Date of Award

Summer 5-15-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Risk Management and Insurance

First Advisor

Liang Peng

Second Advisor

Haci Akcin

Third Advisor

Samuel Harry Cox

Fourth Advisor

Gengsheng Qin

Abstract

Recently the Haezendonck-Goovaerts (H-G) risk measure is receiving much attention in actuarial science with applications to optimal portfolios and optimal reinsurance because of its advantage in well quantifying the tail behavior of losses. This thesis systematically studies statistical inferences of the H-G risk measure under various settings including heavy-tailed losses, fixed and intermediate risk levels.

The thesis starts by proposing an empirical likelihood inference for the H-G risk measure for two different risk levels---fixed risk level and intermediate risk level. More specifically, Chapter 2 considers the case of fixed risk level, and the derived asymptotic limit of a nonparametric inference is employed to construct an interval for the H-G risk measure. Chapter 3 considers the case of intermediate risk level, i.e., the level is treated as a function of the sample size and goes to one as the sample size tends to infinity. The proposed maximum empirical likelihood estimator for the H-G risk measure has a different limit from that for the case of a fixed level. But the proposed empirical likelihood method indeed gives a unified interval estimation for both cases.

Chapter 4 proposes a two-part estimation for the H-G risk measure and the proposed estimators always have an asymptotic normal distribution regardless of the moment conditions. To achieve this, we separately estimate the tail part by extreme value theory and the middle part non-parametrically.

The above chapters focus on independent data. In Chapter 5, we extend our methodology from independent data to dependent data and conduct the sensitivity analysis of a portfolio under the H-G risk measure. We first derive an expression for computing the sensitivity of the H-G risk measure, which enables us to estimate the sensitivity non-parametrically via the H-G risk measure. Second, we derive the asymptotic distributions of the nonparametric estimators for the H-G risk measure and its sensitivity by assuming that loss variables in the portfolio follow from a strictly stationary alpha-mixing sequence. Finally, this estimation combining with a bootstrap method is applied to a real dataset.

Besides the study of the H-G risk measure, we investigate the estimation of the finite endpoint of a distribution function when normally distributed measurement errors contaminate the observations. Under the framework of extreme value theory, we propose a class of estimators for the standard deviation of the measurement errors as well as for the endpoint. Asymptotic properties of the proposed estimators are established and simulations demonstrate their good finite sample performance.

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