Generate bivariate random numbers from joint distribution function

Question

I have an empirical joint distribution function

$ \hat{F}(x_1,x_2) = Pr(X_1 < x_1, X_2 < x_2) $

Can I generate bivariate random number from this distribution with a certain condition such as $ X_1 > c_1, X_2 > c_2 $ ?

Answer 1

Assuming that the only thing that you have is an empirical distribution, the simplest way to go is to draw values from $\hat F$ and reject if $X_1 \leq c_1$ and $X_2 \leq c_2$. Less naive implementation would be to subset $(X_1, X_2)$ values so to drop the values below threshold and draw from $\hat F_\text{trunc}$, the same way as you would do with any other discrete distribution. Simple example in R of such approach can be find below.

set.seed(123)

c1 <- -1
c2 <- 0.5
X <- data.frame(X1 = rnorm(100), X2 = rnorm(100)) # creating fake data
X_trunc <- subset(X, X1 > c1 & X2 > c2) # subset

# draw 1000 samples
X_trunc[sample(nrow(X_trunc), 1000, replace = TRUE), ]

The above example assumes that you have the full data, however if the only thing that you have is the empirical distribution tables with probabilities for $(x_1,x_2)$, than the procedure is the same but you draw the $(x_1,x_2)$ pairs with $\hat F(x_1,x_2)$ probabilities as in the example below.

library(dplyr)

# lets calculate the probabilities for x1,x2 pairs
FX <- group_by(X, X1, X2) %>%
  summarise(n = n()) %>%
  ungroup() %>%
  mutate(prob = n/sum(n))

# next, we subset and sample as above but from F(X1,X2)
FX_trunc <- subset(FX, X1 > c1 & X2 > c2)

# notice that here we sample with parameter prob set to F(x1,x2)
FX_trunc[sample(nrow(FX_trunc), 1000, replace = TRUE, prob = FX_trunc$prob), ]

Drawing from bivariate distribution does not differ in here from drawing from univariate distribution, the values to be drawn are pairs, or more precisely indexes for those pairs.