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I'm an economics graduate student doing some work on a nested logit model.

I am trying to generate random variables that follow the following CDF:

[itex] F(x_1, x_2) =\textrm{exp}[ -(e^{-2x_1}+e^{-2x_2}) ^{1/2}] [/itex]

(This is an extreme-value distribution)

With a single random variable, I know that (assuming you can invert the CDF), you can just draw ##u## from the Uniform [0,1] distribution and do ##x=F^{-1}(u)## to get a random variable that follows the distribution described by ##F(x)##.

With the multivariate case, I think what I need to do is:

1) Find ## F_{x_1}(x_1, x_2)##, the marginal distribution of ##F(x_1, x_2)##. I do this by taking the limit as ##x_2## goes to infinity, so ## F_{x_1}(x_1, x_2)=\textrm{exp}[ -(e^{-2x_1}) ^{1/2}]##

2) Find ## F(x_1, x_2 | x_1)##, the conditional distribution of ##F(x_1, x_2)## given ##x_1##. This is calculated this way: ## F(x_1, x_2 | x_1)= {\frac{F(x_1, x_2)}{F_{x_1}(x_1, x_2)}} ##

3) Invert ## F_{x_1}(x_1, x_2)=u_1## to get ## F_{x_1}^{-1}(u)=x_1 ##. This gives us a random ##x_1## for an value of ##u_1 \in (0,1) ##

4) Use the value of ##x_1## generated in the previous step in this step. Invert ## F(x_1, x_2 | x_1)=u_2 ## to get ## F^{-1}(u_2)=x_2##

Here are the formulas I use for determining the random variables (Sorry they're not all pretty and Latex-y... I pulled them from Excel)

x_1=(LN((LN(u_1))^2))/-2

x_2=(LN(((LN(u_2*(EXP(-1*((EXP(-2*x_1))^(1/2))))))/-1)^2-(EXP(-2*x_1))))/-2

I did all of these steps and, at first,thoughtI got a decent result; As long as I pick ##u##'s that are between 0 and 1, I get a real answer; larger u's generate larger x's; and u's that are arbitrarily close to zero (one) give x's that are very small (large). However, when I ran a simulation and looked at average values of each, my ##x_1##'s tend to be much larger than my ##x_2##'s (about .66 for ##x_1## and -.1 for ##x_2##. Since the CDF is symmetric, I think that these variables should have the same average.

Any help will be much appreciated. This is my first post ever on this site!

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# Generate a Multivariate Random Variable

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