# Multivariate mutual independence test

I have a series of observations $\{X_i,Y_i,Z_i\}$ for random variable $X$, $Y$ and $Z$. Now I want to test if $X$, $Y$ and $Z$ are mutual independent, can anyone help me?

Some pairwise independence tests like this one http://www.gatsby.ucl.ac.uk/~gretton/indepTestFiles/indep.htm are available, but I haven't found such tests for multivariate mutual independence yet?

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• $k$ random variables are mutually independent if $\displaystyle P\left( \cap_i R_i \right) = \Pi_i P(R_i)$ – yayu Jun 17 '11 at 20:07
• The issue here is the distribution of random variables X, Y and Z is unknown, only empirical sample pairs (X_i,Y_i, Z_i) can be observed – sinoTrinity Jun 17 '11 at 20:11

For discrete variables, Pearson's chi-squared test generalizes. Estimate the probabilities in the 3D cells by multiplying the marginal probabilities (which is correct, given the assumption of independence). The degrees of freedom are therefore $(k-1)(m-1)(n-1)$ when the variables have $k$, $m$, and $n$ distinct categories.