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):
What analytic results are there for $(1-X)(1-Y)$ where $X$ and $Y$ both have a lognormal distribution?
Which types of problem is the three parameter lognormal distribution well suited to? [is this problem one of them?]
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:
y=exp(rnorm(1000000,1,0.05)) x=exp(rnorm(1000000,1,0.05)) u=(1-x)*(1-y) hist(u, breaks=100)