Open Access
December 2014 Nonparametric inference in a stereological model with oriented cylinders applied to dual phase steel
K. S. McGarrity, J. Sietsma, G. Jongbloed
Ann. Appl. Stat. 8(4): 2538-2566 (December 2014). DOI: 10.1214/14-AOAS787


Oriented circular cylinders in an opaque medium are used to represent certain microstructural objects in steel. The opaque medium is sliced parallel to the cylinder axes of symmetry and the cut-plane contains the observable rectangular profiles of the cylinders. A one-to-one relation between the joint density of the squared radius and height of the 3D cylinders and the joint density of the squared half-width and height of the observable 2D rectangles is established. We propose a nonparametric estimation procedure to estimate the distributions and expectations of various quantities of interest, such as the cylinder radius, height, aspect ratio, surface area and volume from the observed 2D rectangle widths and heights. Also, the covariance between the radius and height of a cylinder is estimated. The asymptotic behavior of these estimators is established to yield point-wise confidence intervals for the expectations and point-wise confidence sets for the distributions of the quantities of interest. Many of these quantities can be linked to the mechanical properties of the material, and are, therefore, useful for industry. We illustrate the mathematical model and estimation procedures using a banded microstructure for which nearly 90 μm of depth have been observed via serial sectioning.


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K. S. McGarrity. J. Sietsma. G. Jongbloed. "Nonparametric inference in a stereological model with oriented cylinders applied to dual phase steel." Ann. Appl. Stat. 8 (4) 2538 - 2566, December 2014.


Published: December 2014
First available in Project Euclid: 19 December 2014

zbMATH: 06408789
MathSciNet: MR3292508
Digital Object Identifier: 10.1214/14-AOAS787

Keywords: Banded microstructures , grain size distribution , isotonic estimation , stochastic modeling , Wicksell’s problem

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.8 • No. 4 • December 2014
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