EJP Reading Tools

Comparing Fréchet and positive stable laws

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Comparing Fréchet and positive stable laws
2.	Creator	Author's name, affiliation, country	Thomas Simon; Université Lille 1; France
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Convex order; Fréchet distribution; Median; Mittag-Leffler distribution; Mittag-Leffler function; stable distribution; stochastic order
3.	Subject	Subject classification	33E12; 60E05; 60E15; 60G52; 62E15
4.	Description	Abstract	Let ${\bf L}$ be the unit exponential random variable and ${\bf Z}_\alpha$ the standard positive $\alpha$-stable random variable. We prove that $\{(1-\alpha)\alpha^{\gamma_\alpha} {\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha <1\}$ is decreasing for the optimal stochastic order and that $\{(1-\alpha){\bf Z}_\alpha^{ \gamma_\alpha}, 0< \alpha < 1\}$ is increasing for the convex order, with $\gamma_\alpha = \alpha/(1-\alpha).$ We also show that $\{\Gamma(1+\alpha) {\bf Z}_\alpha^{-\alpha}, 1/2\le \alpha \le 1\}$ is decreasing for the convex order, that ${\bf Z}_\alpha^{ \alpha}\,\prec_{st}\, \Gamma(1-\alpha) {\bf L}$ and that $\Gamma(1+\alpha){\bf Z}_\alpha^{-\alpha} \,\prec_{cx}\,{\bf L}.$ This allows to compare ${\bf Z}_\alpha$ with the two extremal Fréchet distributions corresponding to the behaviour of its density at zero and at infinity. We also discuss the applications of these bounds to the strange behaviour of the median of ${\bf Z}_\alpha$ and ${\bf Z}_\alpha^{-\alpha}$ and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of ${\bf Z}_\alpha$ for $\alpha$ rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Agence Nationale de la Recherche
7.	Date	(YYYY-MM-DD)	2014-01-28
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/3058
10.	Identifier	Digital Object Identifier	10.1214/EJP.v19-3058
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
15.	Rights	Copyright and permissions	The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available. Summary of the Creative Commons Attribution License You are free to copy, distribute, display, and perform the work to make derivative works to make commercial use of the work under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author.