The limiting behavior of some infinitely divisible exponential dispersion models

Details The limiting behavior of some infinitely divisible exponential dispersion models The limiting behavior of some infinitely divisible exponential dispersion models

2010-05-18

arxiv.org/abs/1005.3284v1

Mathematics

Statistics Theory

8 pages

Scientific paper

Consider an exponential dispersion model (EDM) generated by a probability $ \mu $ on $[0,\infty )$ which is infinitely divisible with an unbounded L\'{e}vy measure $\nu $. The Jorgensen set (i.e., the dispersion parameter space) is then $\mathbb{R}^{+}$, in which case the EDM is characterized by two parameters: $\theta _{0}$ the natural parameter of the associated natural exponential family and the Jorgensen (or dispersion) parameter $t$. Denote by $EDM(\theta _{0},t)$ the corresponding distribution and let $Y_{t}$ is a r.v. with distribution $EDM(\theta_0,t)$. Then if $\nu ((x,\infty ))\sim -\ell \log x$ around zero we prove that the limiting law $F_0$ of $ Y_{t}^{-t}$ as $t\rightarrow 0$ is of a Pareto type (not depending on $ \theta_0$) with the form $F_0(u)=0$ for $u<1$ and $1-u^{-\ell }$ for $ u\geq 1$. Such a result enables an approximation of the distribution of $ Y_{t}$ for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided.

Affiliated with

Also associated with

No associations

Rating

The limiting behavior of some infinitely divisible exponential dispersion models does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The limiting behavior of some infinitely divisible exponential dispersion models, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The limiting behavior of some infinitely divisible exponential dispersion models will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-58925