The Annals of Statistics

Adaptive goodness-of-fit tests based on signed ranks

Angelika Rohde

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Within the nonparametric regression model with unknown regression function l and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis l=0 against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against Hölder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to 1 in case of homoscedastic Gaussian errors within a broad range of Hölder classes simultaneously.

Article information

Ann. Statist., Volume 36, Number 3 (2008), 1346-1374.

First available in Project Euclid: 26 May 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties 62G35: Robustness

Exact multiple testing exponential inequality multiscale statistic relative asymptotic efficiency signed ranks sharp asymptotic adaptivity


Rohde, Angelika. Adaptive goodness-of-fit tests based on signed ranks. Ann. Statist. 36 (2008), no. 3, 1346--1374. doi:10.1214/009053607000000992.

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