On Cramér–von Mises type test based on local time of switching diffusion process

We consider a Cramér–von Mises type test for hypothesis that the observed diffusion process has sign-type trend coefficient based on empirical density function. It is shown that the limit distribution of the proposed test statistic is defined by the integral type functional of continuous Gaussian process. We provide the Karhunen–Loève expansion of the corresponding limiting process. Approximation of the threshold is given through the representation for the limit statistic.