As others have told you, the convolution of a bunch of gamma's is a gamma,
as long as the scale parameters are the same. If different scale parameters
are involved, or if you want to truncate any of the gammas you can use
Fourier inversion to get the answer. I have a commercially available
program, CrimCalc, that can handle the problem quickly. See my web site
www.crimcalc.com <http://www.crimcalc.com> .
Glenn Meyers
Insurance Services Office, Inc.
Internet: gmeyers@iso.com <mailto:gmeyers@iso.com>
Voice:(212) 898-5938
Fax: (212) 898-6060
----------
From: Arcady A. Novosyolov [SMTP:anov@post.krascience.rssi.ru]
Sent: Sunday, November 14, 1999 4:07 AM
To: DanG3104@aol.com; casnet@lists.casact.org
Subject: Re: Sum of Gammas
Dan:
You wrote:
>I'm hoping someone on the list can help me with a probability
problem. I remember enough
to frame the question, but nowhere near enough to answer it!
>
>I'm modeling a process made up of several independent, consecutive
steps. I'm using a
Gamma distribution for the duration of each step, so the total
duration is the sum of
several independent Gamma variables. Is there a closed form for the
distribution of the
total duration, as a function of the various Gamma parameters?
The answer to this question is partially "yes". Let X(a,b) be a
random variable with Gamma
distribution, i.e. its density function is equal to k(a,b) x^(a-1)
exp(-bx), where
k(a,b)=b^a/G(a) is a normalizing constant, G(a) is Gamma function,
"b" is a scaling
parameter, and "a" is a structural parameter. Then sum of two Gamma
variables
X(a1,b)+X(a2,b) with equal values of scaling parameters is again
Gamma variable
X(a1+a2,b), see Feller "An Introduction to Probability Theory and
its Applications", 1971,
N.Y.:Wiley, vol. II, section II.2. However, if scaling parameters
are different, the
distribution of X(a1,b1)+X(a2,b1) has no closed form. You may use
numeric integration or
Monte Carlo simulation to approximate resulting distribution in the
latter case.
>I'd actually like to use a truncated Gamma (i.e., limited to a
maximum and minimum value)
for the steps. Does that make the distribution of the total
duration too messy?
If the underlying distributions are truncated Gamma, then there's no
closed form for the
distribution of their sum. Numeric integration or Monte Carlo
simulation might be used in
this case as well.
>If there is no closed form, is there a reasonable approximation
that anyone can suggest?
I'm doing this in Visual Basic (Excel was way too slow) so I am
quite flexible about what
kind of algorithm to implement.
>
>Thanks for your help!
>
>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
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