NONPARAMETRIC CONDITIONAL LOCAL INDEPENDENCE TESTING

Conditional local independence is an asymmetric independence relation among continuous time stochastic processes. It describes whether the evolu-tion of one process is directly influenced by another process given the histo-ries of additional processes, and it is important for the description and learn-ing of causal relations among processes. We develop a model-free framework for testing the hypothesis that a counting process is conditionally locally in-dependent of another process. To this end, we introduce a new functional parameter called the Local Covariance Measure (LCM), which quantifies de-viations from the hypothesis. Following the principles of double machine learning, we propose an estimator of the LCM and a test of the hypothesis using nonparametric estimators and sample splitting or cross-fitting. We call this test the (cross-fitted) Local Covariance Test ((X)-LCT), and we show that its level and power can be controlled uniformly, provided that the nonpara-metric estimators are consistent with modest rates. We illustrate the theory by an example based on a marginalized Cox model with time-dependent covari-ates, and we show in simulations that when double machine learning is used in combination with cross-fitting, then the test works well without restrictive parametric assumptions.