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Parameter Estimation for Symmetric Stable Distributions by Probability Integral Transformation

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Applied Information Science, Engineering and Technology

Part of the book series: Topics in Intelligent Engineering and Informatics ((TIEI,volume 7))

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Abstract

In this article a new parameter estimation method for symmetric stable distributions is presented, which is a variation of maximum likelihood type M-estimators. The estimator provides a joint estimation of shape parameter \(\alpha \), scale parameter \(\gamma \) and location parameter \(\delta \). The proposed method possesses all known good robustness performance properties, and is more reliable than other known methods. The estimation procedure does not use the density or the characteristic function (chf) directly, hence is faster than the maximum likelihood or chf based methods. A simulation study is performed to compare the proposed estimator with other methods based on performance properties and assessing convergence of the estimators.

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References

  1. Adler, J.R., Feldman, R.E., Taqqu, M.S. (eds.): A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Birkhäuser, Boston (1998)

    Google Scholar 

  2. Borak, Sz., H\(\ddot{a}\)rdle, W., Weron, R.: Stable Distributions, SFB 649 Discussion Papers, SFB649DP2005-008, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany (2005)

    Google Scholar 

  3. Csörgő, S., Viharos, L.: Estimating the tail index. In: Szyszkowicz, B. (ed.) Asymptotic Methods in Probability and Statistics, pp. 833–881. Elsevier Science, North-Holland (1998)

    Google Scholar 

  4. Davies, P.L.: On locally uniformly linearizable high breakdown location and scale functionals. Ann. Stat. 26, 1103–1125 (1998)

    Article  MATH  Google Scholar 

  5. Fama, E.F., Roll, R.: Parameter estimates of symmetric stable distributions. J. Am. Stat. Assoc. 66, 331–338 (1971)

    Article  MATH  Google Scholar 

  6. Fegyverneki, S.: Robust estimators and probability integral transformations. Math. Comput. Modelling 38, 803–814 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garcia, R., Renault, E., Veredas, D.: Estimation of stable distributions by indirect inference. J. Econometrics 161, 325–337 (2011)

    Article  MathSciNet  Google Scholar 

  8. Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics—The Approach Based on Influence Functions. Wiley, New York (1986)

    MATH  Google Scholar 

  9. Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)

    Article  MATH  Google Scholar 

  10. Huber, P.J.: Robust Statistics. Wiley, New York (1981)

    Book  MATH  Google Scholar 

  11. Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Stat. 35, 73–101 (1964)

    Article  MATH  Google Scholar 

  12. Kogon, S.M., Williams, D.B.: Characteristic function based estimation of stable parameters. In: Adler, R., Feldman, R., Taqqu, M. (eds.) A Practical Guide to Heavy Tails: Statistical Techniques and Applications, pp. 311–335. Birkhauser, Boston (1998)

    Google Scholar 

  13. Koutrouvelis, I.A.: Regression-type estimation of the parameters of stable laws. J. Am. Stat. Assoc. 75, 918–928 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Levy, P.: Calcul des Probabilités. Gauthier-Villars, Paris (1925)

    MATH  Google Scholar 

  15. McCulloch, J.H.: Simple consistent estimators of stable distribution parameters. Commun. Stat.—Simulation 15(4), 1109–1136 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nolan, J.P.: Maximum likelihood estimation of stable parameters. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.) Levy Processes: Theory and Applications, pp. 379–400. Birkhäuser, Boston (2001)

    Chapter  Google Scholar 

  17. Nolan, J.P.: Numerical calculation of stable densities and distribution functions. Commun. Stat.—Stoch. Model. 13, 759–774 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nolan, J.P.: website on stable distributions: http://academic2.american.edu/jpnolan/stable/stable.html

  19. Press, S.J.: Applied Multivariate Analysis. Holt, Rinehart and Winston, New York (1972)

    MATH  Google Scholar 

  20. Rachev, S.T., Mittnik, S.: Stable Paretian Models in Finance. Wiley, New York (2000)

    MATH  Google Scholar 

  21. Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman and Hall, New York (1994)

    MATH  Google Scholar 

  22. Szeidl, L.: Non-normal limit theorem for a new tail index estimation. Annales Univ. Sci. Budapest, Sect. Comp. 24, 307–322 (2004)

    Google Scholar 

  23. Uchaikin, V.V., Zolotarev, V.M.: Chance and Stability—Stable Distributions and their Applications. VSP, Utrecht (1999)

    Book  MATH  Google Scholar 

  24. Weron, R.: Performance of the Estimators of Stable Law Parameters, Hugo Steinhaus Center for Stochastic Methods, Research Report HSC/95/1 (1995)

    Google Scholar 

  25. Zolotarev, V.M.: One-dimensional Stable Distributions. Translations of Mathematical Monographs, vol 65. American Mathematical Society, Providence (1986)

    Google Scholar 

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Acknowledgments

The described work was carried out as part of the TÁMOP-4.2.2/B-10/1-2010-0008 project in the framework of the New Hungarian Development Plan. The realization of this project is supported by the European Union, co-financed by the European Social Fund.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Corvinus University of Budapest, Villányi út 29–33, Budapest, 1118, Hungary

    Csilla Csendes

  2. Department of Applied Mathematics, University of Miskolc, Miskolc-Egyetemvàros H-3515, Miskolc, Hungary

    Sándor Fegyverneki

Authors
  1. Csilla Csendes
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  2. Sándor Fegyverneki
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Correspondence to Csilla Csendes .

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Editors and Affiliations

  1. Faculty of Mechanical Engineering and Informatics, University of Miskolc, Miskolc-Egyetemváros, Hungary

    Gabriella Bognár

  2. Faculty of Mechanical Engineering and Informatics, University of Miskolc, Miskolc-Egyetemváros, Hungary

    Tibor Tóth

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Csendes, C., Fegyverneki, S. (2014). Parameter Estimation for Symmetric Stable Distributions by Probability Integral Transformation. In: Bognár, G., Tóth, T. (eds) Applied Information Science, Engineering and Technology. Topics in Intelligent Engineering and Informatics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-01919-2_1

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  • DOI: https://doi.org/10.1007/978-3-319-01919-2_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-01918-5

  • Online ISBN: 978-3-319-01919-2

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