The method of moments ratio estimator for the tail shape parameter

The so-called Hill estimator for the shape parameter of the tail distribution is known to be downwardly biased. The Hill estimator is a moment estimator, based on the first conditional moment of the highest logarithmically transformed data. We propose a new estimator for the tail index based on the ratio of the second to the first conditional moment. This estimator has a smaller bias than the Hill estimator. We provide simulation results that demonstrate a sizable reduction in bias when a is large, while the MSE is moderated as well. The new estimator is applied to stock return data in order to resolve a long standing issue in economics.