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.


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.

  • $\begingroup$ 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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