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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.

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