Estimation of parameters in heavy - tailed distribution when its second order tail parameter is known ∗ †

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). Several least squares estimators, generalizing rank-based and QQ-estimators, and conditional maximum likelihood estimators, based on the exact form of second order regular variation, are proposed here and their basic asymptotics are established. Several other estimators adapting existing approaches (for example, that of Feuerverger and Hall) are also studied. Numerical performance of all proposed estimators is examined through Monte Carlo simulations.