We consider nonparametric estimation of the distribution function $F$ of squared sphere radii in the classical Wicksell problem. Under smoothness conditions on $F$ in a neighborhood of $x$, in \cite{21} it is shown that the Isotonic Inverse Estimator (IIE) is asymptotically efficient and attains rate of convergence $\sqrt{n / \log n}$. If $F$ is constant on an interval containing $x$, the optimal rate of convergence increases to $\sqrt{n}$ and the IIE attains this rate adaptively, i.e.\ without explicitly using the knowledge of local constancy. However, in this case, the asymptotic distribution is not normal. In this paper, we introduce three \textit{informed} projection-type estimators of $F$, which use knowledge on the interval of constancy and show these are all asymptotically equivalent and normal. Furthermore, we establish a local asymptotic minimax lower bound in this setting, proving that the three \textit{informed} estimators are asymptotically efficient and a convolution result showing that the IIE is not efficient. We also derive the asymptotic distribution of the difference of the IIE with the efficient estimators, demonstrating that the IIE is \textit{not} asymptotically equivalent to the \textit{informed} estimators. Through a simulation study, we provide evidence that the performance of the IIE closely resembles that of its competitors.

翻译：我们考虑经典Wicksell问题中球体半径平方分布函数$F$的非参数估计。在$x$邻域内$F$满足光滑性条件时，文献\cite{21}证明等渗逆估计器（IIE）具有渐近有效性且达到$\sqrt{n / \log n}$收敛速率。若$F$在包含$x$的区间上为常数，最优收敛速率提升至$\sqrt{n}$，且IIE能自适应地达到该速率（即无需显式利用局部常数性信息）。但此时渐近分布不再服从正态分布。本文提出三种基于先验信息的投影型估计器，这些估计器利用常数区间信息，并证明它们具有渐近等价性与正态性。此外，我们建立了该设定下的局部渐近极小极大下界，证明三种先验信息估计器具有渐近有效性，同时通过卷积定理表明IIE不具备有效性。我们还推导了IIE与有效估计器之差的渐近分布，证明IIE与先验信息估计器不存在渐近等价性。通过模拟研究，我们发现IIE的实际表现与其竞争方法极为接近。