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The class of $\alpha$-stable distributions enjoys multiple practicalapplications in signal processing, finance, biology and other areas because itallows to describe interesting and complex data patterns, such as asymmetry orheavy tails, in contrast with the simpler and widely used Gaussiandistribution. The density associated with a general $\alpha$-stabledistribution cannot be obtained in closed form, which hinders the process ofestimating its parameters. A nonlinear population Monte Carlo (NPMC) scheme isapplied in order to approximate the posterior probability distribution of theparameters of an $\alpha$-stable random variable given a set of randomrealizations of the latter. The approximate posterior distribution is computedby way of an iterative algorithm and it consists of a collection of samples inthe 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 frequentisttechniques as well as a Markov chain Monte Carlo (MCMC) and a likelihood-freeBayesian approach. It is shown by means of computer simulations that the NPMCmethod outperforms the existing techniques in terms of parameter estimationerror and failure rate for the whole range of values of $\alpha$, including thesmaller values for which most existing methods fail to work properly.Furthermore, it is shown that accurate parameter estimates can often becomputed based on a low number of observations. Additionally, numerical resultsbased on a set of real fish displacement data are provided.

A nonlinear population Monte Carlo scheme for the Bayesian estimation of parameters of $\alpha$-stable distributions