Adaptive and Efficient Isotonic Estimation in Wicksell's Problem

We consider nonparametric estimation in Wicksell's problem and we show that the isotonized version of the plug-in estimator is asymptotically efficient. The asymptotic variance will depend on the local smoothness at the estimation point and at $0$ of the unknown distribution function $F$ of the ball-squared radii. We prove a corresponding local asymptotic minimax lower bound, as well under some local smoothness condition. This solves in an adaptive way the nonparametric estimation problem.