  In the common nonparametric regression model the problem of testing for a
specific parametric form of the variance function is considered. Recently Dette
and Hetzler (2008) proposed a test statistic, which is based on an empirical
process of pseudo residuals. The process converges weakly to a Gaussian process
with a complicated covariance kernel depending on the data generating process.
In the present paper we consider a standardized version of this process and
propose a martingale transform to obtain asymptotically distribution free tests
for the corresponding Kolmogorov-Smirnov and Cram\'{e}r-von-Mises functionals.
The finite sample properties of the proposed tests are investigated by means of
a simulation study.
