  This paper develops tests for inequality constraints of nonparametric
regression functions. The test statistics involve a one-sided version of
$L_p$-type functionals of kernel estimators $(1 \leq p < \infty)$. Drawing on
the approach of Poissonization, this paper establishes that the tests are
asymptotically distribution free, admitting asymptotic normal approximation. In
particular, the tests using the standard normal critical values have
asymptotically correct size and are consistent against general fixed
alternatives. Furthermore, we establish conditions under which the tests have
nontrivial local power against Pitman local alternatives. Some results from
Monte Carlo simulations are presented.
