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?

Thanks in advance.


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  • $\begingroup$ $k$ random variables are mutually independent if $\displaystyle P\left( \cap_i R_i \right) = \Pi_i P(R_i)$ $\endgroup$ – yayu Jun 17 '11 at 20:07
  • $\begingroup$ 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 $\endgroup$ – 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.

For normally distributed variables, use tests of correlation with adjustments for multiple comparisons. (These are all approximate.)

For other variables, you can bin the values into classes defined a priori and apply a chi-squared test. This is successful when it rejects the null hypothesis of independence; it is not so successful otherwise, because some power is lost in the binning.

  • $\begingroup$ Is there independence test specially for continuous multivariates? $\endgroup$ – sinoTrinity Jun 19 '11 at 21:48
  • $\begingroup$ @sino Not in general (that I know of). There may be some generic independence tests based on copulas, but the example of the multinormal distribution shows that you can expect to obtain more powerful tests by exploiting assumptions about the marginal distributions. $\endgroup$ – whuber Jun 20 '11 at 14:11

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