Excess over Threshold Distribution Function Estimation

The excess over threshold (EOT) distribution is the conditional distribution of the excesses over a threshold, given that the threshold has been exceeded. We propose empirical and kernel-based plug-in estimators of the EOT distribution function and derive their asymptotic bias, variance, and the limiting distributions of the studentized estimators as the sample size increases. We compare the asymptotic efficiency of the proposed estimators and obtain the asymptotically optimal bandwidth for the kernel-based estimator. The kernel estimator with optimal bandwidth is found to be asymptotically more accurate than the empirical estimator. We also prove strong uniform and L_2 convergence of the proposed estimators. Simulations reveal that the kernel estimator outperforms both the empirical and the generalized pareto distribution approximations of the EOT distribution function, especially for sample size up to 500. Using two real data sets, we demonstrate the applications of the estimators in the fields of hydrology and seismology.