还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)探索重尾分布的创新逼近方法，如截断、水平交叉、广义超指数算法等。 2.非参数推断(适用于一般分布) 对于具有复杂分布的数据集，例如多个模式，非参数方法优于参数方法。然而，与参数方法相比，非参数方法可能对分布的尾部缺乏效率或存在技术困难。拟议的计划将开发新的方法来克服这些问题。 (4)研究了一族加权经验分布函数。我将通过利用衡量误差的标准，如Lp范数、超越度量和Hellinger距离，来探索使估计误差最小的最优权重，以改进尾部估计。 (5)经典的核方法存在带宽和核的选择问题。我将利用Hermite正交级数、小波和L统计量来开发一些非核方法，以避免这些困难，并保持良好的性质。 建议的方法(1)至(5)的量度结果如下： A)在效率、一致性、收敛速度和稳健性等方面与现有方法进行比较； B)研究蒙特卡罗模拟，寻找好的方法并验证理论结果； C)将所提出的方法应用于实际问题，找出符合数据的最佳模型；估计风险值、生存函数、在网络中的等待时间并进行拟合优度检验。 科学方法： A)理论方法包括概率论、统计理论、顺序统计量、随机过程、调和分析和逼近理论。 B)计算方法包括：蒙特卡洛法、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。本文的研究成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，并对加拿大极值领域的研究有所裨益。这项研究将有助于解决现实世界的问题，获得准确的解决方案。

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.