[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 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
