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 Gili et al. (2024) it is shown that the Isotonic Inverse Estimator (IIE) is asymptotically efficient and attains rate of convergence n/logn. If F is constant on an interval containing x, the optimal rate of convergence increases to 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 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 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 not asymptotically equivalent to the informed estimators. Through a simulation study, we provide evidence that the performance of the IIE closely resembles that of its competitors, supporting the use of the IIE as the standard choice when no information about F is available.

In this paper, we propose a novel Bayesian approach for nonparametric estimation in Wicksell’s problem. This has important applications in astronomy for estimating the distribution of the positions of the stars in a galaxy given projected stellar positions and in materials science to determine the 3D microstructure of a material, using its 2D cross-sections. We deviate from the classical Bayesian nonparametric approach, which would place a Dirichlet Process (DP) prior on the distribution function of the unobservables, by directly placing a DP prior on the distribution function of the observables. Our method offers computational simplicity due to the conjugacy of the posterior and allows for asymptotically efficient estimation by projecting the posterior onto the L₂ subspace of increasing, right-continuous functions. Indeed, the resulting Isotonized Inverse Posterior (IIP) satisfies a Bernstein–von Mises (BvM) phenomenon with minimax asymptotic variance g₀ (x)/2γ, where γ > 1/2 reflects the degree of Hölder continuity of the true cdf at x. Since the IIP gives automatic uncertainty quantification, it eliminates the need to estimate γ . Our results provide the first semiparametric Bernstein–von Mises theorem for projection-based posteriors with a DP prior in inverse problems.