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Biometrika 2000 87(3):663-673; doi:10.1093/biomet/87.3.663
© 2000 by Biometrika Trust
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Tests for monotonicity of a regression mean with guaranteed level

I GijbelsZ, P HallZZ, MC JonesY and I KochW

Z Institut de Statistique, Université Catholique de Louvain, Voie de Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium E-mail: gijbels@stat.ucl.ac.be ZZ Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia E-mail: peter.hall@anu.edu.au Y Department of Statistics, Open University, Milton Keynes MK7 6AA, UK E-mail: m.c.jones@open.ac.uk W Department of Statistics, University of Newcastle, Newcastle NSW 2308, Australia E-mail: stimk@u2.newcastle.edu.au

In this paper a nonparametric procedure for testing for monotonicity of a regression mean with guaranteed level is proposed. The procedure is based on signs of differences of observations from the response variable. The test is calibrated against the most difficult null hypothesis, when the regression function is constant, and produces an exact test in this context. In general, the test is conservative. The power of the test is good, and comparable with that of other nonparametric tests. It is shown that the testing procedure has asymptotic power 1 against certain local alternatives. The method is also robust against heavy-tailed error distributions, and even maintains good power when the errors are for example Cauchy distributed. A simulation study is provided to demonstrate finite-sample behaviour of the testing procedure.

Key Words: calibration; counts; exact test; lengths of runs; local alternative; Monte Carlo simulation; power; signs of difference


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