  This paper is concerned with the problem of goodness-of-fit for curve
registration, and more precisely for the shifted curve model, whose application
field reaches from computer vision and road traffic prediction to medicine. We
give bounds for the asymptotic minimax separation rate, when the functions in
the alternative lie in Sobolev balls and the separation from the null
hypothesis is measured by the l2-norm. We use the generalized likelihood ratio
to build a nonadaptive procedure depending on a tuning parameter, which we
choose in an optimal way according to the smoothness of the ambient space.
Then, a Bonferroni procedure is applied to give an adaptive test over a range
of Sobolev balls. Both achieve the asymptotic minimax separation rates, up to
possible logarithmic factors.
