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I'd like to know what results there are for $(1-X)(1-Y)$ where $X$ and $Y$ are independent and both have a lognormal distribution. I'd like to find an analytic solution or closed form approximation, since this is part of a larger model. The shape of each $(1-X)$ random variable is a reflected shifted lognormal. Given that the product of two lognormal distributions $Z=XY$ is also lognormal, it "feels" like there might be an analytic result for this shifted reflected problem.

The three parameter lognormal distribution may provide an answer. However, I haven't yet found a good reference to take me through the three parameter lognormal distribution. The three parameter lognormal can deal with shifted lognormal distributions. I think it can deal with reflected lognormal distributions also, but I'm not sure about this second point. I've found a thesis on parameter estimation for three parameter lognormal distribution: http://digitalcommons.fiu.edu/cgi/viewcontent.cgi?article=1677&context=etd

However, I'm really after something as basic as possible. I'm working my way through the thesis references to find the most basic one - I'll update this question if/when I find one.

There are three possible questions (given in order of preference):

  1. What analytic results are there for $(1-X)(1-Y)$ where $X$ and $Y$ both have a lognormal distribution?

  2. Which types of problem is the three parameter lognormal distribution well suited to? [is this problem one of them?]

  3. What references for the three parameter lognormal distribution have you found most useful in your experience?

The graph of $u=(1-x)(1-y)$ is:

(1-x)(1-y)

generated using

y=exp(rnorm(1000000,1,0.05))

x=exp(rnorm(1000000,1,0.05))

u=(1-x)*(1-y)

hist(u, breaks=100)
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  • $\begingroup$ 1. You still don't specify nearly enough; what's the dependence structure between X and Y? 2. I doubt there will be analytic results for it unless there's an problem it relates to that people will have wanted to solve, and at first glance that seems unlikely. 3. I seriously doubt a three parameter lognormal will be adequate, since the result can easily be left skew. 4. I've never read what I'd regard as an introductory reference to the three parameter lognormal (and I doubt such a thing would exist). ... ctd $\endgroup$ – Glen_b Sep 15 '15 at 9:54
  • $\begingroup$ ctd ... If something collects results about the three parameter lognormal of a level that would come close to answering your question above, it would not be introductory. $\endgroup$ – Glen_b Sep 15 '15 at 9:55
  • $\begingroup$ X and Y are independent. I'm not really expecting an answer to (i) since its a slightly niche application. I've read that 3 parameter lognormal can deal with reflected (left skew) as well as shifted, but not certain on that point. I'll edit question again. $\endgroup$ – Ash Sep 15 '15 at 10:09
  • $\begingroup$ Even adding that the distribution can be reflected the problem is that the distribution won't be well approximated by a 3 parameter lognormal in many cases, because in many cases the result is too peaked. I'll address that in an answer. $\endgroup$ – Glen_b Sep 15 '15 at 11:26
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This is only a partial answer.

With independent lognormals, (1-X)(1-Y) is often quite peaked; indeed, sometimes closer to an asymmetric Laplace than a shifted lognormal and sometimes much more strongly peaked (and far more heavy tailed) than Laplace.

Here's an example of a Laplace-like case.

![enter image description here

Where x and y were generated as follows:

y=exp(rnorm(1000000,0,.05))
x=exp(rnorm(1000000,0,.05))

As such three-parameter lognormal distributions won't be a good approximation for this in many cases.


Certainly it can look reasonably lognormal for some parameter values; I thought it was important to show that it often won't be. I don't presently see an easy way of identifying what values of $(\mu_X,\mu_Y,\sigma_X,\sigma_Y)$ would lead to more or less lognormal shaped outcomes.

It would depend in part on what you want to use a three parameter lognormal fit to such a distribution for as well as your tolerance for the impact of deviation from the three-parameter lognormal model.

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    $\begingroup$ The product distribution of two lognormals is also often strongly peaked $\endgroup$ – Ash Sep 15 '15 at 12:00
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    $\begingroup$ Not like this; the product of two independent lognormals is lognormal. Try to get something close to the right hand plot with the log density of a lognormal. $\endgroup$ – Glen_b Sep 15 '15 at 12:14
  • $\begingroup$ I've added a (1-x)(1-y) graph that looks lognormal in shape to the question. When does (1-x)(1-y) stop looking lognormal? I accept for (1-x)(1-y) not to be lognormal you only have to provide one counter example - your example graph does look a counter example but I'm not yet entirely convinced. Anything we can say about the derivatives that would show the (1-x)(1-y) curve has a different character to lognormal dist? $\endgroup$ – Ash Sep 15 '15 at 15:16

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