Robust independence test of functional variables via Gupta angle covariance

In this paper, we propose a novel Gupta angle covariance to test the independence between two functional random variables. The proposed Gupta angle covariance satisfies the independence-zero equivalence property, e.g., it is non-negative and equals zero if and only if the two functional random variables are independent. Since the Gupta angle covariance is well defined without any moment conditions, it is robust to heavy-tailed distributions or outliers in the dataset. In addition, we establish the asymptotic properties of our proposed test statistic. Since the asymptotic null distribution is intractable, we approximate it through the random permutation procedure in practical applications. More importantly, the computational complexity of the proposed test statistic is O{n2}, which significantly alleviates the heavy computational burden of permutation procedure. Numerical studies and a real-world example validate the promising finite-sample performance of our proposed test.