"Maarten Speekenbrink" <[EMAIL PROTECTED]> wrote in message news:<[EMAIL PROTECTED]>... > 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.
As Herman has already said, it won't be logistic. Let's call the variables X1, X2, .... You want the distribution of a1 X1 + a2 X2 + ... + an Xn. Note that if Yi = ai Xi, this is the same as the distribution of the sum of the Yi. The Yi are of course logistic as well (with parameters that are easily computed). Under certain conditions (e.g. constraining the relative variances of the Y's so that a small fraction of the variances don't dominate; that there not be too much dependence) the central limit theorem should begin to kick in once you have a big number of terms. (That is, if the conditions hold and you have enough terms in the sum, the distribution will start getting close to normal.) Clearly the distribution will be symmetric. Herman has already noted that it will be log-concave. I don't know if bounds will help you, but bounds could be computed. An easy bound is to ignore the extra information in it being log-concave and just work with the fact that it is symmetric and unimodal. This leads to a chebyshev-like inequality that is tighter than the usual chebyshev (with 1-1/k^2 replaced by 1-4/9k^2, if memory serves - so 44.4% of the tail area you get in the usual chebyshev). This will give an upper limit on proportions in the tail. To be handwavy for a moment, if the Y's are not too dissimilar in variance, the tails should tend to be lighter than the logistic. Without doing some analysis that's probably beyond me at the moment, this would suggest that under some circumstances the logistic would also be an upper bound on tail area. But since I can't give you a good idea of how wide a range of cicumstances that would happen under**, we can't really call it anything more than a vague approximation which may often tend to be larger. ** of course for specific situations you could do some computations and compare (or even some simulations if that suffices for your purpose). The normal would seem (again without doing any analysis to back it up, so again don't take it seriously unless you have something more to go on than my hand-waving) to provide a lower bound on tail areas. There are a few circumstances where even bounds of "symmetric unimodal" and "normal" might be helpful. Precise answers would probably require doing some form of numerical convolution for each specific circumstance. Glen . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
