Under smoothness conditions, the Isotonic Inverse Estimator (IIE) achieves an asymptotic convergence rate of √n/logn. If F is constant on an interval around x, the rate improves to √n, with the IIE adapting to this without explicit knowledge. However, the IIE is not asymptotically normal or efficient. The paper introduces three projection-type estimators that leverage the constancy of F and proves their asymptotic efficiency and normality. A local minimax lower bound is also established, and simulation results suggest that the IIE performs comparably to the new estimators despite not being asymptotically equivalent.

We consider nonparametric estimation in Wicksell’s problem which has relevant applications in astronomy for estimating the distribution of the positions of the stars in a galaxy given projected stellar positions and in material sciences to determine the 3D microstructure of a material, using its 2D cross sections. We study the isotonized version of the plug-in estimator (IIE) for the underlying cdf F of the spheres’ squared radii and show it is an adaptive, easy-to-compute, and efficient estimator for estimating the underlying distribution function F.

We implemented in Python the Isotonic estimator of the paper “Adaptive and Efficient Isotonic Estimation in Wicksell Problem”. We provide the used code where the user can try the estimator choosing from a set of hidden distribution functions F of the radii distribution.