Kybernetika 50 no. 6, 1065-1076, 2014

Approximated maximum likelihood estimation of parameters of discrete stable family

Lenka Slámová and Lev B. KlebanovDOI: 10.14736/kyb-2014-6-1065


In this article we propose a method of parameters estimation for the class of discrete stable laws. Discrete stable distributions form a discrete analogy to classical stable distributions and share many interesting properties with them such as heavy tails and skewness. Similarly as stable laws discrete stable distributions are defined through characteristic function and do not posses a probability mass function in closed form. This inhibits the use of classical estimation methods such as maximum likelihood and other approach has to be applied. We depart from the $\mathcal{H}$-method of maximum likelihood suggested by Kagan (1976) where the likelihood function is replaced by a function called informant which is an approximation of the likelihood function in some Hilbert space. For this method only some functionals of the distribution are required, such as probability generating function or characteristic function. We adopt this method for the case of discrete stable distributions and in a simulation study show the performance of this method.


parameter estimation, maximum likelihood, discrete stable distribution




  1. L. Devroye: A triptych of discrete distributions related to the stable law. Stat. Probab. Lett. 18 (1993), 349-351.   CrossRef
  2. A. Feuerverger and P. McDunnough: On the efficiency of empirical characteristic function procedure. J. Roy. Stat. Soc. Ser. B 43 (1981), 20-27.   CrossRef
  3. O. V. Gerlein and A. M. Kagan: Hilbert space methods in classical problems of mathematical statistics. J. Soviet Math. 12 (1979), 184-213.   CrossRef
  4. A. M. Kagan: Fisher information contained in a finite-dimensional linear space, and a correctly posed version of the method of moments (in Russian). Problemy Peredachi Informatsii 12 (1976), 20-42.   CrossRef
  5. L. B. Klebanov and I. A. Melamed: Several notes on Fisher information in presence of nuisance parameters. Statistics: J. Theoret. Appl. Stat. 9 (1978), 85-90.   CrossRef
  6. L. B. Klebanov and L. Slámová: Integer valued stable random variables. Stat. Probab. Lett. 83 (2013), 1513-1519.   CrossRef
  7. L. Slámová and L. B. Klebanov: Modelling financial returns with discrete stable distributions. In: Proc. 30th International Conference Mathematical Methods in Economics (J. Ramík and D. Stavárek, eds.), Silesian University in Opava, School of Business Administration in Karviná, 2012, pp. 805-810.   CrossRef
  8. F. W. Steutel and K. van Harn: Discrete analogues of self-decomposability and stability. Ann. Probab. 7 (1979), 893-899.   CrossRef