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This question is related to my previous question in here

So I was trying to simulate correlated $\chi^2(1)$ random variables given the desired co-variance matrix. However, it seems like the only possible route was the following

Given the co-variance matrix $R$, then we perform the cholesky decomposition such that $$ R=LL^t $$ Then we simulate a number of independent vector $A$ with $N\sim(0,1)$, we can then simulate $B$ using $$ LA=B $$ where B should be correlated $N\sim(0,1)$, then we can obtain the correlated $\chi^2(1)$ variables.

When playing around with the variables, I thought that given $$ LA=B $$ and that the correlated $\chi^2$ are simply $B^2$, then by squaring the equation, I should be able to get the corresponding independent $\chi^2$ variables e.g. $$ LAA^tL^t=BB^t $$ However, I note that $AA^t$ and $BB^t$ are both square matrix, so I am not sure whether if they are both $\chi^2$ distributed. What interested me most is that: given a set of correlated random variables that are normally distributed, we can simply use the cholesky distribution to orthogonalizing the variables or un-correlate them. Will there be similar simple equations that we can use to perform the same trick in orthogonalizing $\chi^2$ or even non-centric $\chi^2$ variables?

Thank you

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  • $\begingroup$ For simulating correlated $\chi^2(1)$ variates, why not just square a bivariate Normal distribution? $\endgroup$ – whuber Feb 23 '15 at 18:01
  • $\begingroup$ I can, however, I am interested in whether if there is any method that can simulate correlated $\chi^2(1)$ from independent $\chi^2(1)$ variables. $\endgroup$ – Sam Feb 24 '15 at 4:09

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