| Parameters |
— stability parameter | ||
|---|---|---|---|
| Support | , or if and , or if and | ||
| not analytically expressible, except for some parameter values | |||
| CDF | not analytically expressible, except for certain parameter values | ||
| Median | when | ||
| Mode | when | ||
| Variance | when , otherwise infinite | ||
| Skewness | when , otherwise undefined | ||
| Excess kurtosis | when , otherwise undefined | ||
| MGF | undefined | ||
| CF |
, | ||
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A geometric stable distribution or geo-stable distribution is a type of leptokurtic probability distribution. Geometric stable distributions were introduced in Klebanov, L. B., Maniya, G. M., and Melamed, I. A. (1985). A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables.[1] These distributions are analogues for stable distributions for the case when the number of summands is random, independent of the distribution of summand, and having geometric distribution. The geometric stable distribution may be symmetric or asymmetric. A symmetric geometric stable distribution is also referred to as a Linnik distribution.[2] The Laplace distribution and asymmetric Laplace distribution are special cases of the geometric stable distribution. The Mittag-Leffler distribution is also a special case of a geometric stable distribution.[3]
The geometric stable distribution has applications in finance theory.[4][5][6][7]
- ^ Theory of Probability & Its Applications, 29(4):791–794.
- ^ D.O. Cahoy (2012). "An estimation procedure for the Linnik distribution". Statistical Papers. 53 (3): 617–628. arXiv:1410.4093. doi:10.1007/s00362-011-0367-4.
- ^ D.O. Cahoy; V.V. Uhaikin; W.A. Woyczyński (2010). "Parameter estimation for fractional Poisson processes". Journal of Statistical Planning and Inference. 140 (11): 3106–3120. arXiv:1806.02774. doi:10.1016/j.jspi.2010.04.016.
- ^ Cite error: The named reference
paretianwas invoked but never defined (see the help page). - ^ Trindade, A.A.; Zhu, Y.; Andrews, B. (May 18, 2009). "Time Series Models With Asymmetric Laplace Innovations" (PDF). pp. 1–3. Retrieved 2011-02-27.
- ^ Meerschaert, M.; Sceffler, H. "Limit Theorems for Continuous Time Random Walks" (PDF). p. 15. Archived from the original (PDF) on 2011-07-19. Retrieved 2011-02-27.
- ^ Kozubowski, T. (1999). "Geometric Stable Laws: Estimation and Applications". Mathematical and Computer Modelling. 29 (10–12): 241–253. doi:10.1016/S0895-7177(99)00107-7.