Abstract
A dimension reduction-based adaptive-to-model test is proposed for significance of a subset of covariates in the context of a nonparametric regression model. Unlike existing locally smoothing significance tests, the new test behaves like a locally smoothing test as if the number of covariates was just that under the null hypothesis and it can detect local alternative hypotheses distinct from the null hypothesis at the rate that is only related to the number of covariates under the null hypothesis. Thus, the curse of dimensionality is largely alleviated when nonparametric estimation is inevitably required. In the cases where there are many insignificant covariates, the improvement of the new test is very significant over existing locally smoothing tests on the significance level maintenance and power enhancement. Simulation studies and a real data analysis are conducted to examine the finite sample performance of the proposed test.
Highlights
Consider the nonparametric regression model: Y = m(Z) + ǫ, (1.1)where Y is a scale dependent variable with the covariates Z = (X⊤, W ⊤)⊤, X = (X1, · · ·, Xp1)⊤ ∈ Rp1, W = (W1, · · ·, Wp2)⊤ ∈ Rp2 and p1 + p2 = d, the regression function m(·) : Rd → R is unknown in its form and ǫ is the error term with zero conditional expectation when Z is given: E(ǫ|Z) = 0
There exist several proposals that are based on prevalent local smoothing and global smoothing methodologies in the literature. For the former, Lavergne and Vuong (2000) extended the idea introduced by Fan and Li (1996), proposed a test based on a second conditional moment to check the significance of a subset of covariates
When d = p1 + p2 is large, the convergence rate is very slow because h converges to zero at a certain rate. This implies that these local smoothing methodologies severely suffer from the curse of dimensionality. This problem is caused by using nonparametric estimation for the models under both the null and alternative hypothesis that assumes the significance of all the covariates
Summary
This implies that these local smoothing methodologies severely suffer from the curse of dimensionality This problem is caused by using nonparametric estimation for the models under both the null and alternative hypothesis that assumes the significance of all the covariates. The basic idea is to utilize the dimension reduction structure to adapt the true underlying regression models such that it behaves like a test with univariate covariate under the null hypothesis and adapts to the model structure to make the test omnibus under the alternative hypothesis This approach greatly improves the performance of existing local smoothing tests on significance level maintainance and power enhancement. Zhu et al (2015b) followed the similar idea to develop a dimension reduction global smoothing test for more general regression models Both of these adaptive methods can greatly overcome the curse of dimensionality. All the technical conditions and the proofs of the theoretical results are postponed to the Appendix
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
