On the confidence intervals of parametric functions for Distributions Generated by Symmetric Stable Laws


  • Davood Farbod Quchan Institute of Engineering and Technology
  • Karen V. Gasparian Yerevan State University




In this paper we consider “Discrete Distributions Generated by Standard Symmetric Stable Densities” (DSSD in short) arising in Bioinformatics (Astola and Danielian, 2007). Using well-known asymptotic properties of the maximum likelihood (ML) estimators we obtain the respective asymptotic confidence intervals for some useful parametric functions
of the DSSD.


J. ASTOLA, E. DANIELIAN, (2007), Frequency Distributions in Biomolecular Systems and Growing Networks, Tampere International Center for Signal Processing (TICSP), Series no. 31, Tampere, Finland.

J. ASTOLA, E. DANIELIAN, A. ARAKELYAN, (2007), Frequency distributions in growing biomolecular networks based on stable densities, Journal Reports National Academy of Sciences of Armenia, 107(1), pp. 26-36.

J. ASTOLA, E. DANIELIAN, A. ARAKELYAN, (2008), On distance in variation for frequency distributions generated by stable laws, Journal Reports National Academy of Sciences of Armenia, 108(2), pp. 99-109.

J. ASTOLA, E. DANIELIAN, S. ARZUMANYAN, (2010), Frequency distributions in bioinformatics, A Review, Journal Proceedings Yerevan State University: Phys. Math. Sci., 223(3), pp. 3-22.

A. A. BOROVKOV, (1998), Mathematical Statistics, Gordon and Breach Science Publishers, (translated from original Russian edition).

E. DANIELIAN, J. ASTOLA, (2004), On the steady state of birth-death process with coefficients of moderate growth, Facta Universitatis, Series: Elec. Energ., 17(3), pp. 405-419.

E. DANIELIAN, J. ASTOLA, (2006), On regularly varying hypergeometric distributions, In Astola et al. (eds.), Proceedings International TICSP Workshop on Spectral Methods and Multirate Signal Processing, Florence, Italy, 2-3 Sept. 2006. TICSP Series no. 34, pp. 127-132.

W. H. DUMOUCHEL, (1973), On the asymptotic normality of the maximum likelihood estimate when sampling from a stable distribution, Annals of Statistics, 1(5), pp. 948-957.

D. FARBOD, (2007), The asymptotic properties of some discrete distributions generated by symmetric stable laws, Journal Information Technologies and Management, Armenia, 10, pp. 49-55.

D. FARBOD, (2011), M-estimators as GMM for Stable Laws Discretizations, Journal of Statistical Research of Iran, 8(1), pp. 85-96.

D. FARBOD, K. V. GASPARIAN, (2008), Asymptotic properties of maximum likelihood estimator for some discrete distributions generated by Cauchy stable law, Statistica, Bologna, 68(3), pp. 321-326.

V. A. KUZNETSOV, (2003), Family of skewed distributions associated with the gene expression and proteome evolution, Signal Processing, Elsevier, 33(4), pp. 889-910.

E. L. LEHMANN, (1983), Theory of Point Estimation, Wiley & Sons.

M. MATSUI, A. TAKEMURA, (2004), Some improvements in numerical evaluation of symmetric stable density and its derivatives, Technical Report, CIRJE-F University of Tokyo, http://arxiv.org/abs/math/0408321.

J. P. NOLAN, (2010), Stable Distributions - Models for Heavy Tailed Data, Boston: Chapter 1 online at academic2.american.edu/~jpnolan.

V. M. ZOLOTAREV, (1986), One-dimensional Stable Distributions, Amer. Math. Soc., (translated from original 1983 Russian edition: Odnomernye Ustoichivye Raspredeleniya).




How to Cite

Farbod, D., & Gasparian, K. V. (2012). On the confidence intervals of parametric functions for Distributions Generated by Symmetric Stable Laws. Statistica, 72(4), 405–413. https://doi.org/10.6092/issn.1973-2201/3699