Estimate covariance matrix using R without knowing the joint distribution

Question

n <- 10000
X <- rnorm(n, mean = 3, sd = 3)
Y <- rexp(n, rate = 2)
Z <- rnorm(n, mean = -2, sd = 0.5)

cov(X, Y)
cov(X, Z)
cov(Y, Z)

Suppose I know the marginal distributions of the random variables $X \sim N(3, 9)$, $Y \sim Exp(2)$, and $Z \sim N(-2, 0.025)$. However, I do not have any information on the joint distribution of $(X, Y, Z)$ and I do not know whether these r.v.s are independent or not.

I'm interested in estimating the covariance matrix $Cov\begin{bmatrix}X\\ Y \\ Z\end{bmatrix}$. Is it appropriate to do what I did above: generate observations from each marginal distribution and simply use the sample covariance as an estimate? i.e., cov(X, Y), cov(X, Z), cov(Y, Z) in R?

Answer 1

No. What you did above is simulated $(X,Y,Z)$ where $X$, $Y$ and $Z$ are independent. The sample covariances will simply converge to 0 as $n$ converges to infinity. To simulate any random variables jointly, you have to have an idea how they are related