Optimized quantile-based estimation of the parameters of a stable distribution

The quantile-based estimator proposed by McCulloch is a commonly used estimator of the parameters of a stable distribution. The quantile-based estimator is fast and has no convergence issues when used in the recommended parameter range. It can be used to provide an initial estimate for the maximum likelihood estimator. Disadvantages of the quantile-based estimator include its poor effciency at some values of the shape parameter α ∈ [0, 2] and the skewness parameter β ∈ [–1, 1] and that it is a consistent estimator only where α ≥ 0.6 (McCulloch).

In this paper, we identify alternative quantile levels for the quantile-based estimator that optimize its asymptotic efficiency for the joint estimation of α and β at selected values of α and β Asymptotic variances of these optimized estimators are calculated and compared with those of the maximum likelihood estimator. A two-step estimator is defined to demonstrate the practical use of the optimized estimators. We also investigate the invertibility of the quantile-based estimator functions and show that the quantile-based estimator is actually consistent over the wider range of parameters, (Formula presented.).