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Nonparametric comparison of regression functions. (English) Zbl 1194.62056
Summary: We provide a new methodology for comparing regression functions \(m_{1}\) and \(m_{2}\) from two samples. Since apart from smoothness no other (parametric) assumptions are required, our approach is based on a comparison of nonparametric estimators \(\widehat m_1\) and \(\widehat m_2\) of \(m_{1}\) and \(m_{2}\), respectively. The test statistics \(\widehat T\) incorporate weighted differences of \(\widehat m_1\) and \(\widehat m_2\) computed at selected points. Since the design variables may come from different distributions, a crucial question is where to compare the two estimators.
As our main results we obtain the limit distribution of \(\widehat T\) (properly standardized) under the null hypothesis \(H_{0}:m_{1}=m_{2}\) and under local and global alternatives. We are also able to choose the weight function so as to maximize the power. Furthermore, the tests are asymptotically distribution free under \(H_{0}\) and both shift and scale invariant. Several such \(\widehat T\)’s may then be combined to get maximin tests when the dimension of the local alternative is finite. In a simulation study we found out that our tests achieve the nominal level and already have excellent power for small to moderate sample sizes.

62G08 Nonparametric regression and quantile regression
60G10 Stationary stochastic processes
62E20 Asymptotic distribution theory in statistics
65C60 Computational problems in statistics (MSC2010)
Full Text: DOI
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