Adaptive confidence sets for kink estimation

Abstract

We consider estimation of the location and the height of the jump in the $\gamma $-th derivative - a kink of order $\gamma $ - of a regression curve, which is assumed to be Hölder smooth of order $s\geq \gamma +1$ away from the kink. Optimal convergence rates as well as the joint asymptotic normal distribution of estimators based on the zero-crossing-time technique are established. Further, we construct joint as well as marginal asymptotic confidence sets for these parameters which are honest and adaptive with respect to the smoothness parameter $s$ over subsets of the Hölder classes. The finite-sample performance is investigated in a simulation study, and a real data illustration is given to a series of annual global surface temperatures.