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.


60E07 Infinitely divisible distributions; stable distributions
60E10 Characteristic functions; other transforms
65C60 Computational problems in statistics (MSC2010)
62F12 Asymptotic properties of parametric estimators


Zbl 0379.62004
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