Testing for a constant coefficient of variation in nonparametric regression

In the common nonparametric regression model Y_i=m(X_i)+sigma(X_i)epsilon_i we consider the problem of testing the hypothesis that the coefficient of the scale and location function is constant. The test is based on a comparison of the observations Y_i=\hat{sigma}(X_i) with their mean by a smoothed empirical process, where \hat{sigma} denotes the local linear estimate of the scale function. We show weak convergence of a centered version of this process to a Gaussian process under the null hypothesis and the alternative and use this result to construct a test for the hypothesis of a constant coefficient of variation in the nonparametric regression model. A small simulation study is also presented to investigate the finite sample properties of the new test. AMS Subject Classi cation: 62G10, 62F3