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**Approximated maximum likelihood estimation of parameters of discrete stable family.**
*(English)*
Zbl 1308.60022

Summary: 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 A. M. Kagan [Probl. Peredaci Inform. 12, No. 2, 20–42 (1976; Zbl 0379.62004)] 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.

### MSC:

60E07 | Infinitely divisible distributions; stable distributions |

60E10 | Characteristic functions; other transforms |

65C60 | Computational problems in statistics (MSC2010) |

62F12 | Asymptotic properties of parametric estimators |

### Citations:

Zbl 0379.62004
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\textit{L. Slámová} and \textit{L. B. Klebanov}, Kybernetika 50, No. 6, 1065--1076 (2014; Zbl 1308.60022)

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### References:

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[4] | Kagan, A. M.: 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. · Zbl 0379.62004 |

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[7] | Slámová, L., Klebanov, L. B.: Modelling financial returns with discrete stable distributions. Proc. 30th International Conference Mathematical Methods in Economics (J. Ramík and D. Stavárek, Silesian University in Opava, School of Business Administration in Karviná, 2012, pp. 805-810. |

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