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How do you write down the likelihood for random vectors when each component follows a different distribution with a dependence structure?

For example, Suppose there are n-random vectors, mutually independent, $X_1,\ldots,X_n$ with 3 components of each vector, i.e. $X_i=[X_{i,1},X_{i,2},X_{i,3}],i=1,\ldots,n$. Suppose $X_{i,1:2}\sim N(\mu,\Sigma)$ and $X_{i,3}\sim \text{Exp}(\lambda)$. Also $\text{Cov}(X_{i,1},X_{i,3})=a_1$ and $\text{Cov}(X_{i,2},X_{i,3})=a_2$. Then what is the likelihood of this model?

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  • $\begingroup$ Are X_{i,1} and X_{i,2} independent? Are X_{i} independent for all i? By the likelihood, do you mean the probability density for this three-variate pdf? $\endgroup$
    – Alex
    Commented Sep 25, 2022 at 8:30
  • $\begingroup$ @Alex X_{i,1} and X_{i,2} need not be independent, $\Sigma$ could have non-zero off-diagonal elements. $X_1,\ldots,X_n$ are mutually independent. Yes, I am looking for the three-variate pdf. $\endgroup$ Commented Sep 25, 2022 at 8:36
  • $\begingroup$ You cannot derive joint distribution simply by knowing marginal distributions. $\endgroup$ Commented Sep 25, 2022 at 9:46
  • $\begingroup$ @StubbornAtom There is also information about the covariance between each component. What is the sufficient amount of information needed to obtain the joint distribution? $\endgroup$ Commented Sep 25, 2022 at 10:00
  • $\begingroup$ Reading up on copulas might give you a place to start. $\endgroup$ Commented Sep 25, 2022 at 10:17

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