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Distance-covariance-based tests for heteroscedasticity in nonlinear regressions

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Abstract

We use distance covariance to introduce novel consistent tests of heteroscedasticity for nonlinear regression models in multidimensional spaces. The proposed tests require no user-defined regularization, which are simple to implement based on only pairwise distances between points in the sample and are applicable even if we have non-normal errors and many covariates in the regression model. We establish the asymptotic distributions of the proposed test statistics under the null and alternative hypotheses and a sequence of local alternatives converging to the null at the fastest possible parametric rate. In particular, we focus on whether and how the estimation of the finite-dimensional unknown parameter vector in regression functions will affect the distribution theory. It turns out that the asymptotic null distributions of the suggested test statistics depend on the data generating process, and then a bootstrap scheme and its validity are considered. Simulation studies demonstrate the versatility of our tests in comparison with the score test, the Cramér-von Mises test, the Kolmogorov-Smirnov test and the Zheng-type test. We also use the ultrasonic reference block data set from National Institute for Standards and Technology of USA to illustrate the practicability of our proposals.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11901006 and 11601008) and Natural Science Foundation of Anhui Province (Grant No. 1908085QA06).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241002, China

    Kai Xu & Mingxiang Cao

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  1. Kai Xu
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  2. Mingxiang Cao
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Correspondence to Kai Xu.

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Xu, K., Cao, M. Distance-covariance-based tests for heteroscedasticity in nonlinear regressions. Sci. China Math. 64, 2327–2356 (2021). https://doi.org/10.1007/s11425-020-1759-5

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