General tests of conditional independence based on empirical processes indexed by functions

This paper focuses on nonparametric procedures for testing conditional independence between random vectors using Möbius transformation. We derive a method predicated on general empirical processes indexed by a specific class of functions. Conditional half-space and conditional empirical characteristic processes are used to demonstrate two abstract approximation theorems and their applications in real-world situations. We conclude by describing the limiting behavior of the Möbius transformation of the empirical conditional processes indexed by functions under contiguous sequences of alternatives. Our results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. Monte Carlo simulation results indicate that the suggested statistical test for independence behaves reasonably well in finite samples.