Glen wrote: > [EMAIL PROTECTED] (David C. Howell) wrote in message > news:<[EMAIL PROTECTED]>... >> My daughter just asked me a question that I should be able to >> answer, but >> can't. Any help would be appreciated. >> >> It is well know that the sum of normally distributed random >> variables is >> itself normally distributed. But I want to know the sampling >> distribution >> of the sum of lognormal distributions. > > It's not anything that has a name, or even a nice closed form. > >> My first thought would be that the central limit theorem would >> suggest that >> the sampling distribution would approach normal as N increases. > > True - when the CLT applies, it applies. But don't expect it to come > in very quickly! > >> On the >> other hand, when I generate multiple independent lognormal >> distributions >> and then take their sum, a P-P plot tells me that the sum is nicely >> lognormal, not normal. > > You have rediscovered something many, many people have noticed before. > I have seen it called the permanence of the lognormal - the tendency > of sums of lognormal random variables to have an approximately > lognormal distribution. (I've seen it mentioned in papers in > geostatistics and in optometry - or some similar eye-related research > area, perhaps opthamology - for example.) > > In many other application areas where the use of the lognormal is > common, it doesn't have a name but people have still noticed it. I > think it's one of those things you tend to hear about by word of mouth > a lot more than by reading it in papers. > > I've seen it often (it's relevant to an application I've been involved > with) - and it happens quite nicely even when the lognormals aren't > independent. I've dealt with cases which include several /hundred/ > correlated lognormals (the underlying normals being linearly > correlated), and in the cases we deal with - perhaps 99% of the time - > the lognormal was an extremely good fit to the sum of lognormals > (substantially better than gamma, and much, much better than normal), > even right into the tail (looking at p-p plots) > > (NB: This is not inconsistent with CLT - lognormals with small sigma > can look pretty normalish, so a sum of lognormals can continue to be > approximately lognormal while it's approaching normality.) > > I have played around with some algebra and found out when it's not > going to work so well, but those circumstances don't tend to occur in > the applications we deal with. > > Glen
The usual book "Continuous Univariate Distributions" vol 1 (Johnson, Kotz & Balakrishnan, 1994) has a short discussion of this, with references to a few publications, but there seems to be nothing much extra that can be done in this loose context. David Jones . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
