Kybernetika 50 no. 6, 929-949, 2014

Parameter estimation of sub-Gaussian stable distributions

Vadym OmelchenkoDOI: 10.14736/kyb-2014-6-0929

Abstract:

In this paper, we present a parameter estimation method for sub-Gaussian stable distributions. Our algorithm has two phases: in the first phase, we calculate the average values of harmonic functions of observations and in the second phase, we conduct the main procedure of asymptotic maximum likelihood where those average values are used as inputs. This implies that the main procedure of our method does not depend on the sample size of observations. The main idea of our method lies in representing the partial derivative of the density function with respect to the parameter that we estimate as the sum of harmonic functions and using this representation for finding this parameter. For fifteen summands we get acceptable precision. We demonstrate this methodology on estimating the tail index and the dispersion matrix of sub-Gaussian distributions.

Keywords:

characteristic function, maximum likelihood, stable distribution, sub-Gaussian distribution

Classification:

93E12, 62A10

paper.pdf

References:

  1. M. Carrasco and J. Florens: Generalization of GMM to a continuum moment condition. Econom. Theory 16 (2000), 767-834.   CrossRef
  2. J. M. Chambers, C L. Mallows and B. W. Stuck: A method for simulating stable random variables. J. Amer. Statist. Assoc. 71 1976), 340-344.   CrossRef
  3. B. N. Cheng and S. Rachev: Multivariate stable securities in financial markets. Math. Finance 5 (1995), 133-153.   CrossRef
  4. W. H. DuMouchel: Stable Distributions in Statistical Inference. PhD. Thesis, University of Ann Arbor, Ann Arbor 1971.   CrossRef
  5. E. Fama: Portfolio analysis in a stable Paretian market. Management Sci. 11 (1965), 404-419.   CrossRef
  6. B. M. Hill: A simple general approach to inference about the tail of a stable distribution. Ann. Stat. 3 (1975), 5, 1163-1174.   CrossRef
  7. R. A. Horn and C. R. Johnson: Matrix Analysis. Cambridge University Press 1985.   CrossRef
  8. A. Kagan: Fisher information contained in a finite-dimensional linear space, and a properly formulated version of the method of moments (in Russian). Problemy Peredachi Informatsii 12 (2009), 15-29.   CrossRef
  9. L. Klebanov: Heavy Tailed Distributions. Matfyzpress, Prague 2003.   CrossRef
  10. I. A. Koutrovelis: Regression-type estimation of the parameters of stable laws. J. Amer. Statist. Assoc. 75 (1980), 918-928.   CrossRef
  11. S. Kring, S. Rachev, M. H{ö}chst{ö}tter and F. J. Fabozzi: Estimation of Alpha-Stable Sub-Gaussian Distributions for Asset Returns. In: Risk Assessment: Decisions in Banking and Finance. Physica-Verlag, Heidelberg 2008, pp. 111-152.   CrossRef
  12. D. B. Madan and E. Seneta: The variance gamma model from shared market returns. J. Bus. 63 (1990), 511-524.   CrossRef
  13. B. Mandelbrot: The variation of certain speculative prices. J. Bus. 26 (1963), 394-419.   CrossRef
  14. J. H. McCulloch: Simple consistent estimators of stable distribution parameters. Commun. Statist. -- Simula 15 (1986), 1109-1136.   CrossRef
  15. J. H. McCulloch: Estimation of the bivariate stable spectral representation by the projection method. Comput. Econom. 16 (2000), 47-62.   CrossRef
  16. S. Mittnik and S. Rachev: Tail estmation of the stable index alpha. Applied mathematics. Letters 9 (1996), 3, 53-56.   CrossRef
  17. S. Mittnik and M. S. Paolella: Prediction of Financial Downside-Risk with Heavy-Tailed Conditional Distributions    CrossRef
  18. J. P. Nolan: Modeling Financial Data with Stable Distributions. In: Handbook of Heavy Tailed Distributions in Finance, Handbooks in Finance: Book 1 (2003), pp. 105-130.   CrossRef
  19. J. P. Nolan: Maximum likelihood estimation and diagnostics for stable distributions. In: L{é}vy Processes (O. E. Barndorff-Nielsen, T. Mikosch, and S. Resnick, eds.), Brikhauser, Boston 2001.   CrossRef
  20. J. P. Nolan and A. K. Panorska: Data analysis for heavy tailed multivariate samples. Commun. Statist.: Stochastic Models (1997), 687--702.   CrossRef
  21. V. Omelchenko: Elliptical stable distributions. In: Mathematical Methods in Economics 2010 (M. Houda and J. Friebelova, eds.), pp. 483-488.   CrossRef
  22. S. Ortobelli, I. Huber, S. Rachev and E. S. Schwarz: Portfolio Choice Theory with Non-Gaussian Distributed Return. In: Handbook of Heavy Tailed Distributions in Finance, Handbooks in Finance: Book 1 (2003), pp. 547-594.   CrossRef
  23. M. Pivato and L. Seco: Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis. J. Multivariate Anal. 87 (2003), 2, 219-240.   CrossRef
  24. S. T. Rachev, E. S. Schwarz and I. Khindanova: Stable Modeling of Market and Credit Value at Risk. In: Handbook of Heavy Tailed Distributions in Finance, Handbooks in Finance: Book 1 (2003), pp. 255-264.   CrossRef
  25. G. Samorodnitsky and M. S. Taqqu: Stable Non-Gaussian Random Processes. Chapman and Hall 1994.   CrossRef
  26. P. Schmidt: An improved version of Quandt-Ramsey MGF estimator for mixtures of normal distributions and switching regressions. Econometrica 50 (1982), 501-524.   CrossRef
  27. L. Slámová and L. Klebanov: Modeling financial returns by discrete stable distributions. In: Proc. 30th International Conference Mathematical Methods in Economics 2012.   CrossRef
  28. K. C. Tran: Estimating mixtures of normal distribution via empirical characteristic function. Econom. Rev. 17 (1998), 167-83.   CrossRef
  29. V. Zolotarev: On representation of stable laws by integrals selected translation. Math. Statist. Probab. 6 (1986), 84-88.   CrossRef