Adaptive minimax test of independence

The paper is concerned with the adaptive minimax problem of testing the independence of the components of a d-dimensional random vector. The functions under alternatives consist of smooth densities supported on 0, 1]d and separated away from the product of their marginals in L2-norm. We are interested in finding the adaptive minimax rate of testing and a test that attains this rate. We focus mainly on the tests for which the error of the first kind an can decrease to zero as the number of observations increases. We show also how this property of the test affects its error of the second kind.