Domain of Attraction of -stable Distributions under Finite Mixture Models

Authors

  • C.E.G. Otiniano
  • C.R. Gonçalves

DOI:

https://doi.org/10.5540/tema.2010.011.01.0069

Abstract

In this work, we study the asymptotic distribution of the normalized sum of independent, identically distributed random variables under the finite mixture models. In the Theorem we give necessary conditions for a distribution function of a mixed population with k components to belong to the domain of attraction of an α-stable distribution, by assuming that each component of the mixture also pertains to the domain of attraction of an α-stable distribution. Examples are given to illustrate the result.

References

P. Embrechts, T. Mikosch, C. Kluppelberg, “Modelling Extremal Events for Insurance and Finance”, Springer - Verlag, Berlin, 1997.

B.S. Everitt, D.J. Hand, “Finite Mixture Distributions”, Chapman and Hall, London, 1981.

W. Feller, “An Introduction to Probability Theory and its Applications II”, 3rd edition, Wiley, New York, 1968.

I.A. Ibragimov, Yu. V. Linnik, “Independent and Stationary Sequences of Ran- dom Variables”, Wolters-Noordhoff, Groningen, 74–85, 1971.

B. G. Lindsay, “Mixture Models: Theory, Geometry and Applications”, Insti- tute of Mathematical and Statistics, Hayward, CA, 1995.

G.J. McLachlan, K.E.Basford, “Mixture Models: Aplications to Clustering”, Marcel Dekker, New York, 1988.

N. Ravishanker, D.K. Dey, Multivariate survival models with a mixture of positive stable frailties, Journal Methodology and Computing in Applied Probability, 2 (2004), 293–308.

G. Samorodnitsky, M. Taqqu, “Stable Non-Gaussian Random Process”, Chap- man and Hall, New York, 1994.

D.M. Titterington, A.F.M. Smith, U.E. Makov, “Statistical Analysis of Finite Mixture Distributions”, Wiley, New York, 1985.

V. M. Zolotarev, “One Dimensional Stable Distributions”, American Mathematical Society, Providence, 1986.

Downloads

Published

2010-06-01

How to Cite

Otiniano, C., & Gonçalves, C. (2010). Domain of Attraction of -stable Distributions under Finite Mixture Models. Trends in Computational and Applied Mathematics, 11(1), 69–76. https://doi.org/10.5540/tema.2010.011.01.0069

Issue

Vol. 11 No. 1 (2010)

Section

Original Article

License

Authors who publish in this journal agree to the following terms: 

  1. Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.

  2. Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
     

  3. Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).

  4. This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
    author. This is in accordance with the BOAI definition of open access

 

Intellectual Property

All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.