Topics on Statistical Inference for Extreme Values
Open Access
- Author:
- Zhang, Likun
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- July 07, 2020
- Committee Members:
- Benjamin Shaby, Dissertation Advisor/Co-Advisor
Benjamin Shaby, Committee Chair/Co-Chair
Ephraim Mont Hanks, Committee Member
Lin Lin, Committee Member
Eric B Ford, Outside Member
Martin Tingley, Special Member
Ephraim Mont Hanks, Program Head/Chair - Keywords:
- Global maximum
Profile likelihood
Noninformative
Objective Bayes
Asymptotic dependence
Censored likelihood - Abstract:
- The topics of this research include many aspects of statistical inference for extreme values, and they are summarized in three different chapters. In the first chapter, we study the classic asymptotic theory on the maximum likelihood estimator of the three-parameter generalized extreme value (GEV) distribution. GEV distribution arises from classical univariate extreme value theory and is in common use for analyzing the far tail of observed phenomena. Curiously, important asymptotic properties of likelihood-based estimation under this standard model have yet to be established. In this chapter, we formally prove that the maximum likelihood estimator is global and unique. An interesting secondary result entails the uniform consistency of a class of limit relations in a tight neighborhood of the shape parameter. In the second chapter, we derive a collection of reference prior distributions for Bayesian analysis under the three-parameter GEV distribution. These priors are based on an established formal definition of noninformativeness. They depend on the ordering of the three parameters, and we show that the GEV is unusual in that some orderings fail to yield proper posteriors for any sample size. We also consider a reparametrization that explicitly regards return level estimation, which is the most common goal of GEV analysis, to be the most important inferential task. We investigate the properties of the derived priors using simulation and apply them to an analysis of a fire threat index in California. In the third chapter, we present work on flexible modeling for spatial extremes. Flexible spatial models that allow transitions between tail dependence classes have recently appeared in the literature. However, inference for these models is computationally prohibitive, even in moderate dimensions, due to the necessity of repeatedly evaluating the multivariate Gaussian distribution function. In this work, we attempt to achieve truly high-dimensional inference for extremes of spatial processes, while retaining the desirable flexibility in the tail dependence structure, by modifying an established class of models based on scale mixtures of Gaussian processes. We show that the desired extremal dependence properties from the original models are preserved under the modification, and demonstrate that the corresponding Bayesian hierarchical model does not involve the expensive computation of the multivariate Gaussian distribution function. We fit our model to exceedances of a high threshold, and perform coverage analyses and cross-model checks to validate its ability to capture different types of tail characteristics. We use a standard adaptive Metropolis algorithm for model fitting, and further accelerate the computation via parallelization and Rcpp. Lastly, we apply the model to a dataset of a fire threat index on the Great Plains region of the US, which is vulnerable to massively destructive wildfires. We find that the joint tail of the fire threat index exhibits a decaying dependence structure that cannot be captured by limiting extreme value models.