apparently, the topic of this question did not generate too much
interest. Nevertheless, I got a few interesting replies, which are
summarized below.
Thanks and Best Regards
Marek Brabec
1.
Omiros wrote:
your question relates to infinite divisibility and self-decomposable
distributions, but what often is true is that the random variable X can
be written as X=Y1+...+Yn, for any n>=1, where the Yjs are iid, but not
necesarily from the same distribution as X
checking the literature on self-decomposability can be a start to your
problem
O.
2.
Jim wrote:
Is it even possible if a<0 and b>0? (i.e. can you guarantee support
(0,infinity) in such a case?).
Good luck,
Jim Burridge.
Comment:
That is just my omission - I ment (but did not say) that a>0 and b>0.
3.
Kjetil wrote:
closure under convolution is msatisfied by any exponential family.
Kjetil Halvorsen
Original querry:
>dear list,
>I would like to have a parametric family of continuous distributions,
>say G(theta), indexed by a (vector) parameter theta is.element.of Theta,
>with the following properties:
>i) all family members have (0,infinity) support
>ii) (tractable) density exists (to be able to use it for likelihood
>estimation realtively easily)
>iii) the family is closed under convolution (or even better, under
>linear combination)
>in the sense that if: X~G(theta_1), Y~G(theta_2), then (X+Y)~G(theta)
>for some theta from Theta
>(or even if: X~G(theta_1), Y~G(theta_2), a,b are real constants then
>(a.X+b.Y)~G(theta) for some theta from Theta)
>
>Obviously, gamma with a fixed scale parameter can satisfy convolution
>closure (but not the closure under lin. combination), what about other
>possibilities?
>
>Thanks for any hints.
>Best regards
>Marek Brabec
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