# On a Nonparametric Notion of Residual and its Applications

arXiv: https://arxiv.org/abs/1409.3886 · HTML: https://mlsanity.com/paper/1409.3886/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2014.

## Built on by (most cited first, 2 of 2)

- [DIET: Conditional independence testing with marginal dependence measures of residual information](https://mlsanity.com/paper/2208.08579/) (2022)
- [A survey of some recent developments in measures of association](https://mlsanity.com/paper/2211.04702/) (2022)

## Abstract (arXiv)

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.