Testing for a constant coefficient of variation in nonparametric regression

In the common nonparametric regression model Yi = m(Xi) + σ(Xi) 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 Yi/σ̂(Xi) with their mean by a smoothed empirical process, where σ̂ 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.