Acta Mathematicae Applicatae Sinica, English Series

, Volume 19, Issue 3, pp 511–520 | Cite as

A Note on the Nonparametric Least-squares Test for Checking a Polynomial Relationship

Original papers


Recently, Gijbels and Rousson[6] suggested a new approach, called nonparametric least-squares test, to check polynomial regression relationships. Although this test procedure is not only simple but also powerful in most cases, there are several other parameters to be chosen in addition to the kernel and bandwidth. As shown in their paper, choice of these parameters is crucial but sometimes intractable. We propose in this paper a new statistic which is based on sample variance of the locally estimated pth derivative of the regression function at each design point. The resulting test is still simple but includes no extra parameters to be determined besides the kernel and bandwidth that are necessary for nonparametric smoothing techniques. Comparison by simulations demonstrates that our test performs as well as or even better than Gijbels and Rousson’s approach. Furthermore, a real-life data set is analyzed by our method and the results obtained are satisfactory.


Local polynomial fitting polynomial regression derivative estimation p-value 

2000 MR Subject Classification

62J05 62G05 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.LMAM, Institute of MathematicsPeking UniversityBeijingChina
  2. 2.School of SciencesXi’an Jiaotong UniversityXi’anChina

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