Cumulative divergence revisited

Abstract

We revisit cumulative divergence, a robust metric to measure the conditional mean independence between two random variables. First, we prove that if ${\\rm~E}(Y\\mid~~X)$ does not varies with $X$, then the corresponding sample estimator of cumulative divergence is $n$ consistent; otherwise, it is root-$n$ consistent. We propose a wild bootstrap procedure to approximate the asymptotic null distribution and prove that this bootstrap method is consistent. Second, we propose an independent screening procedure using cumulative divergence to filter out unimportant covariates in ultrahigh dimensional mean regressions. We establish the ranking consistency property and the sure screening property for our proposal and demonstrate its robust performance through comprehensive numerical studies.