Probabilistic Properties and Statistical Inference for Regularly Varying Time Series

规律变化的时间序列的概率性质和统计推断

• 批准号: RGPIN-2018-04156
• 负责人: Kulik, Rafal
• 金额: $ 2.04万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712756
• 关键词: Probabilistic; Properties; Statistical; Inference; Regularly

A classical assumption in probability and statistics is that data (random variables) are independent and approximately Gaussian. However, in many practical applications (like finance or insurance), the observed data are dependent (weakly or strongly) and heavy-tailed, that is, large positive or negative values occur with a higher probability than in a Gaussian case. Furthermore, extremal observations tend to form clusters, meaning that large positive or negative values occur at consecutive time points. As such, time series models suitable to capture extremal behaviour have to be used. My long-term vision is to advance probabilistic theory for clusters of extremes and use it to develop statistical techniques for dependent, heavy-tailed data, with the aim of answering very practical questions on how to estimate extremal characteristics and cluster indices. In this program, I will focus on the following aspects of time-series extremes. 1) Probabilistic properties of clusters. My goal is to develop a probabilistic description of clusters of large observations for heavy-tailed time series. The research program will involve three subtopics: 1A) Cluster properties under weak dependence; 1B) Cluster properties under strong dependence; 1C) Time series with extremal independence. Each of these subtopics requires significantly different mathematical approaches, however, such advanced tools as regular variation, weak and vague convergence of measures on infinite dimensional spaces will play a major role. Equipped with these tools, I will obtain new exciting results that will deepen our understanding of extremal behaviour of time series. Several new concepts, like a link between a tail process and clusters will be developed. 2) Statistical inference for clusters. Statistical inference for extremes presents many theoretical and practical challenges even in the case of independence. My goal is to tackle these challenging problems in the context of weak dependence, strong dependence as well as extremal independence, complementing the probabilistic development in the first part of the proposal. I will use advanced tools from the theory of empirical processes and resampling techniques. As such, I will further develop statistical theory and methodology for estimation of extremal characteristics and risk measures, extending or simplifying many existing limit theorems and providing new results. Furthermore, equipped with the probabilistic tools developed in the first part of the research program, I will establish new methodology for estimation of cluster indices in case of weak dependence. This is a completely new research direction. The limiting theory for estimators will be complemented with the corresponding resampling techniques. As such, my research will be not only of interest to academics, but also to practitioners, who will be able to use my software to perform statistical analysis.

概率和统计的经典假设是数据（随机变量）是独立的，并且大致是高斯。但是，在许多实际应用（例如金融或保险）中，观察到的数据取决于（弱或强）且重尾，即比高斯案例更高的概率出现大的正值或负值。此外，极端观测倾向于形成簇，这意味着在连续时间点发生大的正值或负值。因此，必须使用适合捕获极端行为的时间序列模型。我的长期愿景是推进极端群体的概率理论，并使用它来开发依赖，重尾数据的统计技术，以回答有关如何估计极端特征和集群指数的非常实用的问题。 在此计划中，我将重点介绍极端时间序列的以下方面。 1）集群的概率特性。 我的目标是对重尾时间序列的大型观察群制定概率描述。该研究计划将涉及三个子主题： 1a）依赖性较弱的群集特性； 1b）在强依赖性下的群集特性； 1C）时间序列具有极端独立性。 这些子主题中的每一个都需要明显不同的数学方法，但是，无限尺寸空间的定期变化，弱和模糊融合等高级工具将发挥主要作用。配备了这些工具，我将获得新的令人兴奋的结果，以加深我们对时间序列的极端行为的理解。将开发几个新概念，例如尾部过程和簇之间的链接。 2）集群的统计推断。 极端的统计推论即使在独立性的情况下也提出了许多理论和实际挑战。我的目标是在弱依赖性，强大的依赖和极端独立性的背景下解决这些具有挑战性的问题，并补充提案第一部分的概率发展。我将使用经验过程和重新采样技术理论中的先进工具。因此，我将进一步开发统计理论和方法，以估计极端特征和风险措施，扩展或简化许多现有的限制定理并提供新的结果。此外，在研究计划的第一部分中配备了概率工具，我将在依赖较弱的情况下建立新的方法来估算集群指数。这是一个全新的研究方向。估计器的限制理论将与相应的重采样技术相辅相成。因此，我的研究不仅对学者感兴趣，而且还会引起从业人员，他们将能够使用我的软件进行统计分析。