Title: Distance Covariance in Metric Spaces: Non-Parametric Independence Testing in Metric Spaces (Master's thesis),
Abstract: The aim of this thesis is to find a solution to the non-parametric
independence problem in separable metric spaces. Suppose we are given finite
collection of samples from an i.i.d. sequence of paired random elements, where
each marginal has values in some separable metric space. The non-parametric
independence problem raises the question on how one can use these samples to
reasonably draw inference on whether the marginal random elements are
independent or not. We will try to answer this question by utilizing the
so-called distance covariance functional in metric spaces developed by Russell
Lyons. We show that, if the marginal spaces are so-called metric spaces of
strong negative type (e.g. seperable Hilbert spaces), then the distance
covariance functional becomes a direct indicator of independence. That is, one
can directly determine whether the marginals are independent or not based
solely on the value of this functional. As the functional formally takes the
simultaneous distribution as argument, its value is not known in the posed
non-parametric independence problem. Hence, we construct estimators of the
distance covariance functional, and show that they exhibit asymptotic
properties which can be used to construct asymptotically consistent statistical
tests of independence. Finally, as the rejection thresholds of these
statistical tests are non-traceable we argue that they can be reasonably
bootstrapped.