On goodness of fit testing for a diffusion type process with a delay parameter

The issue of goodness-of-fit testing for diffusion processes with small noise is addressed in this article. The basic hypothesis is composite parametric with an unknown delay in the drift. The primary purpose is to obtain an asymptotically distribution-free (ADF) test. In order to address this problem, it was decided to carry out a test statistic of Cramér-von Mises type in some sense. This statistic is written using two estimators of the delay parameter, namely a maximum likelihood estimator and a minimum distance estimator. Next, we examine the weak convergence of the proposed test statistic to prove that the limiting process, resulting from a specific transformation of the score function process, is a Brownian bridge. Then an asymptotically distribution-free (ADF) test is proposed. Afterwards, the consistency of the above test is established against any fixed alternative hypothesis.