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 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 $F$.

[1]  G. Jongbloed,et al.  Existence and approximation of densities of chord length- and cross section area distributions , 2023, Image Analysis and Stereology.

[2]  Cun-Hui Zhang,et al.  Confidence intervals for multiple isotonic regression and other monotone models , 2020, The Annals of Statistics.

[3]  Aad van der Vaart,et al.  Fundamentals of Nonparametric Bayesian Inference , 2017 .

[4]  J. Cuzzi,et al.  Recovering 3D particle size distributions from 2D sections , 2017, Meteoritics & planetary science.

[5]  S. Llana-Fúnez,et al.  An extension of the Saltykov method to quantify 3D grain size distributions in mylonites , 2016 .

[6]  J. Qin,et al.  Nonparametric maximum likelihood estimation for the multisample Wicksell corpuscle problem , 2016, Biometrika.

[7]  B. Sen,et al.  Bootstrap confidence intervals for isotonic estimators in a stereological problem , 2012, 1211.5420.

[8]  G. Jongbloed.  Sieved Maximum Likelihood Estimation in Wicksell's Problem and Related Deconvolution Problems , 2001 .

[9]  Golubev G K,et al.  Asymptotically efficient estimation in the Wicksell problem , 1998 .

[10]  T. Mattfeldt.  Stochastic Geometry and Its Applications , 1996 .

[11]  P. Groeneboom,et al.  Isotonic estimation and rates of convergence in Wicksell's problem , 1995 .

[12]  R. Durrett.  Probability: Theory and Examples , 1993 .

[13]  D. Pollard,et al.  Cube Root Asymptotics , 1990 .

[14]  Peter Hall,et al.  The kernel method for unfolding sphere size distributions , 1988 .

[15]  B. Levit,et al.  On a Non-Parametric Analogue of the Information Matrix , 1977 .

[16]  Geoffrey S. Watson,et al.  Estimating functionals of particle size distributions , 1971 .

[17]  D. V. Lindley,et al.  An Introduction to Probability Theory and Its Applications. Volume II , 1967, The Mathematical Gazette.

[18]  J. Lamperti.  ON CONVERGENCE OF STOCHASTIC PROCESSES , 1962 .

[19]  W. Feller.  An Introduction to Probability Theory and Its Applications , 1959 .

[20]  Soumendu Sundar Mukherjee,et al.  Weak convergence and empirical processes , 2019 .

[21]  Geurt Jongbloed,et al.  Nonparametric Estimation under Shape Constraints , 2014 .

[22]  A. V. D. Vaart,et al.  Lectures on probability theory and statistics , 2002 .

[23]  G. K. Golubev,et al.  Asymptotically Efficient Smoothing in the Wicksell Problem under Squared Losses , 2001, Probl. Inf. Transm..