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A nonlinear population Monte Carlo scheme for the Bayesian estimation of parameters of α-stable distributions

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  • Koblents, Eugenia
  • Míguez, Joaquín
  • Rodríguez, Marco A.
  • Schmidt, Alexandra M.

Abstract

The class of α-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 α-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 α-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 α-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 α, 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.

Suggested Citation

  • Koblents, Eugenia & Míguez, Joaquín & Rodríguez, Marco A. & Schmidt, Alexandra M., 2016. "A nonlinear population Monte Carlo scheme for the Bayesian estimation of parameters of α-stable distributions," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 57-74.
    • Handle: RePEc:eee:csdana:v:95:y:2016:i:c:p:57-74
    DOI: 10.1016/j.csda.2015.09.007
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    References listed on IDEAS

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    1. Peters, G.W. & Sisson, S.A. & Fan, Y., 2012. "Likelihood-free Bayesian inference for α-stable models," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3743-3756.
    2. Lombardi, Marco J., 2007. "Bayesian inference for [alpha]-stable distributions: A random walk MCMC approach," Computational Statistics & Data Analysis, Elsevier, vol. 51(5), pages 2688-2700, February.
    3. repec:dau:papers:123456789/6072 is not listed on IDEAS
    4. Menn, Christian & Rachev, Svetlozar T., 2006. "Calibrated FFT-based density approximations for [alpha]-stable distributions," Computational Statistics & Data Analysis, Elsevier, vol. 50(8), pages 1891-1904, April.
    5. Mark A. Beaumont & Jean-Marie Cornuet & Jean-Michel Marin & Christian P. Robert, 2009. "Adaptive approximate Bayesian computation," Biometrika, Biometrika Trust, vol. 96(4), pages 983-990.
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    Cited by:

    1. Eugenia Koblents & Inés P. Mariño & Joaquín Míguez, 2019. "Bayesian Computation Methods for Inference in Stochastic Kinetic Models," Complexity, Hindawi, vol. 2019, pages 1-15, January.
    2. Jürgen Kampf & Georgiy Shevchenko & Evgeny Spodarev, 2021. "Nonparametric estimation of the kernel function of symmetric stable moving average random functions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(2), pages 337-367, April.
    3. Marc S. Paolella, 2016. "Stable-GARCH Models for Financial Returns: Fast Estimation and Tests for Stability," Econometrics, MDPI, vol. 4(2), pages 1-28, May.
    4. Marcin Pitera & Aleksei Chechkin & Agnieszka Wyłomańska, 2022. "Goodness-of-fit test for $$\alpha$$ α -stable distribution based on the quantile conditional variance statistics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(2), pages 387-424, June.

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