Estimation of Parameters in Heavy-Tailed Distribution When Its Second Order Tail Parameter is Known

Journal of Statistical Planning and Inference, 140(7) (2010), 1957-1967
http://dx.doi.org/10.1016/j.jspi.2010.01.046

Estimating parameters in heavy-tailed distribution plays a central role in extreme value theory. It is well known that classical estimators based on the first order asymptotics such as the Hill, rank-based and QQ estimators are seriously biased under finer second order regular variation framework. To reduce the bias, many authors proposed the so-called second order reduced bias estimators for both first and second order tail parameters. In this work, estimation of parameters in heavy-tailed distributions are studied under the second order regular variation framework when the second order parameter in the distribution tail is known. This is motivated in large part by a recent work by the authors showing that the second order tail parameter is known for a large class of popular random difference equations (for example, ARCH models). The focus is on least squares estimators that generalize rank-based and QQ estimators. Though other possible estimators are also briefly discussed, the least squares estimators are most simple to use and perform best for finite samples in Monte Carlo simulations.