  The class of stable distributions is used in practice to model data that
exhibit heavy tails and/or skewness. The stability index $\alpha$ of a stable
distribution is a measure of tail heaviness and is often of primary interest.
Existing methods for estimating the index parameter include maximum likelihood
and methods based on the sample quantiles. In this paper, a new approach for
estimating the index parameter of a stable distribution is proposed. This new
approach relies on the location-scale family representation of the class of
stable distributions and involves repeatedly partitioning the single observed
sample into two independent samples. An asymptotic likelihood method based on
sample order statistics, previously used for estimating location and scale
parameters in two independent samples, is adapted for estimating the stability
index. The properties of the proposed method of estimation are explored and the
resulting estimators are evaluated using a simulation study.
