极值事件数学建模中的参数估计问题

The appearance of extremal events in the related fields always brings catastrophic consequences, such as bankruptcies in financial industry, earth quakes and typhoons in geoscience, insurance claims in insurance, breakdowns of servers in data network and epileptic attack in medical science. Heavy-tailed models are widely used in modelling extremal events, which usually has the characteristic of the clustering of extremes. This characteristic is the key to model extremal events and then predict the impact of extremal events to reduce or prevent the potential losses. Unfortunately, the classical risk measures in extreme value theory, such as Value-at-Risk, fail to capture the clustering of extremes, which is the main reason why the consequences of extremal events are underestimates. The applicants and his coauthors applied the techniques of Fourier analysis for the study of extremal events in heavy-tailed times series based on the proposed concept, named extremogram, in 2009. The results in this research has been used to perform goodness-of-fit tests to heavy-tailed models. This project is to continue the previous research and to focus on problems of parameter estimation in modelling extremal events.

极值事件一旦在相关领域里出现会带来灾难性结果:金融产业 (如破产等)，地球科学(如地震、台风等)，保险(保险索赔等 )，数据网络(如服务器崩溃等)以及医学领域(如癫痫发作等)。这些领域都与国计民生有很大的关系。厚尾模型被广泛应用于极值事件建模中，常有“极值事件连续地集群出现”这个关键特征。这一特征可以被用来进行对极值事件的数学建模，并用来对极值事件进行预测，从而避免或减少极值事件带来的损失。传统的极值理论中一些著名的风险度量，如风险价值(VaR)等，很难描述极值事件集群现象，从而导致低估极值事件的影响。申请人与其合作者在2009年提出的极值图概念基础上，引入傅立叶分析的方法研究时间序列中极值事件的相依关系，并应用于模型拟合优度检验等。本项目的目标就是在之前工作的基础上对极值事件数学建模中的参数估计问题加深研究。

最大稳定模型广泛应用于极端气候的建模，其参数估计一直是一个有挑战性的问题。通过对随机场数据使用了傅立叶变化构建了Whittle估计量用于最大稳定模型的参数估计。与传统的组合似然方法相比，Whittle估计量收敛速度更快，误差更小。进一步，我们把其中傅立叶变化的方法做推广，应用到三角矩阵型数据，并研究相关渐近性质。

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