Independence tests based on the conditional expectation

In this paper we propose a new procedure for testing independence of random variables, which is based on the conditional expectation. As it is well known, the behaviour of the conditional expectation may determine a necessary condition for stochastic independence, that is, the so called mean independence. We provide a necessary and sufficient condition for independence in terms of conditional expectation and propose an alternative method to test independence based on this result. Consequently, we provide general class of tests. Observe that generally some non-parametric methods are needed to approximate the conditional expectation, since its exact expression (given the joint distribution) is usually unknown, except for few trivial cases (e.g. Gaussian): we generalize this well known result to the family of elliptical distributions. In order to obtain a sufficiently accurate approximation of the conditional expectation, we propose to use the kernel method or, alternatively, the recently introduced OLP method.