In article <[EMAIL PROTECTED]>,
Maarten Speekenbrink <[EMAIL PROTECTED]> wrote:
>Oops, I've made a mistake in my former post. the corrected message is:
>Hi there,
>I would like to know the distribution of a linear combination of (not
>identical) logistically distributed variables, i.e.
>f(x) = exp( (x - alpha)/beta )/(1+exp( (x-alpha)/beta ))^2 .
>Does anybody now what this distribution would look like, or know a good
>reference? I've looked in the 'Handbook of the logistic distribution' by
>Balakrishnan, as well as in Johnson & Kotz, but couldn't find anything.
>Since the distribution is very similar to the normal, I would expect a
>similar result to that for the normal distributed (it is logistically
>distributed itself). Thanks for your help.
The question is rather vague. One can compute the moment
generating function of a logistic random variable as
m(t) = exp(alpha*t)*pi*beta*t/sin(pi*beta*t)
for |beta*t| < 1, and the moment generating function of a sum
of independent random variables is the product of the moment
generating functions.For numerical calculation, I would suggest
using the complex integral for the density or the cdf, whichever
is desired, and using steepest descent on it.
As it is a convolution of log-concave distributions, the density
is log-concave. It is also infinitely divisible, with an easily
computed Levy measure, although it is hard to see what one can do
with this.
It is very definitely NOT logistically distributed.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
.
.
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