NSF Award
2413516
With the increasing availability of high-resolution data, curve-type data have emerged in various fields. Examples include daily precipitation curves, daily pollution level patterns, and intraday stock return curves. As highly fluctuating patterns in these curves become increasingly common due to the growing impact of extreme events, such as unusual weather or financial downturns, it is crucial to effectively analyze and predict these extreme patterns for risk management across all domains. Currently, there is a notable lack of appropriate statistical tools for analyzing extremal behavior in such data, primarily due to its complexity. To address this critical gap, this project aims to develop innovative methodologies for accurate modeling and quantifying extremal patterns in curve-type data. The outcomes of the project have a potential to enhance preparedness for natural disasters and to advances risk assessment in the financial sector. For instance, the tools developed can forecast the likelihood of simultaneous extreme precipitation patterns in different locations, thereby aiding in developing more efficient risk mitigation strategies for natural disasters like flash floods. This capability is crucial in regions with diverse topographies, such as West Virginia, where mountainous terrain with numerous creeks and rivers is susceptible to flash floods during intense rainfall, as seen in the rare 2016 event that caused significant damage and loss of life. The risk assessment tools are adaptable beyond West Virginia, benefiting other states facing similar challenges in managing extreme weather events. Additionally, these tools can help financial institutions manage risk exposure by determining the likelihood of concurrent extreme losses in intraday return patterns across different sectors. Given that millions of Americans have savings in retirement plans, accurately quantifying the risk of catastrophic financial losses is essential. By providing precise measurements of risks associated with extreme market conditions, this project supports national efforts to safeguard economic security. Furthermore, it will contribute to workforce development by training undergraduate and graduate students in statistics and mathematics research.<br/><br/>This project introduces a new framework for analyzing and modeling extremal behavior in functional data. The research agenda aims to develop statistical tools for quantifying extremal dependence in paired functional samples and to create statistical hypothesis tests for the independence of heavy-tailed functional time series. Specifically, this project will develop a novel tool—the extremal correlation coefficient—to measure how likely extreme curves exhibit similar patterns simultaneously. For instance, it can answer questions such as: how likely is it for location A to experience heavy precipitation patterns similar to those observed in location B on the same day? Or, during a stock market crisis, do returns of different sectors exhibit similar extreme daily trajectories? Additionally, based on the extremal correlation coefficient, the project will propose a new autocorrelation function and a portmanteau white noise test tailored for heavy-tailed functional time series. Currently, no method exists to detect serial dependence structures in such functional time series. These tools will evaluate the serial correlation of heavy-tailed functional time series and validate the independence of the model residuals. By leveraging the mathematical theory of regularly varying measures for functional objects, the project aims to ensure the asymptotic properties of the proposed estimator for the extremal correlation coefficient and establish theoretical results for the portmanteau white noise test.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
