Confidence Intervals for the Tail Index

One of the best-known estimators for the tail index of a heavy-tailed distribution is the Hill estimator. In this paper, confidence intervals based on the asymptotic normal approximation of Hill estimator are studied. The coverage accuracy is evaluated and the theoretical optimal choice of the sample fraction for the one-sided confidence interval is given. One surprising finding is that the order of optimal coverage accuracy for the one-sided confidence interval depends on the sign of the second-order regular variation.