Minimax hypothesis testing for curve registration

Abstract

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 $l_{2}$-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.