Skip to main content
SpringerLink
Log in

Wavelet-Based Estimation for Univariate Stable Laws

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Stable distributions are characterized by four parameters which can be estimated via a number of methods, and although approximate maximum likelihood estimation techniques have been proposed, they are computationally intensive and difficult to implement. This article describes a fast, wavelet-based, regression-type method for estimating the parameters of a stable distribution. Fourier domain representations, combined with a wavelet multiresolution approach, are shown to be effective and highly efficient tools for inference in stable law families. Our procedures are illustrated and compared with other estimation methods using simulated data, and an application to a real data example is explored. One novel aspect of this work is that here wavelets are being used to solve a parametric problem, rather than a nonparametric one, which is the more typical context in wavelet applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abramovich F., Bailey T.C., Sapatinas T. (2000). Wavelet analysis and its statistical applications. The Statistician 49:1–29

    Google Scholar 

  • Antoniadis A. (1997). Wavelets in statistics: a review (with discussion). Journal of Italian Statistical Society 6:97–144

    Article  Google Scholar 

  • Antoniadis A., Grégoire G., McKeague I. (1994). Wavelet methods for curve estimation. Journal of American Statistical Association 89(428):1340–1353

    Article  MATH  Google Scholar 

  • Antoniadis A., Grégoire G., Nason G. (1999). Density and hazard rate estimation for right- censored data using wavelet methods. Journal of Royal Statistical Society, B 61:63–84

    Article  MATH  Google Scholar 

  • Bruce, A. G., Gao, H. -Y. (1994). S+Wavelets, users manual. Seattle: StatSci.

  • Buckheit J.B., Donoho D. (1995). Wavelab and reproducible research. In: Antoniadis A., Oppenheim G. (eds). Wavelets and statistics. Lecture Notes in Statistics (vol. 103). Springer, Berlin Heidelberg New york

    Google Scholar 

  • Chambers J.M., Mallows C.L., Stuck B.W. (1976). A method for simulating stable random variables. Journal of American Statistical Association 71:340–344

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Y. (1991). Distributions for Asset Returns. PhD Thesis, Department of Economics, SUNY-Stony Brook.

  • Christof G., Wolf G. (1992). Convergence theorems with a stable limit law. Akademie Verlag, Berlin

    Google Scholar 

  • Chui K. (1992). Wavelets: A tutorial in theory and applications. Academic, Boston

    MATH  Google Scholar 

  • Daubechies I. (1992). Ten lectures on wavelets. CBMS-NSF Regional Conferences Series in Applied Mathematics. SIAM, Philadelphia

    MATH  Google Scholar 

  • Donoho D.L., Johnstone I.M., Kerkyacharian G., Picard D. (1996). Density estimation by wavelet thresholding. The Annals of Statistics 24(2):508–539

    Article  MathSciNet  MATH  Google Scholar 

  • DuMouchel, W. H. (1971). Stable distributions in statistical inference. PhD Dissertation, Yale University.

  • DuMouchel W.H. (1973a). On the asymptotic normality of maximum likelihood estimates when sampling from a stable distribution. The Annals of Statistics 1:948–957

    MathSciNet  MATH  Google Scholar 

  • DuMouchel W.H. (1973b). Stable distributions in statistical inference I: Symmetric stable distributions compared to other symmetric long-tailed distributions. Journal of American Statistical Association 68:469–482

    Article  MathSciNet  MATH  Google Scholar 

  • DuMouchel W.H. (1975). Stable distributions in statistical inference II: Information from stably distributed samples. Journal of American Statistical Association 70:386–393

    Article  MathSciNet  MATH  Google Scholar 

  • DuMouchel W.H. (1983). Estimating the stable index α in order to measure tail thickness: a critique. The Annals of Statistics 11:1019–1031

    MathSciNet  MATH  Google Scholar 

  • Fama E. (1965). The behaviour of stock prices. Journal of Business 38:34–105

    Article  Google Scholar 

  • Fama E., Roll R. (1968). Some properties of symmetric stable distributions. Journal of American Statistical Association 63:817–836

    Article  MathSciNet  Google Scholar 

  • Fama E., Roll R. (1971). Parameter estimates for symmetric stable distributions. Journal of American Statistical Association 66:331–338

    Article  MATH  Google Scholar 

  • Feller W. (1971). An introduction to probability theory and its applications (Vol 2, 2nd ed). Wiley, New York

    Google Scholar 

  • Feuerverger A., McDunnough P. (1981a). On some Fourier methods for inference. Journal of American Statistical Association 76:379–387

    Article  MathSciNet  MATH  Google Scholar 

  • Feuerverger A., McDunnough P. (1981b). On efficient inference in symmetric stable laws and processes. In: A.K. Md. E. Saleh, et al. (Eds.), Proceedings of the International Symposium on Statistics and Related Topics, Ottawa, May 1980, (pp. 109–122). Amsterdam: North Holland.

  • Feuerverger A., McDunnough P. (1984). On statistical transform methods and their efficiency. Canadian Journal of Statistics 12:303–317

    MathSciNet  MATH  Google Scholar 

  • Flandrin P. (1992). Wavelet analysis and synthesis of fractional Brownian motion. IEEE Transactions on Information Theory 38:910–917

    Article  MathSciNet  Google Scholar 

  • FracLab. (2002). INRIA Project Fractales. Open source freeware distributed by INRIA-Fractales. http://www-rocq.inria.fr/fractales.

  • Gao H.-Y. (1993). Wavelet estimation of spectral densities in time series analysis. PhD Thesis, University of California, Berkeley

    Google Scholar 

  • Kogon S.M., Williams D.B. (1998). Characteristic function based estimation of stable distribution parameters. In: Adler R.J., Feldman R.E., Taqqu M.S. (eds). A practical guide to heavy tails. Birkhauser, Boston, pp. 311–335

    Google Scholar 

  • Gnedenko V.B., Kolmogorov A.N. (1954). Limit distributions for Sums of Independent Random Variables. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Hall P., Patil P. (1995). On wavelet methods for estimating smooth functions. Bernoulli 1:41–58

    Article  MathSciNet  MATH  Google Scholar 

  • Hill B.M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3:1163–1174

    MathSciNet  MATH  Google Scholar 

  • Holschneider M. (1995). Wavelets: An analysis tool. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Ibragimov I.A., Linnik Yu.V. (1971). Independent and stationary sequences of random variables. Wolters-Noordhoff, Groningen

    MATH  Google Scholar 

  • Janicki A., Weron A. (1994). Simulation and chaotic behaviour of α-stable stochastic processes. Marcel Dekker, New York

    Google Scholar 

  • Johnstone I.M., Kerkyacharian G., Picard D. (1992). Estimation d’une densité de probabilité par méthode d’ondelettes. Comptes Rendus de l’Academie Sciences, Paris A, 315:211–216

    MathSciNet  MATH  Google Scholar 

  • Koutrouvelis I.A. (1980). Regression-type estimation of the parameters of stable laws. Journal of American Statistical Association 75:918–928

    Article  MathSciNet  MATH  Google Scholar 

  • Koutrouvelis I.A. (1981). An iterative procedure for the estimation of the parameters of the stable law. Communications in Statistics Part B–Simulation and Computation 10:17–28

    MathSciNet  Google Scholar 

  • Lukacs E. (1970). Characteristic functions. Hafner, Connecticut

    MATH  Google Scholar 

  • Mallat S.G. (1989). A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11:674–693

    Article  MATH  Google Scholar 

  • Mallat S.G. (1998). A wavelet tour of signal processing. Academic, San Diego

    MATH  Google Scholar 

  • Mandelbrot B.B. (1963). The variation of certain speculative prices. Journal of Business, 36:394–419

    Article  Google Scholar 

  • Mandelbrot B.B. (1972). The variation of certain speculative prices. Journal of Business 45:542–543

    Article  Google Scholar 

  • McCulloch J.H. (1986) Simple consistent estimators of stable distribution parameters. Communications in Statistics Part B—Simulation and Computation 15:1109–1136

    MathSciNet  MATH  Google Scholar 

  • McCulloch J.H. (1998) Linear regression with stable disturbances. In: Adler R., Feldman R.E., Taqqu M.S. (eds). A practical guide to heavy tails: Statistical techniques and applications. Birkhäuser, Boston, pp. 359–376

    Google Scholar 

  • Mittnik M., Rachev S. (1993). Modeling asset returns with alternative stable distributions. Econometric Review 12:261–330

    MathSciNet  MATH  Google Scholar 

  • Mittnik M., Rachev S. (eds) (2000). Stable Paretian models in finance. Wiley, New York

    MATH  Google Scholar 

  • Nason G.J., Silverman B.W. (1994). The discrete wavelet transform in S. Journal of Computational and Graphical Statistics 3:163–191

    Article  Google Scholar 

  • Nolan J. (1997). Numerical calculation of stable densities and distribution functions. Communications in Statistics—Stochastic Models 13:759–774

    MathSciNet  MATH  Google Scholar 

  • Nolan J. (2001). Maximum likelihood estimation and diagnostics for stable distributions. In: Barndorff-Nielson O.E., Mikosch T., Resnick S.I. (eds). Lévy processes. Birkhauser, Boston, pp. 379–400

    Google Scholar 

  • Ogden T.R. (1997). Essential wavelets for statistical applications and data analysis. Birkhäuser, Basel

    MATH  Google Scholar 

  • Paulson A.S., Halcomb E.W., Leitch R.A. (1975). The estimation of the parameters of the stable laws. Biometrika 62:163–170

    Article  MathSciNet  MATH  Google Scholar 

  • Press S.J. (1972). Estimation in univariate and multivariate stable distributions. Journal of American Statistical Association 67:842–846

    Article  MathSciNet  MATH  Google Scholar 

  • Samorodnitsky G., Taqqu M. (1994). Stable non-Gaussian random processes: Stochastic models with infinite variance. Chapman & Hall, New York

    MATH  Google Scholar 

  • Strang G. (1986). Introduction to Applied Mathematics. Addison-Wellesley, Cambridge

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Vidakovic B. (1999). Statistical modeling by wavelets. Wiley, New York

    MATH  Google Scholar 

  • Zolotarev, V. M. (1986). One-dimensional stable distributions. Translations of Mathematical Monographs (Vol. 65). Providence, Rhode Island: American Mathematical Society (Translation of the original Russian edition of 1983).

Download references

Author information

Authors and Affiliations

  1. Laboratoire IMAG-LMC, University Joseph Fourier, BP 53, 38041, Grenoble Cedex 9, France

    Anestis Antoniadis

  2. Department of Statistics, University of Toronto, Toronto, ON, Canada, M5S 3G3

    Andrey Feuerverger

  3. INRIA Rhône-Alpes, ZIRST, 655 Avenue de l’Europe, 38330, Monbonnot Saint Martin, France

    Paulo Gonçalves

Authors
  1. Anestis Antoniadis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrey Feuerverger
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paulo Gonçalves
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrey Feuerverger.

About this article

Cite this article

Antoniadis, A., Feuerverger, A. & Gonçalves, P. Wavelet-Based Estimation for Univariate Stable Laws. AISM 58, 779–807 (2006). https://doi.org/10.1007/s10463-006-0042-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-006-0042-z

Keywords

Search

Navigation