Comparing Fréchet and positive stable laws
By Thomas Simon. 2013
110 citations recorded by the source corpus.
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 Frechet 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.
Subfields: Applied Mathematics, Finance, Management Science and Operations Research
View Paper: https://www.semanticscholar.org/paper/b14a01094c6f3aec3900b83f4d0e84043180aa43
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In Applied Mathematics
In Finance
In Management Science and Operations Research
Citations in the corpus
- Some Properties of the Kilbas-Saigo Function (2020, methodology): “ property holds for m→ Eα,m,m−l(x) and any l≤ 1/α.In the case l∈ {1,1/α},this would require from (2.2) a monotonicity analysis of the mapping m→ Xα,m,m−l,which does not seem easy at first sight. As in [22], our analysis to obtain the uniform bounds will use some notions of stochastic ordering. Recall”