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. We propose a bootstrap scheme to approximate the critical value of this test. We compare the proposed test, which is easily implementable, with some of the existing procedures through a simulation study.