Date of Award

August 2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Vytaras Brazauskas

Committee Members

Lei Wang, Daniel Gervini, Chudamani Poudyal, Chao Zhu

Keywords

Goodness-of-fit, Least Squares, Loss Models, Quantiles, Relative Efficiency, Robustness

Abstract

Actuaries and statisticians use statistical models to predict future losses for pricing and other purposes. However, a key challenge in modeling is estimating the unknown parameters that index these distributions. Ensuring both efficiency and robustness of the chosen method is crucial, especially given the prevalence of outliers or extreme losses in insurance claims data. The primary objective of this dissertation is to introduce a robust, efficient, and computationally easy parameter estimation method that can be applied to various loss modeling scenarios. The proposed method exploits the joint asymptotic normality of sample quantiles (of i.i.d. random variables) to construct both ordinary and generalized Quantile Least Squares (QLS) estimators. We assess the method's efficiency using the asymptotic relative efficiency tool and its robustness through breakdown points and the influence function properties. In particular, the QLS estimators are constructed for the location-scale, log-location-scale, and Pareto I-IV families. These distributions were chosen due to their relevance in modeling claim size distributions in the insurance industry, as well as their widespread application in economics and numerous other fields. We establish the joint asymptotic normality of the proposed estimators and explore their finite-sample performance using simulations. Comparisons with the maximum likelihood estimator (MLE) are emphasized. Additionally, we demonstrate the QLS estimators' capability to handle outliers, particularly in scenarios where data are contaminated by atypical claims. We further demonstrate the computational advantage of the QLS estimators for big data scenarios, by evaluating the computational costs of these estimators and comparing them with that of MLEs. Furthermore, we develop two goodness-of-fit tests and study their performance using simulations and real data. For real data examples, we first consider Google stock returns from January 2, 2020 to December 29, 2023, and identify a logistic distribution as most appropriate for this data. Additionally, we analyze hurricane damages in the United States from 1925 to 1995 for which a Cauchy model offers the best fit.

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