Bootstrap Test for Difference between Means in Nonparametric Regression

Abstract A bootstrap test is proposed for detecting a difference between two mean functions in the setting of nonparametric regression. Error distributions in the regression model are permitted to be arbitrary and unequal. The test enjoys power properties akin to those in a parametric setting, in the sense that it can distinguish between regression functions distant only n −1/2 apart, where n is the sample size. It has exceptional level accuracy, with level error of only n −2, and uses a very accurate estimate of the critical point of an exact test, being in error by only n −3/2 under the null hypothesis. The test admits several generalizations, for example to the case of testing for differences between several regression means. (This is a nonparametric regression analog of analysis of variance.) A simulation study using n as small as 15 corroborates the asymptotic result on level accuracy of the bootstrap test. Applications are illustrated with an example involving acid rain data.