Algorithm for statistical estimation of the parameters of fractional-stable distributions is described in this article. This algorithm is constructed on the basis of the method of distance minimization between the empirical and theoretical distributions. As the distance between the two distributions, the χ distance is considered. The main difficulty in dealing with fractional-stable distributions is the absence of explicit expressions for the probability density function. That is why the theoretical density is estimated by the histogram method. The results of the test calculations and the results of the estimation of the quadratic deviation are presented. The results obtained by this estimator are compared with the results obtained by the method of moments. An example of the use of the estimator for the approximation of the experimental data obtained in the investigation of gene expression by RNA sequencing technology is given as well. It is shown that the probability density function of the gene expression in a wide-enough domain can be described by fractional-stable distributions.
Similar content being viewed by others
References
C. R. Dance and E. E. Kuruglu, “Estimation of the Parameters of Skewed a-Stable Distributions,” Technical Report (1999).
E. E. Kuruoglu, “Density parameter estimation of skewed amp;alpha;-stable distributions,” IEEE Trans. Signal Process., 49, No. 10, 2192–2201 (2001).
V. M. Zolotarev, “Statistical estimates of the parameters of stable laws,” Math. Stat.: Banach Center Publ., 6, 359–376 (1980).
V. M. Zolotarev, One-dimensional stable Distributions. Amer. Mat. Soc., Providence (1986).
V. E. Bening, V. Y. Korolev, V. N. Kolokoltsov, V. V. Uchaikin, V. V. Saenko, and V. M. Zolotarev, “Estimation of parameters of fractional stable distributions,” J. Math. Sci., 123, No. 1, 3722 – 3732 (2004).
A. S. Paulson, E. W. Holcomb, and R. A. Leitch, “The estimation of the parameters of the stable laws,” Biometrika, 62, No. 1, 163–170 (1975).
G. J. Worsdale, “The Estimation of the Symmetric Stable Distribution Parameters,” Metrika, 23, No. 1, 55–63 (1981).
B.Wade Brorsen and S. R. Yang, “Maximum Likelihood Estimates of Symmetric Stable Distribution Parameters,” Commun. Stat. Simul. Comput., 19, 1459–1464 (1990).
J. P. Nolan, “Maximum likelihood estimation and diagnostics for stable distributions,” L´evy Processes: Theory and Applications, Springer, New York, 379–400 (2001).
N. Ravishanker, “Monte Carlop EM Estimation for Stable Distribution,” in: Heavy Tails’99, (1999), p. Tails45.
V. V. Saenko, “Maximum likelihood algorithm for approximation of local fluctuational fluxes at the plasma periphery by fractional stable distributions,” arXivID: 1209.2297 (2012).
V. V. Uchaikin and V. V. Saenko, “Simulation of random vectors with isotropic fractional stable distributions and calculation of their probability density function,” J. Math. Sci., 112, No. 2, 4211 – 4228 (2002).
E. F. Fama and R. Roll, “Parameter estimates for symmetric stable distributions,” J. Am. Stat. Assoc., 66, No. 334, 331–338 (1971).
S. Maymon, J. Friedman, E. Fisher, and H. Messer-Yaron, “Estimation of the Parameters of a Stable Distribution Based on Order Statistics,” in: Heavy Tails’99, (1999), p. Tails37.
V. N. Kolokoltsov, V. Y. Korolev, and V. V. Uchaikin, “Fractional Stable Distributions,” J. Math. Sci., 105, No. 6, pp. 2569–2576 (2001).
B. Bunday, Basic Optimization Methods, Hodder Arnold (1984).
H. R. Ueda, S. Hayashi, S. Matsuyama, T. Yomo, S. Hashimoto, S. A. Kay, J. B. Hogenesch, and M. Iino, “Universality and flexibility in gene expression from bacteria to human.,” Proc. Natl. Acad. Sci. USA, 101, 3765–3769 (2004).
L. S. Liebovitch, V. K. Jirsa, and L. A. Shehadeh, “Structure of genetic regulatory networks: evidence for scale free networks,” in Complexus Mundi - Emergent Patterns in Nature, World Scientific Publishing Co. Pte. Ltd., Singapore (2006), pp. 1–8.
C. Furusawa and K. Kaneko, “Zipfs Law in Gene Expression,” Phys. Rev. Lett., 90, 8–11 (2003).
V. A. Kuznetsov, G. D. Knott, and R. F. Bonner, “General statistics of stochastic process of gene expression in eukaryotic cells,” Genetics, 161, 1321–1332 (2002).
D. C. Hoyle, M. Rattray, R. Jupp, and A. Brass, “Making sense of microarray data distributions,” Bioinformatics (Oxford, England), 18, 576–84 (2002).
C. Lu and R. D. King, “An investigation into the population abundance distribution of mRNAs, proteins, and metabolites in biological systems,” Bioinformatics (Oxford, England), 25, 2020–2027 (2009).
J. M. Chambers, C. L. Mallows, and B. W. Stuck, “A method for simulating stable random variables,” J. Am. Stat. Assoc., 71, No. 354, 340–344 (1976).
M. Kanter, “Stable Densities Under Change of Scale and Total Variation Inequalities,” Ann. Probab., 3, No. 4, 697–707 (1975).
Author information
Authors and Affiliations
Corresponding author
Additional information
*The work was supported by the Ministry of Education and Science of the Russian Federation (grant No 6.1617.2014/K)
Proceedings of the XXXII International Seminar on Stability Problems for Stochastic Models, Trondheim, Norway, June 16–21, 2014.
Rights and permissions
About this article
Cite this article
Saenko, V.V. Estimation of the Parameters of Fractional-Stable Laws by the Method of Minimum Distance*. J Math Sci 214, 101–114 (2016). https://doi.org/10.1007/s10958-016-2760-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-016-2760-y