  The class of $\alpha$-stable distributions enjoys multiple practical
applications in signal processing, finance, biology and other areas because it
allows to describe interesting and complex data patterns, such as asymmetry or
heavy tails, in contrast with the simpler and widely used Gaussian
distribution. The density associated with a general $\alpha$-stable
distribution cannot be obtained in closed form, which hinders the process of
estimating its parameters. A nonlinear population Monte Carlo (NPMC) scheme is
applied in order to approximate the posterior probability distribution of the
parameters of an $\alpha$-stable random variable given a set of random
realizations of the latter. The approximate posterior distribution is computed
by way of an iterative algorithm and it consists of a collection of samples in
the parameter space with associated nonlinearly-transformed importance weights.
A numerical comparison of the main existing methods to estimate the
$\alpha$-stable parameters is provided, including the traditional frequentist
techniques as well as a Markov chain Monte Carlo (MCMC) and a likelihood-free
Bayesian approach. It is shown by means of computer simulations that the NPMC
method outperforms the existing techniques in terms of parameter estimation
error and failure rate for the whole range of values of $\alpha$, including the
smaller values for which most existing methods fail to work properly.
Furthermore, it is shown that accurate parameter estimates can often be
computed based on a low number of observations. Additionally, numerical results
based on a set of real fish displacement data are provided.
