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Commonality of cusum, von Neumann and smoothing-based goodness-of-fit tests. (English) Zbl 0792.62042
Summary: Recent papers by P. J. Munson and R. J. Jernigan [ibid. 76, No. 1, 39-47 (1989; Zbl 0664.62068)] and M. J. Buckley [ibid. 78, 253-262 (1991)] propose nonparametric tests for the hypothesis of no predictor effect in regression. The Munson-Jernigan test is similar to the J. von Neumann test [Ann. Math. Statist. 12, 367-395 (1941; Zbl 0060.299)], 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.

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics
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