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.
Highlights
Our concern is the statistical problem of curve registration, which appears naturally in a large number of applications, when the available data consist of a set of noisy, distorted signals that possess a common structure or pattern
As a matter of fact, if adaptation to the unknown smoothness typically entails a loglog-factor, other types of adaptation can bring simple logarithmic ones: it is proved in Lepski and Tsybakov [30] that the asymptotic minimax separation rate for signal detection when the signal to be detected belongs to a Sobolev or Holder ball and the separation from 0 is measured by the sup-norm is (σ2 log σ−1)s/2s+1, while it is σ2s/2s+1 when the separation from 0 is measured by the value of the signal at a fixed point
The choice of our model was inspired by practical considerations, and we intend to apply it to a problem in computer vision: that of keypoint matching as briefly discussed in Collier and Dalalyan [11]
Summary
Our concern is the statistical problem of curve registration, which appears naturally in a large number of applications, when the available data consist of a set of noisy, distorted signals that possess a common structure or pattern. The problem considered in the present work is qualitatively different from the aforementioned works on the minimax separation rate, since our null hypothesis is composite and semiparametric It seems that the finite-dimensional parameter cannot be uniformly consistently estimated, which contrasts with the situation of Horowitz and Spokoiny [24]. As a matter of fact, if adaptation to the unknown smoothness typically entails a loglog-factor, other types of adaptation can bring simple logarithmic ones: it is proved in Lepski and Tsybakov [30] that the asymptotic minimax separation rate for signal detection when the signal to be detected belongs to a Sobolev or Holder ball and the separation from 0 is measured by the sup-norm is (σ2 log σ−1)s/2s+1, while it is σ2s/2s+1 when the separation from 0 is measured by the value of the signal at a fixed point.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
