A Bayesian approach for estimating the parameters of an α-stable distribution

References

• Mandelbrot B. The variation of certain speculative prices. J Bus. 1963;36:394–419.
   Web of Science ®Google Scholar
• Fama EF. The behaviour of stock market prices. J Bus. 1965;38:35–105.
   Google Scholar
• Fama EF, Roll R. Parametric estimates for symmetric stable distributions. J Am Stat Assoc. 1971;68:331–338.
   Google Scholar
• DuMouchel WH. Stable distributions in statistical inference: 1. Symmetric stable distributions compared to other long-tailed distribution. J Am Stat Assoc. 1973;68:469–477.
   Web of Science ®Google Scholar
• Crato N, Linhares RR, Lopes SRC. α-stable laws for noncoding regions in DNA sequences. J Appl Stat. 2011;38(2):261–271.
   Web of Science ®Google Scholar
• Achcar JA, Lopes SRC, Mazucheli J, et al. A Bayesian approach for stable distributions: some computational aspects. Open J Stat. 2013;3:268–277.
   Google Scholar
• Buckle D. Bayesian inference for stable distributions. J Am Stat Assoc. 1995;90:605–613.
   Web of Science ®Google Scholar
• Achcar JA, Lopes SRC. Linear and non-linear regression models assuming a stable distribution. Rev Colomb Estad. 2016;39(1):109–128.
   Google Scholar
• Prass TS, Lopes SRC, Achcar JA. MCMC Bayesian estimation in FIEGARCH models. Commun Stat Simul Comput. 2015;45(9):3238–3258.
   Web of Science ®Google Scholar
• Stein J, Lopes SRC, Medino AV. Continuous process derived from the solution of generalized Langevin equation: theoretical properties and some simulations. J Stat Comput Simul. 2016;86:2819–2845.
   Web of Science ®Google Scholar
• Zolotarev VM. One-dimensional stable distributions. Providence: American Mathematical Society; 1986. (Translations of mathematical monographs; vol. 65).
   Google Scholar
• Nolan JP. Parameterizations and modes of stable distributions. Stat Probab Lett. 1998;38(2):187–195.
   Web of Science ®Google Scholar
• Lombardi MJ. Bayesian inference for α-stable distributions: a random walk MCMC approach. Comput Stat Data Anal. 2007;51(5):2688–2700.
   Web of Science ®Google Scholar
• Menn C, Rachev ST. Calibrated FFT-based density approximations for α-stable distributions. Comput Stat Data Anal. 2006;50(8):1891–1904.
   Web of Science ®Google Scholar
• Nolan JP. Numerical calculation of stable densities and distribution functions. Comm Statist Stoch Models. 1997;13(4):759–774.
   Google Scholar
• Casarin R. Bayesian inference for mixtures of stable distributions. CEREMADE, University Paris IX; 2004. (Working paper No. 0428).
   Google Scholar
• Samorodnitsky G, Taqqu MS. Stable non-gaussian random processes: stochastic models with infinite variance. Boca Raton: Chapman & Hall/CRC; 1994.
   Google Scholar
• Garoni TM, Frankel NE. Lévy flights: exact results and asymptotics beyond all orders. J Math Phys. 2002;43(5):2670–2689.
   Web of Science ®Google Scholar
• Holtsmark J. Über die Verbreiterung von Spektrallinien. Ann Phys. 1919;363(7):577–630.
   Google Scholar
• Andrews GE, Askey R, Roy R. Special functions. Cambridge: Cambridge University Press; 1999. (Encyclopedia of mathematics and its applications; vol. 71).
   Google Scholar
• Lambert P, Lindsey JK. Analysing financial returns by using regression models based on non-symmetric stable distributions. J R Stat Soc Ser C. 1999;48(3):409–424.
   Google Scholar
• Swihart B, Lindsey J, Lambert P. stable: probability functions and generalized regression models for stable distributions. R package version 1.1.4. 2019. Available from: https://CRAN.R-project.org/package=stable.
   Google Scholar
• Wuertz D, Maechler M, and Rmetrics core team members. stabledist: stable distribution functions. R package version 0.7-1. 2016. Available from: https://CRAN.R-project.org/package=stabledist.
   Google Scholar
• Ament S, O'Neil M. Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics. Stat Comput. 2018;28(1):171–185.
   Web of Science ®Google Scholar
• Mittnik S, Rachev ST, Doganoglu T, et al. Maximum likelihood estimation of stable Paretian models. Math Comput Model. 1999;29(10-12):275–293.
   Google Scholar
• Nolan JP. Maximum likelihood estimation and diagnostics for stable distributions. In: Barndorff-Nielsen OE, Resnick SI, Mikosch T, editors. Lévy processes: theory and applications. Boston: Birkhäuser; 2001. p. 379–400.
   Google Scholar
• Bergström H. On some expansions of stable distributions. Ark Mat. 1952;2(28):375–378.
   Google Scholar
• D'Agostino RB, Stephens MA. Goodness-of-fit techniques. New York: Marcel Dekker; 1986.
   Google Scholar
• Beaulieu M-C, Dufour J-M, Khalaf L. Exact confidence sets and goodness-of-fit methods for stable distributions. J Econom. 2014;181:3–14.
   Web of Science ®Google Scholar
• Gelfand AE, Smith AFM. Sampling-based approaches to calculating marginal distributions. J Am Stat Assoc. 1990;85(410):398–409.
   Web of Science ®Google Scholar
• Chib S, Greenberg E. Understanding the metropolis-hastings algorithm. Am Stat. 1995;49(4):327–335.
   Web of Science ®Google Scholar
• Lunn DJ, Thomas A, Best N, et al. WinBUGS – a Bayesian modelling framework: concepts, structure, and extensibility. Stat Comput. 2000;10:325–337.
   Web of Science ®Google Scholar
• Plummer M. JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling. 2003.
   Google Scholar
• Devroye L. Non- uniform random variate generation. New York: Springer-Verlag; 1986.
   Google Scholar
• Varian HR. A Bayesian approach to real estate assessment. In: Fienberg SE, Zellner A, editors. Studies in Bayesian econometrics and statistics: In honor of Leonard J. Savage. Amsterdam: North-Holland Publishing Company; 1975. p. 195–208.
   Google Scholar
• Zellner A. Bayesian estimation and prediction using asymmetric loss functions. J Am Stat Assoc. 1986;81(394):446–451.
   Web of Science ®Google Scholar
• Ali S. On the Bayesian estimation of the weighted Lindley distribution. J Stat Comput Simul. 2015;85(5):855–880.
   Web of Science ®Google Scholar
• Yousaf R, Aslam M, Ali S. Bayesian estimation of the transmuted Frechet distribution. Iran J Sci Technol Trans A Sci. 2019;43(4):1629–1641. DOI:10.1007/s40995-018-0581-1
   Google Scholar
• Dey S, Ali S, Kumar D. Weighted inverted Weibull distribution: properties and estimation. J Stat Manag Syst. 2020;23(5):843–885. DOI:10.1080/09720510.2019.1669344
   Web of Science ®Google Scholar
• Kumar D, Kumar P, Singh SK, et al. A new asymmetric loss function: estimation of parameter of exponential distribution. J Stat Appl Probab Lett. 2019;6(1):37–50.
   Google Scholar
• Spiegelhalter DJ, Best NG, Carlin BP, et al. Bayesian measures of model complexity and fit. J R Stat Soc Ser B. 2002;64(4):583–639.
   Google Scholar
• Borak S, Härdle WK, Weron R. Stable distributions. 2005. (SFB 649 discussion paper; Vol. 2005(8)).
   Google Scholar
• Dufour J-M, Kurz-Kim J-R. Exact inference and optimal invariant estimation for the stability parameter of symmetric α-stable distributions. J Empir Finance. 2010;17:180–194.
   Web of Science ®Google Scholar
• Moreira RC. The effect of case management method on the glycemia control of people with type 2 diabetes mellitus [doctoral thesis]. Curitiba: Nursing Post-Graduation Program, Federal University of Paraná; 2013. Available from: http://hdl.handle.net/1884/31778.
   Google Scholar