Applications Of Nonparanetric Statistical Methods And EVT In Finance And Insurance

Extreme Value Theory is an important part of statistics, also an important tool in risk manage-ment so that it has many applications in finance, insurance and other related fields. In this thesis,we apply empirical likelihood method to the statistical inferences of extreme value index and high quantile and study the optimal multivariate quota-share reinsurance under a nonparametric mean-CVaR framework by nonparametric methods. This thesis includes several components as follows.First, we consider the estimation of the confidence intervals for the extreme value index when it is less than -1/2. In that case, we replace the unknown right endpoint by the maximum order statistic, and adopt the methodology of Lu and Peng (2002) to prove that the log empirical likelihood ratio function is asymptotic x2(1) distributed. The results of the simulation study indicate that the empirical likelihood is better than the normal approximation (Falk, 1995) in the sense of coverage probability and less sensitive to the selection of sample fraction k.Second, we further consider the problem of testing the equivalence of the risks in the left and the right tail of a distribution. Based on the Hill estimators of the extreme value indices and the two sample empirical likelihood method, we propose two different test statistics and prove the convergence results rigorously. The results of the simulation study and real data example show that our methods have lower empirical Type I errors and higher powers than the maximum likelihood ratio test (Jondeau and Rockinger, 2003).Third, we study the estimation of the confidence intervals for the high quantile. Extrapolating from two modified empirical likelihood ratio functions for intermediate quantiles, we then obtain the empirical likelihood ratio function for the extreme value index and the high quantile. By the profile method, a confidence interval for the high quantile can be constructed.Finally, we propose two nonparametric optimal multivariate quota-share reinsurance models under the mean-CVaR framework based on the empirical measure and kernel estimation method,which can be solved by linear programming and convex programming respectively. Statistical consistency of the resulting estimators for the best CVaR is established for both nonparametric models under mild conditions and the results of numerical examples verify our theoretical results.