Smooth estimation of size distributions in an oriented cylinder model
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Smooth estimation of size distributions in an oriented cylinder model Geurt Jongbloed1
· Kimberly S. McGarrity2 · Jilt Sietsma1
Received: 10 March 2017 / Accepted: 5 August 2020 © The Author(s) 2020
Abstract Kernel estimators are proposed for estimating the cumulative distribution functions and the probability density functions of several quantities of interest in a stereological oriented cylinder model. This oriented cylinder model was developed to represent anisotropic microstructural features in materials. The asymptotic properties of these estimators are studied, and the estimators are applied to two banded dual phase steel microstructures. The estimation method is quite general and can also be applied to distributions of other univariate quantities of interest. Keywords Nonparametric estimation · Asymptotics · Materials science · Inverse problem · Stereology Mathematics Subject Classification 62G05 · 62G20 · 62P30
1 Introduction The so-called Wicksell problem introduced in Wicksell (1925) is a classical inverse problem in statistics. The original motivation was medical. A postmortem examination of spleens containing approximately spherical tumors was performed. Based on cross sections of the spleens (showing circular profiles of the tumors), the aim was to estimate the distribution of tumor sizes based on the observed circle radii. Wicksell’s problem is a typical example of a stereological problem, where one aims to infer ‘three-dimensional properties’ from ‘two-dimensional information’. Not only within the field of anatomy, but also in materials science and astronomy, this type of problem
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Geurt Jongbloed [email protected] Kimberly S. McGarrity [email protected]
1
Delft University of Technology, Delft, The Netherlands
2
Delft University of Technology and Materials innovation institute, Delft, The Netherlands
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G. Jongbloed et al.
is frequently encountered. See, e.g., Sen and Woodroofe (2011) for an astronomical application of the model. Over the years, quite some stereological problems related to Wicksell’s problem have been introduced and studied; see, e.g., Ohser and Mücklich (2000) for problems related to different shapes of the three-dimensional objects and Feuerverger and Hall (2000) for a problem where the data are obtained slightly differently. In this paper, we study another related model, specifically designed for a materials science problem. In this model, circular cylinders (all with the same orientation, say vertical axes) are distributed within an opaque medium which is cut vertically (parallel to the axes). The problem then is to estimate distributional properties of various threedimensional quantities related to size (volume, surface area, e.g.,) only based on data obtained from the two-dimensional section. This model was introduced in McGarrity et al. (2014), where also the relations between the distributions of (unobservable) threedimensional quantities and (observable) two-dimensional quantities are derived. These will be reviewed in Sect. 2. In that paper, estimator
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