  We propose a test of independence of two multivariate random vectors, given a
sample from the underlying population. Our approach, which we call MINT, is
based on the estimation of mutual information, whose decomposition into joint
and marginal entropies facilitates the use of recently-developed efficient
entropy estimators derived from nearest neighbour distances. The proposed
critical values, which may be obtained from simulation (in the case where one
marginal is known) or resampling, guarantee that the test has nominal size, and
we provide local power analyses, uniformly over classes of densities whose
mutual information satisfies a lower bound. Our ideas may be extended to
provide a new goodness-of-fit tests of normal linear models based on assessing
the independence of our vector of covariates and an appropriately-defined
notion of an error vector. The theory is supported by numerical studies on both
simulated and real data.
