# Estimating Distortion Risk Measures Under Truncated and Censored Data Scenarios

August 2020

Dissertation

## Degree Name

Doctor of Philosophy

## Department

Mathematics

Vytaras Brazauskas

## Committee Members

Daniel Gervini, David Spade, Wei Wei, Chao Zhu

## Keywords

Censored Data, Distortion Risk, Estimating Risk Measure, Risk Estimation, Risk Measure, Truncated Data

## Abstract

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ABSTRACT\\

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ESTIMATING DISTORTION RISK MEASURES UNDER TRUNCATED AND CENSORED DATA SCENARIOS

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~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.

COinS