Adaptive and Efficient Isotonic Estimation in Wicksell's Problem

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 (cf. Sen, B. and Woodroofe, F. N. (2012)) and in material sciences to determine the 3D microstructure of a material, using its 2D cross sections (cf. Cuzzi, J. and Olson, D. (2017) or Lopez-Sanchez, M. A. and LLana-Fúnez, S. (2016)). In the classical setting, an opaque medium that contains randomly positioned 3D spherical particles, that cannot be observed directly, is considered. Taking a planar section of the medium, we obtain a sample of observed 2D section profile areas of the intersected particles. The distribution function F of interest is the underlying distribution of the 3D particle squared radii. In this setting, we study the isotonized version of the plug-in estimator (IIE cf. Groeneboom, P. and Jongbloed, G. (1995)) 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 √logn/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. Our results cover several novel theoretical findings, which prove essentially that the isotonic estimator is naturally adaptive and efficient in the classical statistical sense. No other estimators for the classical Wicksell’s problem have been shown to have all these properties. Therefore this work is of relevance also from the application point of view, in fields like materials science, astronomy and biosciences, because it shows that the isotonic estimator is the unequivocal choice for practitioners, surpassing alternative estimators previously proposed in Wicksell’s problem.