Tail dependence: application, exploration, and development of novel methods - Wixson, Troy P., author; Cooley, Daniel, advisor; Shaby, Benjamin, advisor; Huang, Dongzhou, committee member; Wang, Tianying, committee member; Barnes, Elizabeth, committee member (Colorado State University. Libraries, 2025)
The study of multivariate extreme events is largely concerned with modeling the dependence in the tail of the joint distribution. The understanding of extremal dependence and methodology for modeling that dependence has been an active research field over the past few decades and we contribute to that literature with three projects that are detailed in this dissertation. In the first project we consider the challenge of assessing the changing risk of wildfires. Wildfire risk is greatest during high winds after sustained periods of dry and hot conditions. This chapter is a statistical extreme event risk attribution study which aims to answer whether extreme wildfire seasons are more likely now than under past climate. This requires modeling temporal dependence at extreme levels. We propose the use of transformed-linear time series models which are constructed similarly to traditional ARMA models while having a dependence structure that is tied to a widely used framework for extremes (regular variation). We fit the models to the extreme values of the seasonally adjusted Fire Weather Index (FWI) time series to capture the dependence in the upper tail for past and present climate. Ten-thousand fire seasons are simulated from each fitted model and we compare the proportion of simulated high-risk fire seasons to quantify the increase in risk. Our method suggests that the risk of experiencing an extreme wildfire season in Grand Lake, Colorado under current climate has increased dramatically compared to the risk under the climate of the mid-20th century. Our method also finds some evidence of increased risk of extreme wildfire seasons in Quincy, California, but large uncertainties do not allow us to reject a null hypothesis of no change. In the second project we explore a fundamental characterization of tail dependence and develop a method to classify data into the two regimes. Classifying a data set as asymptotically dependent (AD) or asymptotically independent (AI) is a necessary early choice in the modeling of multivariate extremes. These two dependence regimes are defined asymptotically which complicates inference as practitioners have finite samples. We perform a series of experiments to determine whether a finite sample has enough information for a convolutional neural network to reliably distinguish between these regimes in the bivariate case. Along the way we develop a new classification tool for practitioners which we call nnadic as it is a Neural Network for Asymptotic Dependence/Independence Classification. This tool accurately classifies 95\% of test datasets and is robust to a wide range of sample sizes. The datasets which we are unable to correctly classify tend to either be nearly exactly independent or exhibit near perfect dependence, which are boundary cases for both the AD and AI models used for training. In the third project we consider the challenge of using likelihood methods for models developed for the tail of the distribution. Many multivariate extremes models have intractable likelihoods thus practitioners must use alternative fitting methods and likelihood-based methods for uncertainty quantification and model selection are unavailable. We develop a proxy-likelihood estimator for multivariate extremes models. Our method is based on the tail pairwise dependence (TPD) which is a summary measure of the dependence in the tail of any multivariate extremes model. The TPD parameter has a one-to-one relationship with the dependence parameter of the HR distribution. We use the HR distribution as a proxy for the likelihood in a composite likelihood approach. The method is demonstrated using the transformed linear extremes time series (TLETS) models of Mhatre & Cooley (2024).
Keywords: extreme value theory; statistics; wildfire; machine learning; asymptotic dependence; time series