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Statistica Sinica 31 (2021), 1489-1508


Zhihua Sun, Feifei Chen, Hua Liang and David Ruppert

University of Chinese Academy of Sciences, Beijing Normal University at Zhuhai,
George Washington University and Cornell University

Abstract: We propose a projection-based test to check partially linear models. The proposed test achieves a reduction in dimension and, in the presence of multiple linear regressors, behaves as though only a single covariate is present. The test is shown to be consistent and can detect Pitman local alternative hypothetical models. We further derive the asymptotic distributions of the proposed test under the null hypothesis and the local and global alternatives. Most importantly, the test�s numerical performance is consistently and remarkably superior to that of its competitors. Real examples are presented for illustration. Although we assume that the nonparametric component of the model has a univariate covariate, our model can be generalized to partially linear additive models, partially linear single-index models, and other models with linear and nonparametric components.

Key words and phrases: Consistent test, curse of dimensionality, dimensionality reduction, empirical process, integrated conditional moment, projection, uncountable moments restriction.

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