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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.

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