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Bayesian inference for alpha-stable distributions: a random walk MCMC approach

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Abstract

The alpha-stable family of distributions constitutes a generalization of the Gaussian distribution, allowing for asymmetry and thicker tails. Its practical usefulness is coupled with a marked theoretical appeal, given that it stems from a generalized version of the central limit theorem in which the assumption of the finiteness of the variance is replaced by a less restrictive assumption concerning a somehow regular behavior of the tails. The absence of the density function in a closed form and the associated estimation difficulties have however hindered its diffusion among practitioners. In this paper I introduce a novel approach for Bayesian inference in the setting of alpha-stable distributions that resorts to a FFT of the characteristic function in order to approximate the likelihood function; the posterior distributions of the parameters are then produced via a random walk MCMC method. Contrary to the other MCMC schemes proposed in the literature, the proposed approach does not require auxiliary variables, and so it is less computationally expensive, especially when large sample sizes are involved. A simulation exercise highlights the empirical properties of the sampler; an application on audio noise data demonstrates how this estimation scheme performs in practical applications.

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  • Marco J. Lombardi, 2004. "Bayesian inference for alpha-stable distributions: a random walk MCMC approach," Econometrics Working Papers Archive wp2004_11, Universita' degli Studi di Firenze, Dipartimento di Statistica, Informatica, Applicazioni "G. Parenti".
  • Handle: RePEc:fir:econom:wp2004_11
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    1. Marco J. Lombardi & Simon J. Godsill, 2004. "On-line Bayesian estimation of AR signals in symmetric alpha-stable noise," Econometrics Working Papers Archive wp2004_05, Universita' degli Studi di Firenze, Dipartimento di Statistica, Informatica, Applicazioni "G. Parenti".
    2. Tsionas, Efthymios G., 1998. "Monte Carlo inference in econometric models with symmetric stable disturbances," Journal of Econometrics, Elsevier, vol. 88(2), pages 365-401, November.
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    1. Lombardi, Marco J. & Calzolari, Giorgio, 2009. "Indirect estimation of [alpha]-stable stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2298-2308, April.
    2. Adam Misiorek & Rafal Weron, 2010. "Heavy-tailed distributions in VaR calculations," HSC Research Reports HSC/10/05, Hugo Steinhaus Center, Wroclaw University of Technology.
    3. 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.
    4. Szymon Borak & Adam Misiorek & Rafał Weron, 2010. "Models for Heavy-tailed Asset Returns," SFB 649 Discussion Papers SFB649DP2010-049, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    5. Marco J. Lombardi & Simon J. Godsill, 2004. "On-line Bayesian estimation of AR signals in symmetric alpha-stable noise," Econometrics Working Papers Archive wp2004_05, Universita' degli Studi di Firenze, Dipartimento di Statistica, Informatica, Applicazioni "G. Parenti".
    6. Strid, Ingvar, 2010. "Efficient parallelisation of Metropolis-Hastings algorithms using a prefetching approach," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2814-2835, November.
    7. Marco J. Lombardi & Giorgio Calzolari, 2004. "Indirect estimation of alpha-stable distributions and processes," Econometrics Working Papers Archive wp2004_07, Universita' degli Studi di Firenze, Dipartimento di Statistica, Informatica, Applicazioni "G. Parenti".
    8. Dominicy, Yves & Veredas, David, 2013. "The method of simulated quantiles," Journal of Econometrics, Elsevier, vol. 172(2), pages 235-247.

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    Keywords

    Alpha-stable distributions; Infinite variance; MCMC.;

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