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Biometrika 1993 80(1):89-98; doi:10.1093/biomet/80.1.89
© 1993 by Biometrika Trust
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Commonality of cusum, von Neumann and smoothing-based goodness-of-fit tests

R. L. EUBANK and JEFFREY D. HART

Department of Statistics, Texas A&M University College Station, Texas 77843, U.S.A.

Recent papers by Munson & Jernigan (1989) and Buckley (1991) propose nonparametric tests for the hypothesis of no predictor effect in regression. The Munson-Jernigan test is similar to the von Neumann (1941) test, while that of Buckley is based on a functional of cusums. These tests are shown to be special cases of a wider class of tests based on nonparametric function estimation ideas. Fourier analysis is used to qualitatively compare the Munson & Jernigan and Buckley tests with two new tests constructed from nonparametric smoothers. Their relative powers are then studied by means of large-sample analysis and simulation. The cusum test is the most powerful for very smooth departures from the no-effect hypothesis, while the new tests based on smoothing ideas are clearly superior when the alternative is high frequency.

Key Words: Fourier series estimators • Local alternatives • Nonparametric regression • Pitman relative efficiency • Smoothing splines


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