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. In the classical setting, we study the isotonized version of the plug-in estimator (IIE) for the underlying cdf $F$ of the spheres' squared radii. This estimator is fully automatic, in the sense that it does not rely on tuning parameters, and we show it is adaptive to local smoothness properties of the distribution function $F$ to be estimated. Moreover, we prove a local asymptotic minimax lower bound in this non-standard setting, with $\sqrt{\log{n}/n}$-asymptotics and where the functional $F$ to be estimated is not regular. Combined, our results prove that the isotonic estimator (IIE) is an adaptive, easy-to-compute, and efficient estimator for estimating the underlying distribution function $F$.

翻译：我们考虑Wicksell问题中的非参数估计，该问题在天文学中具有重要应用——根据投影恒星的分布估计星系中恒星位置分布，以及在材料科学中利用二维截面确定材料的三维微观结构。在经典设定下，我们研究了球体半径平方的累积分布函数$F$的插入估计量（IIE）的等渗化版本。该估计量完全自动化（即不依赖调节参数），并且我们证明它能够自适应被估计分布函数$F$的局部光滑性质。此外，我们在这一非标准设定下证明了局部渐近极小极大下界，其渐近速率为$\sqrt{\log{n}/n}$，且待估泛函$F$是非正则的。综合而言，我们的结果证明了等渗估计量（IIE）是估计底层分布函数$F$的自适应、易计算且高效的估计量。