Detecting practically significant dependencies in infinite dimensional data via distance correlations

In this paper we take a different look on the problem of testing the independence of two infinite dimensional random variables using the distance correlation. Instead of testing if the distance correlation vanishes exactly, we are interested in the null hypothesis that it does not exceed a certain threshold. Our formulation of the testing problem is motivated by the observation that in many cases it is more reasonable to test for a practically significant dependency since it is rare that a null hypothesis of perfect independence is exactly satisfied. This point of view also reflects statistical practice, where one often classifies the strength of the association in categories such as 'small', 'medium' and 'large' and the precise definitions depend on the specific application. To address these problems we develop a pivotal test for the hypothesis that the distance correlation $\mathrm{dcor}(X,Y)$ between two random variables $X$ and $Y$ does not exceed a pre-specified threshold $\Delta$, that is $H_0 : \mathrm{dcor}(X,Y) \leq \Delta$ versus $H_1 : \mathrm{dcor}(X,Y) > \Delta$. We also determine a minimum value $\hat \Delta_\alpha$ from the data such that $H_0$ is rejected for all $\Delta \leq \hat \Delta_\alpha$ at controlled type I error $\alpha$. This quantity can be interpreted as a measure of evidence against the hypothesis of independence. The new test is applicable to data modeled by a strictly stationary and absolutely regular process with components taking values in separable metric spaces of negative type, which includes Euclidean as well as functional data. Our approach is based on a new functional limit theorem for the sequential distance correlation process.