Optimal index estimation of heavy-tailed distributions

Abstract

The optimal parameter estimation problem is considered. The optimization problem is solved in the general problem statement. A model-free approach is applied and supposes no knowledge of the model that the parameter to be estimated belongs to. Optimality of the considered estimators in the sense of a special type risk function is established. The considered risk function makes it possible to optimize the asymptotic variances of the estimators and is used for sample size estimation. Applications for optimization of the truncated parameter estimators of heavy-tailed indexes of distributions, such as Pareto type, Cauchy, and log-gamma, are presented. A class of these estimators is introduced having guaranteed accuracy based on a sample of fixed size. Simulation results confirm theoretical results.

Keywords:

• Cauchy distribution
• heavy tails
• log-gamma distribution
• optimal convergence rate
• optimal parameter estimation
• Pareto-type distributions