On a Nonparametric Notion of Residual and its Applications

Let $(X, \mathbf{Z})$ be a continuous random vector in $\mathbb{R} \times \mathbb{R}^d$, $d \ge 1$. In this paper, we define the notion of a nonparametric residual of $X$ on $\mathbf{Z}$ that is always independent of the predictor $\mathbf{Z}$. 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 $(X, Y, \mathbf{Z})$ in $\mathbb{R} \times \mathbb{R} \times \mathbb{R}^d$, we use this notion of residual to show that the conditional independence between $X$ and $Y$, given $\mathbf{Z}$, is equivalent to the mutual independence of the residuals (of $X$ on $\mathbf{Z}$ and $Y$ on $\mathbf{Z}$) and $\mathbf{Z}$. 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.