>Thanks to everyone for your responses. I've done some empirical testing that
>I'd like to share.
>
>First, since there are a variety of parameterizations, I'll use Excel &
>@Risk's f(x,a,b), where b=variance/mean and a=mean/b. (That's an easier way
>to type out which parameter is which, and those two formulas are going to be
>useful below.)
>
>After I sent my message, I decided to fire up @Risk and see what a few
>thousand simulations of a sum of Gammas might look like. It turns out to
>look a lot like a Gamma. After a couple of guesses, I figured out that it
>apparently is a Gamma whose mean is the sum of the component means (which is
>what I expected), and whose variance is the sum of the variances. Using the
>formulas above, you can then get a and b for the total distribution.
>
>If the b's are the same, you get the result that a couple of my respondents
>gave: just add the a's together and leave b as is.
>
>However, not knowing any better, I had gone ahead and added Gammas with
>different b values, which both respondents told me does not have a closed
>form. But it sure looks like a Gamma to me. Generalized, I get the
>following hypothesis:
>
>If x1...xn are random variables with Gamma distributions with parameters
>a1...an and b1...bn, then the sum of the x's has a Gamma distribution with
>parameters
> b = Sum of (ai bi^2) / Sum of (ai bi), and
> a = Sum of (ai bi) / b
>
>(The i's are supposed to be the subscripts that you're summing over)
>
>For the more mathematically inclined (or those that Fred gives homework
>assignments to), is this in fact true, or is it just a very close
>approximation? If the latter, is it generally close, or just for the range
>of parameters I happen to be using (a from 3 to 9, b from 50 to 200; what's
>probably important is that b is large compared to a).
This is only approximation that may be good or bad,
depending on parameters values. Below there are some qualitative
assertions for the sum of two Gamma variables with parameters
a1, b1 and a2, b2. Denote D the distribution of their sum,
"d" approximation error and r = b1/b2.
- if a1, a2 tend to infinity, then D (being normalized) approaches
to the Gamma distribution you calculated (the latter being properly
normalized too); both sum distribution and approximating Gamma
distribution tend to normal distribution;
- if r tends to 1, then d tends to 0;
- if a1, a2 are small and r <> 1, then d may be arbitrary large.
To get illustration of the latter try a1 = a2 = 0.1 or less, and some
different b's, say b1 = 50, b2 = 200.
>
>Of course, truncation messes up tidy results like this. I plan to do some
>more testing to see how much.
>
>Dan Goddard
Arcady
=============================================
Arcady A. Novosyolov, Ph.D.
Institute of Computational Modeling, SB RAS,
19-149, Academgorodok, Krasnoyarsk, Russia, 660036,
tel. +7 3912 495382 (office), +7 3912 498596 (home)
fax (603) 688-4664 (U.S.A. number)
mailto: anov@cc.krascience.rssi.ru
Home page: http://www.geocities.com/novosyolov/
Forum: http://www.delphi.com/risktheory/
Visit the CAS Web Site at http://www.casact.org
===============================================
To subscribe or unsubscribe from CASNET:
Send an e-mail to caslists@lists.casact.org
Type in the body join casnet to subscribe
or leave casnet to unsubscribe.