Moment estimators for the two-parameter M-Wright distribution

Computational Statistics volume 27, pages487–497(2012)Cite this article

Abstract

A formal parameter estimation procedure for the two-parameter M-Wright distribution is proposed. This procedure is necessary to make the model useful for real-world applications. Note that its generalization of the Gaussian density makes the M-Wright distribution appealing to practitioners. Closed-form estimators are also derived from the moments of the log-transformed M-Wright distributed random variable, and are shown to be asymptotically normal. Tests using simulated data indicated favorable results for our estimation procedure.

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

References

  1. Beghin L, Orsingher E (2009) Fractional Poisson processes and related planar random motions. Electron J Probab 15: 1790–1826

    MathSciNet  Google Scholar 

  2. Beghin L, Orsingher E (2010) Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron J Probab 15: 684–709

    MathSciNet  MATH  Article  Google Scholar 

  3. Cahoy DO (2007) Fractional Poisson process in terms of alpha-stable densities. PhD Thesis Case Western Reserve University

  4. Cahoy DO (2011) Estimation and simulation for the M-Wright function. Commun Stat Theory Methods (To appear)

  5. Cahoy DO, Polito F (2011) Simulation and estimation for the fractional Yule process. Methodol Comput Appl Probab (To appear)

  6. Ferguson T (1996) A course in large sample theory. Chapman & Hall, Great Britain

    Google Scholar 

  7. Germano G, Politi M, Scalas E, Schilling RL (2009) Stochastic calculus for uncoupled continuous-time random walks. Physical Review E 79, http://pre.aps.org/abstract/PRE/v79/i6/e066102 and http://arxiv.org/abs/0802.3769

  8. Gorenflo R, Mainardi F (2009) Some recent advances in theory and simulation of fractional diffusion processes. J Comput Appl Math 229: 400–415

    MathSciNet  MATH  Article  Google Scholar 

  9. Gorenflo R, Luchko Y, Mainardi F (1999) Analytical properties and applications of the Wright function. Fractional Calc Appl Anal 2: 383–414

    MathSciNet  MATH  Google Scholar 

  10. Gorenflo G, Mainardi F, Vivoli A (2007) Continuous-time random walk and parametric subordination in fractional diffusion. Chaos, Solitons & Fractals 34: 87–103

    MathSciNet  MATH  Article  Google Scholar 

  11. Haubold HJ, Mathai AM, Saxena RK (2009) Mittag-Leffler functions and their applications. http://arxiv.org/abs/0909.0230

  12. Mainardi F, Luchko Y, Pagnini G (2001) The fundamental solution of the space-time fractional diffusion equation. Fractional Calc Appl Anal 4:153–192, E-print http://arxiv.org/abs/cond-mat/0702419

    Google Scholar 

  13. Mainardi F, Pagnini G (2003) The Wright functions as solutions of the time-fractional diffusion equation. Appl Math Comput 141: 51–62

    MathSciNet  MATH  Article  Google Scholar 

  14. Mainardi F, Pagnini G, Gorenflo R (2003) Mellin transform and subordination laws in fractional diffusion processes. Fractional Calc Appl Anal 6:441–459, E-print http://arxiv.org/abs/math/0702133

    Google Scholar 

  15. Mainardi F, Mura A, Pagnini G (2010) The M-Wright function in time-fractional diffusion processes: a tutorial survey. Int J Differ Equ 2010:29. doi:10.1155/2010/104505

  16. Mura A, Pagnini G (2008) Characterizations and simulations of a class of stochastic process to model anomalous diffusion. J Phys A: Math Theor 41: 285003

    MathSciNet  Article  Google Scholar 

  17. Mura A, Taqqu MS, Mainardi F (2008) Non-Markovian diffusion equations and processes: analysis and simulations. Physica A 387:5033–5064, E-print http://arxiv.org/abs/0712.0240

    Google Scholar 

  18. Orsingher E, Polito F (2010) Fractional pure birth processes. Bernoulli 16: 858–881

    MathSciNet  MATH  Article  Google Scholar 

  19. Orsingher E, Polito F, Sakhno L (2010) Fractional non-linear, linear and sublinear death processes. J Stat Phys 141: 68–93. doi:10.1007/s10955-010-0045-2

    MathSciNet  MATH  Article  Google Scholar 

  20. Podlubny I (1999) Fractional differential equations. Mathematics in science and engineering, vol 198. Academic Press, San Diego

    Google Scholar 

  21. Uchaikin VV, Cahoy DO, Sibatov RT (2008) Fractional processes: from Poisson to branching one. Int J Bifurcation and Chaos 18: 2717–2725

    MathSciNet  MATH  Article  Google Scholar 

  22. Uchaikin VV, Zolotarev VM (1999) Chance and stability: stable distributions and their applications. VSP, Utrecht, The Netherlands

    Google Scholar 

  23. Zolotarev VM (1986) One-dimensional stable distributions: translations of mathematical monographs. American Mathematical Society, United States of America

    Google Scholar 

Download references

Author information

Affiliations

  1. Program of Mathematics and Statistics, College of Engineering and Science, Louisiana Tech University, Ruston, LA, USA

    Dexter O. Cahoy

Authors
  1. Dexter O. Cahoy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dexter O. Cahoy.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cahoy, D.O. Moment estimators for the two-parameter M-Wright distribution. Comput Stat 27, 487–497 (2012). https://doi.org/10.1007/s00180-011-0269-x

Download citation

Keywords

  • Wright function
  • M-Wright
  • Mittag-Leffler
  • Financial modeling
  • Economics