# On Testing Independence and Goodness-of-fit in Linear Models

arXiv: https://arxiv.org/abs/1302.5831 · HTML: https://mlsanity.com/paper/1302.5831/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Feb 2013.

## Built on by (most cited first, 3 of 3)

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- [Kernel Integrated R2: A Measure of Dependence](https://mlsanity.com/paper/2602.22985/) (2026)

## Abstract (arXiv)

We consider a linear regression model and propose an omnibus test to simultaneously check the assumption of independence between the error and the predictor variables, and the goodness-of-fit of the parametric model. Our approach is based on testing for independence between the residual obtained from the parametric fit and the predictor using the Hilbert--Schmidt independence criterion (Gretton et al. (2008)). The proposed method requires no user-defined regularization, is simple to compute, based merely on pairwise distances between points in the sample, and is consistent against all alternatives. We develop distribution theory for the proposed test statistic, both under the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution. We prove the consistency of the bootstrap scheme. A simulation study shows that our method has better power than its main competitors. Two real datasets are analyzed to demonstrate the scope and usefulness of our method.