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Bootstrapping the order selection test. (English) Zbl 1044.62042
Summary: We consider bootstrap versions of the order selection tests of R. L. Eubank and J. D. Hart [Ann. Stat. 20, 1412–1425 (1992; Zbl 0776.62045)] and M. Kuchibhatla and J. D. Hart [J. Nonparametric Stat. 7, 1–22 (1996; Zbl 0877.62041)] for testing lack-of-fit of regression models. For homoscedastic data, conditions are established under which the bootstrap level error is smaller (asymptotically) than that of the large sample test. A new statistic is proposed to deal with the case of heteroscedastic data. The limiting distribution of this test statistic is derived and shown to depend on the unknown error variance finction. This dependency makes using the large sample test a formidable task in practice.
An alternative approximation is to apply bootstrap procedures. We propose various bootstrap tests, including ones based on the wild bootstrap. Simulation studies indicate that the wild bootstrap generally has good level and power properties, although sometimes power can be increased by appropriate smoothing of squared residuals. A real-data example is also considered to further illustrate the methodology.
MSC:
62G09 Nonparametric statistical resampling methods
62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
Software:
bootstrap
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