A novel martingale difference correlation via data splitting with applications in feature screening

In this paper, we introduce a novel sample martingale difference correlation via data splitting to measure the departure of conditional mean independence between a response variable $Y$ and a vector predictor $X$. The proposed correlation converges to zero and has an asymptotically symmetric sampling distribution around zero when $Y$ and $X$ are conditionally mean independent. In contrast, it converges to a positive value when $Y$ and $X$ are conditionally mean dependent. Leveraging these properties, we develop a new model-free feature screening method with false discovery rate (FDR) control for ultrahigh-dimensional data. We demonstrate that this screening method achieves FDR control and the sure screening property simultaneously. We also extend our approach to conditional quantile screening with FDR control. To further enhance the stability of the screening results, we implement multiple splitting techniques. We evaluate the finite sample performance of our proposed methods through simulations and real data analyses, and compare them with existing methods.