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We consider a nonparametric goodness of fit test problem for the driftcoefficient of one-dimensional small diffusions. Our test is based on discreteobservation of the processes, and the diffusion coefficient is a nuisancefunction which is estimated in our testing procedure. We prove that the limitdistribution of our test is the supremum of the standard Brownian motion, andthus our test is asymptotically distribution free. We also show that our testis consistent under any fixed alternatives.

Goodness of fit test for small diffusions by discrete observations