High-Efficiency Estimation for the Positive Stable Laws

Abstract

A moment method based on the sample averages of X ^-t and X ^-2t (with suitably chosen t) is suggested for estimating the index a and scale parameter c ^1/α of the positive stable distribution with Laplace transform E exp(-ΛX) = exp(-cΛ^α). The corresponding Cramér-Rao lower bounds are computed by using a new and simple Fourier technique, which is based on an idea of Jarrett. It is found that the asymptotic efficiencies of the proposed estimators exceed .97 for all values of α between zero and .994. A comparison is made with the optimal asymptotic efficiencies attainable by using a different moment method based on e ^{-t ₁ X} and e ^{2-t ₂ X} .

Key Words:

• Stable distribution
• Information matrix
• Cramér-Rao lower bound
• Moment estimation
• Asymptotic efficiency