Abstract
We propose estimators for the parameters of the Linnik L(α, γ) distribution. The estimators are derived from the moments of the log-transformed Linnik distributed random variable, and are shown to be asymptotically unbiased. The estimation algorithm is computationally simple and less restrictive. Our procedure is also tested using simulated data.
Similar content being viewed by others
References
Anderson DN (1992) A multivariate Linnik distribution. Stat Probab Lett 14(4): 333–336
Anderson DN, Arnold BC (1993) Linnik distributions and processes. J Appl Probab 30(2): 330–340
Arslan O (2008) An alternative multivariate skew Laplace distribution: properties and estimation, Stat Papers 30(2). doi:10.1007/s00362-008-0183-7
Bening VE, Korolev VY, Kolokol’tsov VN, Saenko VV, Uchaikin VV, Zolotarev VV (2004) Estimation of parameters of fractional stable distributions. J Math Sci 123(1): 3722–3732
Cahoy DO, Uchaikin VV, Woyczynski WA (2010) Parameter estimation in fractional Poisson processes. J Stat Plan Inference 140(11): 3106–3120
Devroye L (1990) A note on Linnik’s distribution. Stat Probab Lett 9: 305–306
Ferguson T (1996) A course in Large sample theory. Chapman & Hall, London
Haan LD, Resnick SI (1980) A simple asymptotic estimate for the index of a stable distribution. J R Stat Soc B 42: 83–87
Hall P (1982) On some simple estimates of an exponent of regular variation. J R Stat Soc B 44: 37–42
Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3: 1163–1174
Jacques C, Rémillard B, Theodorescu R (1999) Estimation of Linnik law parameters. Stat Decis 17(3): 213–236
Kotz S, Ostrovskii IV (1996) A mixture representation of the Linnik distribution. Stat Probab Lett 26(1): 61–64
Kozubowski TJ (1999) Geometric stable laws: estimation and applications. Math Comput Model 29: 241–253
Kozubowski TJ (2001) Fractional moment estimation of Linnik and Mittag–Leffler parameters. Math Comput Model 34: 1023–1035
Kuttykrishnan AP, Jayakumar K (2006) Bivariate semi α-Laplace distribution and processes. Stat Papers 49(2): 303–313
Leitch RA, Paulson AS (1975) Estimation of stable law parameters: stock price behavior application. J Am Stat Assoc 70: 690–697
Lin GD (1998) A note on the Linnik distributions. J Math Anal Appl 217: 701–706
Linnik JV (1963) Linear forms and statistical criteria. I. II. Select Trans Math Stat Probab 3: 1–90
Pakes AG (1998) Mixture representations for symmetric generalized Linnik laws. Stat Probab Lett 37: 213–221
Paulson AS, Holcomb EW, Leitch RA (1975) The estimation of the parameters of the stable laws. Biometrika 62: 163–170
Press RN (1972) Estimation in univariate and multivariate stable distribution. J Am Stat Assoc 67: 842–846
Zolotarev VM (1986) One-dimensional stable distributions: translations of mathematical monographs, Vol 65. American Mathematical Society, Providence
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cahoy, D.O. An estimation procedure for the Linnik distribution. Stat Papers 53, 617–628 (2012). https://doi.org/10.1007/s00362-011-0367-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-011-0367-4