On a Nonparametric Notion of Residual and its Applications

Let (X,Z) be a continuous random vector in R × Rd, d ≥ 1. In this paper, we define the notion of a nonparametric residual of X on Z that is always independent of the predictor Z. We study its properties and show that the proposed notion of residual matches with the usual residual (error) in a multivariate normal regression model. Given a random vector (X,Y,Z) in R × R × Rd, we use this notion of residual to show that the conditional independence between X and Y , given Z, is equivalent to the mutual independence of the residuals (of X on Z and Y on Z) and Z. This result is used to develop a test for conditional independence.