#StackBounty: #correlation #mathematical-statistics #simulation #copula If I have a vector of $N$ correlated probabilities. How can I t…

Bounty: 300

My ultimate goal is to be able to have a way to generate a vector of size $N$ of correlated Bernoulli random variables. One way I am doing this is to use the Gaussian Coupla approach. However, the Gaussian Coupla approach just leaves me with a vector:

$$
(p_1, ldots, p_N) in [0,1]^N
$$

Suppose that I have generated $(p_1, ldots, p_N)$ such that the common correlation between them is $rho$. Now, how can I transform these into a new vector of $0$ or $1$’s? In other words, I would like:

$$
(X_1, ldots, X_N) in {0,1}^N
$$

but with the same correlation $rho$.

One approach I thought of was to assign a hard cutoff rule such that if $p_i < 0.5$, then let $X_i = 0$ and if $p_i geq 0.5$, then let $X_i = 1$.

This seems to work well in simulations in that it retains the correlation structure but it is very arbitrary to me what cutoff value should be chosen aside from $0.5$.

Another way is to treat each $X_i$ as a Bernoulli random variable with success probability $p_i$ and sample from it. However this approach seems to cause loss of correlation and instead of $rho$, I may get $frac{rho}{2}$ or $frac{rho}{3}$.

Does anyone have any thoughts or inputs into this? Thank you.


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