On a Nonparametric Notion of Residual and its Applications

Abstract

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.

19 pages, 2 figures

References in corpus (5)

• Brownian distance covariance
• Kernel-based Conditional Independence Test and Application in Causal Discovery
• Uniform in bandwidth consistency of kernel-type function estimators
• DISCO analysis: A nonparametric extension of analysis of variance
• Nonparametric testing of conditional independence by means of the partial copula