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Science Forum Index  »  Statistics - Math Forum  »  Approximate a Lognormal by sum of Gaussian
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Guest
Posted: Sat Jan 20, 2007 2:26 am
My friend told me that I can approximate a lognormal distribution
(acutally any function) by a sum of several Gaussian functions. I tried
to google the detail, but returned so many unrelated links. I want to
know how to find out the parameters for the Gaussian functions and the
error term. If you happen to know good references or some keywords,
please leave a message. Thank you.
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Jack Tomsky
Posted: Mon Jan 22, 2007 1:18 pm
Guest
Quote:
My friend told me that I can approximate a lognormal
distribution
(acutally any function) by a sum of several Gaussian
functions. I tried
to google the detail, but returned so many unrelated
links. I want to
know how to find out the parameters for the Gaussian
functions and the
error term. If you happen to know good references or
some keywords,
please leave a message. Thank you.




The sum of several Gaussians is itself a Gaussian. Therefore, you need only a single Gaussian. What your friend probably meant was that the log of a lognormal is a Gaussian. That Gaussian could then be decomposed into several Gaussians if that is of interest.

Jack
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danheyman@yahoo.com
Posted: Mon Jan 22, 2007 6:20 pm
Guest
I have never heard of this result, but since X has a lognormal dst.
means that log(X) has a Gaussian dst., there is certainly a functional
connection that's easy to write down.
Since the lognormal dst. lives on[0,oo), it can be approximated
arbitrarily closely by sums and mixtures of gamma distributions. This
is because the Laplace transform of the latter are polynomials that can
be made arbitrarily close to the Laplace transform of the density of
any given non-negative r.v. You can also use phase-type distributions.
However, finding the parameters for a "good fit" is a poorly understood
art.

Dan Heyman

mailcwc@gmail.com wrote:
Quote:
My friend told me that I can approximate a lognormal distribution
(acutally any function) by a sum of several Gaussian functions. I tried
to google the detail, but returned so many unrelated links. I want to
know how to find out the parameters for the Gaussian functions and the
error term. If you happen to know good references or some keywords,
please leave a message. Thank you.
Back to top
Robert Dodier
Posted: Mon Jan 22, 2007 11:12 pm
Guest
mailcwc@gmail.com wrote:

Quote:
My friend told me that I can approximate a lognormal distribution
(acutally any function) by a sum of several Gaussian functions. I tried
to google the detail, but returned so many unrelated links. I want to
know how to find out the parameters for the Gaussian functions and the
error term. If you happen to know good references or some keywords,
please leave a message. Thank you.

Try googling for "mixture density approximation" or some subset of
those words.
Also try "Gaussian mixture model" or something like that.

Since the support of a Gaussian density is (-oo, +oo) and that of a
lognormal is [0, +oo), a Gaussian mixture approximation isn't the best;
maybe a mixture of gamma densities (as suggested by another poster).

Hope this helps,
Robert Dodier
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David Jones
Posted: Wed Jan 24, 2007 6:28 am
Guest
Herman Rubin wrote:
Quote:
In article
32124213.1169507947458.JavaMail.jakarta@nitrogen.mathforum.org>,
Jack Tomsky <jtomsky@ix.netcom.com> wrote:
My friend told me that I can approximate a lognormal
distribution
(acutally any function) by a sum of several Gaussian
functions. I tried
to google the detail, but returned so many unrelated
links. I want to
know how to find out the parameters for the Gaussian
functions and the
error term. If you happen to know good references or
some keywords,
please leave a message. Thank you.




The sum of several Gaussians is itself a Gaussian. Therefore, you
need only a single Gaussian. What your friend probably meant was
that the log of a lognormal is a Gaussian. That Gaussian could
then
be decomposed into several Gaussians if that is of interest.

The question is not approximating a lognormal random variable
by a sum of normal random variable, but a lognormal distribution
by a sum (rather, linear combination) of normal distributions.

AFAIK, there is no good way of doing this.

... to say anything more specific, the OP would need to say whether
the use of negative coefficients in the sum of distribution functions
would be acceptable.

David Jones
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