Threshold Selection In Tail Index Estimation

The tail index is an important parameter to decide the degree of heaviness of the tail of a distribution. In the analysis of financial data the tail of the distribution imply the serious loss of a financial institution led by potential disastrous events.Most of the loss/profit distributions of financial assets have heavy tails .So how to estimate the tail index better is very important for the risk managers and supervisors of financial institutions.Hill estimator is significant to estimate the tail of a heavy distribution.It appeals by virtue of its simplicity and interesting asymptotic properties.However ,the performance of the Hill estimator depends on the adaptive choice of its threshold .How to select the optimal threshold is a delicate matter. This paper provide four methods for estimation of the optimal threshold which minimize the AMSE (asymptotic mean squared error) of the corresponding Hill estimators bootstrap methods,direct estimation of the AMSE of the Hill estimators,the special method for the Hall class and a sequential procedure.This paper estimates the optimal threshold for the estimation of the tail index of daily exchange rate returns of the U.S. dollar relative to yen with the different threshold selection methods which are then compared.