Testing independence in contamination models

We consider a bivariate contamination model in which the distribution of a gold standard is known, while the mixing weight and the distribution of the contaminant component are unknown, and we test whether the two coordinates of the unknown component are independent. Classical independence tests are meaningless here, since a mixture of product distributions is not a product distribution. Following the Inversion-Best Matching principle, we invert the model for every candidate weight and compare the inverted distribution of the unknown component with the product of its marginals. The weight is estimated where they are closest in a Cramer-von Mises sense, and the scaled minimal distance is the test statistic. An exact quadratic structure makes the estimator computable without numerical optimisation and yields a transparent asymptotic theory under both hypotheses. The null distribution is calibrated by a multiplier bootstrap. Simulations and applications to gene expression data and spam filtering illustrate the method.