Generating data from exponential distribution by incorporating correlation between two random variables

Question

Suppose X~exp(.67) , Y~exp(.45) and Z~exp(.8). Now X is correlated with Y with a correlation coefficient -0.6. Again, X is correlated with Z with a correlation coefficient -0.6. How can I incorporate this correlations to generate random variables X, Y and Z? I know if there were no correlation among them, then I could simply generate data by X <- rexp(n=10, rate=.67), Y <- rexp(10, .45) and Z <- rexp(10, .8).

Answer 1

To do this, you can use the Iman and Conover method from the package mc2d.

First, create your settings. I have assumed that Y and Z are uncorrelated, given the absence of a stated correlation above. (If they are not, just change the correlation matrix accordingly.)

library(mc2d)
x1 <- rexp(n = 1000, rate = 0.67)
x2 <- rexp(n = 1000, rate = 0.45)
x3 <- rexp(n = 1000, rate = 0.8)
mat <- cbind(x1, x2, x3)
corr <- matrix(c(1, -0.6, -0.6, -0.6, 1, 0, -0.6, 0, 1), ncol=3)

We can now test the actual correlations of the random samples ... that all seem independent, as expected.

cor(mat, method="spearman")

... which generates:

            x1         x2         x3
x1  1.00000000 0.01602557 -0.0493488
x2  0.01602557 1.00000000  0.0124209
x3 -0.04934880 0.01242090  1.0000000

We now apply the Iman and Conover method to the data.

matc <- cornode(mat, target=corr, result=TRUE)

Doing so yields the following correlations:

Spearman Rank Correlation Post Function
           x1          x2          x3
x1  1.0000000 -0.59385975 -0.56201396
x2 -0.5938597  1.00000000 -0.04115543
x3 -0.5620140 -0.04115543  1.00000000

Finally, by running head(matc), we see the initial rows of your revised sample:

             x1        x2         x3
[1,]  1.1375395 0.3081750 2.26850817
[2,]  2.9387996 0.4434207 0.08940867
[3,]  1.0918648 0.4175625 2.29498679
[4,] 10.0273879 1.1478072 0.16099230
[5,]  1.5093832 0.4023230 2.57870672
[6,]  0.9474039 2.1134685 0.95268424