Estimating Distortion Risk Measures Under Truncated and Censored Data Scenarios

Abstract

\begin{center} ABSTRACT\\ \vspace{0.4in} ESTIMATING DISTORTION RISK MEASURES UNDER TRUNCATED AND CENSORED DATA SCENARIOS \end{center} \doublespacing \noindent ~In insurance data analytics and actuarial practice, a broad class of risk measures -- {\em distortion risk measures\/} -- are used to capture the riskiness of the distribution tail. Point and interval estimates of the risk measures are then employed to price extreme events, to develop reserves, to design risk transfer strategies, and to allocate capital. When solving such problems, the main statistical challenge is to choose an appropriate estimate of a risk measure and to assess its variability. In this context, the empirical nonparametric approach is the simplest one to use, but it lacks efficiency due to the scarcity of data in the tails. On the other hand, parametric estimators, although prone to model mis-specification, can improve estimators' efficiency significantly. Moreover, they can easily accommodate data truncation and censoring that are common features of insurance loss data. The first objective of this dissertation is to derive the asymptotic distributions of empirical and parametric estimators of distortion risk measures under the truncated and censored data scenarios. For parametric estimation, we use maximum likelihood (ML) and percentile matching (PM) procedures. The risk measures we consider include: {\em value-at-risk\/} (VaR), {\em conditional tail expectation\/} ({\sc cte}), {\em proportional hazards transform\/} ({\sc pht}), {\em Wang transform\/} ({\sc wt}), and {\em Gini shortfall\/} ({\sc gs}). Conditions under which these measures are finite are studied rigorously. The ML and PM estimators of the risk measures are derived for three severity models (with identical support): shifted exponential, Pareto I, and shifted lognormal. Their asymptotic properties are established and compared with those of the empirical estimators. Then, the second objective of the dissertation is to cross-validate and augment the theoretical results using simulations. Finally, the third objective is to provide a few numerical examples involving applications of the new estimators to actual reinsurance data.