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I have two sets of continuous multivariate observations $X=\{x_1, x_2, ..., x_d\}$ and $Y=\{y_1, y_2, ..., y_d\}$. How can I justify if they are statistically independent or not?

For simplicity, I assume another variable $Z=\{x_1, x_2, ..., x_d, y_1, y_2, ..., y_d\}$ by concatenating $X$ and $Y$. Moreover, I assume multivariate normal distribution for all random variables $X$, $Y$ and $Z$.

Now, to prove the statistical independence, is it sufficient to verify: $p(Z) = p(X)p(Y)$, where $p(Z) = p(X,Y)$ according to the concept of multivariate distribution and joint distribution discussed here.

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If you can really assume multivariate normality you simply need to have the correlation matrix be block diagonal (corresponding to the correlations between X and Y being 0) for X and Y to be independent of each other.

However, if you don't assume multivariate normality, there are many ways to be dependent; it's quite difficult to identify general unspecified dependence unless you have a lot of data. If you can specify the form of dependence, it can be relatively straightforward.

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  • $\begingroup$ Thank you for your answer. I am wondering if for real data case it will really happen. Moreover, I would like to know if there is a way to measure the degree of independence. $\endgroup$ – Hasnat Aug 6 '14 at 8:11
  • $\begingroup$ My answer is a response to your above question; did you want to change your question, or add a new question? $\endgroup$ – Glen_b Aug 6 '14 at 9:48
  • $\begingroup$ I appreciate your answer. However, I am interested to know more and therefore added additional point in my comment. Please let me know your opinion on "to measure the degree of independence"? $\endgroup$ – Hasnat Aug 11 '14 at 19:44
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    $\begingroup$ The first part about whether it will really happen: I'd almost never expect variables to be truly independent. It's an approximation. Even if variables were independent, their sample covariances won't be 0. The second part - the thing about measurement - is a whole new question. Please understand that while I am happy to deal with clarifying issues, I choose to answer questions on the basis of what the actual posted question asks, not on the basis of what other questions might occur to you after I answer the question that was posted. Ask a new question as a new question. $\endgroup$ – Glen_b Aug 11 '14 at 19:48
  • $\begingroup$ However, you should modify it to ask about degree of dependence rather than of independence (dependence has degrees, independence is sort of 'it is or it isn't'). $\endgroup$ – Glen_b Aug 11 '14 at 19:53

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