×
van der Jagt, Thomas; Jongbloed, Geurt; Vittorietti, Martina

Stereological determination of particle size distributions for similar convex bodies. (English) Zbl 07823227

Electron. J. Stat. 18, No. 1, 742-774 (2024).
Summary: Consider an opaque medium that contains 3D particles. All particles are convex bodies of the same shape, but they vary in size. The particles are randomly positioned and oriented within the medium and cannot be observed directly. Taking a planar section of the medium we obtain a sample of observed 2D section profile areas of the intersected particles. In this paper, the distribution of interest is the underlying 3D particle size distribution for which an identifiability result is obtained. Moreover, a nonparametric estimator is proposed for this size distribution. The estimator is proven to be consistent and its performance is assessed in a simulation study.

MSC:

62G05 Nonparametric estimation
60D05 Geometric probability and stochastic geometry
60E10 Characteristic functions; other transforms

Keywords:

stereology; consistency; Mellin transform; particle system; iterative convex minorant; EM
Cite Review PDF
Full Text: DOI arXiv Link

References:

[1] ARRATIA, R., GOLDSTEIN, L. and KOCHMAN, F. (2019). Size bias for one and all. Probab. Surv. 16 1-61. Digital Object Identifier: 10.1214/13-ps221 Google Scholar: Lookup Link MathSciNet: MR3896143 · Zbl 1427.60002 · doi:10.1214/13-ps221
[2] BADDELEY, A. and JENSEN, E. B. V. (2004). Stereology for Statisticians. Chapman and Hall/CRC. Digital Object Identifier: 10.1201/9780203496817 Google Scholar: Lookup Link MathSciNet: MR2107000 · doi:10.1201/9780203496817
[3] BENEŠ, V. and RATAJ, J. (2004). Stochastic Geometry: Selected Topics. Kluwer Academic Publishers. Digital Object Identifier: 10.1007/b129604 Google Scholar: Lookup Link MathSciNet: MR2068607 · Zbl 1052.60001 · doi:10.1007/b129604
[4] BISSANTZ, N. and HOLZMANN, H. (2008). Statistical inference for inverse problems. Inverse Probl. 24 034009. Digital Object Identifier: 10.1088/0266-5611/24/3/034009 Google Scholar: Lookup Link MathSciNet: MR2421946 · Zbl 1137.62325 · doi:10.1088/0266-5611/24/3/034009
[5] BUTZER, P. L. and JANSCHE, S. (1997). A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3 325-376. Digital Object Identifier: 10.1007/BF02649101 Google Scholar: Lookup Link MathSciNet: MR1468369 · Zbl 0885.44004 · doi:10.1007/BF02649101
[6] CAVALIER, L. (2008). Nonparametric statistical inverse problems. Inverse Probl. 24 034004. Digital Object Identifier: 10.1088/0266-5611/24/3/034004 Google Scholar: Lookup Link MathSciNet: MR2421941 · Zbl 1137.62323 · doi:10.1088/0266-5611/24/3/034004
[7] CHIU, S. N., STOYAN, D., KENDALL, W. S. and MECKE, J. (2013). Stochastic Geometry and its Applications. John Wiley & Sons, Ltd. Digital Object Identifier: 10.1002/9781118658222 Google Scholar: Lookup Link MathSciNet: MR3236788 · Zbl 1291.60005 · doi:10.1002/9781118658222
[8] DAVY, P. and MILES, R. E. (1977). Sampling theory for opaque spatial specimens. J. R. Stat. Soc. Ser. B Methodol. 39 56-65. Digital Object Identifier: 10.1111/j.2517-6161.1977.tb01605.x Google Scholar: Lookup Link MathSciNet: MR0501482 · Zbl 0355.62010 · doi:10.1111/j.2517-6161.1977.tb01605.x
[9] DEMPSTER, A. P., LAIRD, N. M. and RUBIN, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B Methodol. 39 1-22. Digital Object Identifier: 10.1111/j.2517-6161.1977.tb01600.x Google Scholar: Lookup Link MathSciNet: MR0501537 · doi:10.1111/j.2517-6161.1977.tb01600.x
[10] GROENEBOOM, P., JONGBLOED, G. and MICHAEL, S. (2013). Consistency of maximum likelihood estimators in a large class of deconvolution models. Can. J. Stat. 41 98-110. Digital Object Identifier: 10.1002/cjs.11152 Google Scholar: Lookup Link MathSciNet: MR3030787 · Zbl 1273.62069 · doi:10.1002/cjs.11152
[11] GROENEBOOM, P. and WELLNER, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser Basel. Digital Object Identifier: 10.1007/978-3-0348-8621-5 Google Scholar: Lookup Link MathSciNet: MR1180321 · Zbl 0757.62017 · doi:10.1007/978-3-0348-8621-5
[12] JONGBLOED, G. (1998). The iterative convex minorant algorithm for nonparametric estimation. J. Comput. Graph. Stat. 7 310-321. Digital Object Identifier: 10.2307/1390706 Google Scholar: Lookup Link MathSciNet: MR1646718 · doi:10.2307/1390706
[13] JONGBLOED, G. (2001). Sieved maximum likelihood estimation in Wicksell’s problem and related deconvolution problems. Scand. J. Stat. 28 161-183. Digital Object Identifier: 10.1111/1467-9469.00230 Google Scholar: Lookup Link MathSciNet: MR1844355 · Zbl 0965.62028 · doi:10.1111/1467-9469.00230
[14] KAWATA, T. (1972). Fourier Analysis in Probability Theory. Academic Press. Digital Object Identifier: 10.1016/C2013-0-07405-1 Google Scholar: Lookup Link MathSciNet: MR0464353 · Zbl 0271.60022 · doi:10.1016/C2013-0-07405-1
[15] KISEL’ÁK, J. and BALUCHOVÁ, G. (2021). On particle-size distribution of convex similar bodies in \(\mathbb{R}^3\). J. Math. Imaging Vision 63 108-119. Digital Object Identifier: 10.1007/s10851-020-00997-y Google Scholar: Lookup Link MathSciNet: MR4205280 · Zbl 1517.45012 · doi:10.1007/s10851-020-00997-y
[16] MCGARRITY, K. S., SIETSMA, J. and JONGBLOED, G. (2014). Nonparametric inference in a stereological model with oriented cylinders applied to dual phase steel. The Annals of Applied Statistics 8 2538-2566. Digital Object Identifier: 10.1214/14-AOAS787 Google Scholar: Lookup Link MathSciNet: MR3292508 · Zbl 1454.62516 · doi:10.1214/14-AOAS787
[17] OHSER, J. and MÜCKLICH, F. (1995). Stereology for some classes of polyhedrons. Adv. Appl. Prob. 27 384-396. Digital Object Identifier: 10.2307/1427832 Google Scholar: Lookup Link MathSciNet: MR1334820 · Zbl 0822.60009 · doi:10.2307/1427832
[18] OHSER, J. and MÜCKLICH, F. (2000). Statistical Analysis of Microstructures in Materials Science. John Wiley & Sons, Inc. · Zbl 0960.62129
[19] OHSER, J. and NIPPE, M. (1997). Stereology of cubic particles: various estimators for the size distribution. J. Microsc. 187 22-30. Digital Object Identifier: 10.1046/j.1365-2818.1997.2020762.x Google Scholar: Lookup Link · doi:10.1046/j.1365-2818.1997.2020762.x
[20] SANTALÓ, L. A. and KAC, M. (2004). Integral Geometry and Geometric Probability. Cambridge University Press. Digital Object Identifier: 10.1017/CBO9780511617331 Google Scholar: Lookup Link MathSciNet: MR2162874 · Zbl 1116.53050 · doi:10.1017/CBO9780511617331
[21] VAN DER JAGT, T., JONGBLOED, G. and VITTORIETTI, M. (2023). Existence and approximation of densities of chord length- and cross section area distributions. Image Analysis and Stereology 42 171-184. Digital Object Identifier: 10.5566/ias.2923 Google Scholar: Lookup Link · Zbl 1535.60026 · doi:10.5566/ias.2923
[22] Vardi, Y. (1989). Multiplicative censoring, renewal processes, deconvolution and decreasing density: nonparametric estimation. Biometrika 76 751-761. Digital Object Identifier: 10.1093/biomet/76.4.751 Google Scholar: Lookup Link MathSciNet: MR1041420 · Zbl 0678.62051 · doi:10.1093/biomet/76.4.751
[23] WELLNER, J. A. and ZHAN, Y. (1997). A hybrid algorithm for computation of the nonparametric maximum likelihood estimator from censored data. J. Am. Stat. Assoc. 92 945-959. Digital Object Identifier: 10.1080/01621459.1997.10474049 Google Scholar: Lookup Link MathSciNet: MR1482125 · Zbl 0889.62026 · doi:10.1080/01621459.1997.10474049
[24] WICKSELL, S. D. (1925). The corpuscle problem. A mathematical study of a biometric problem. Biometrika 17 84-99. Digital Object Identifier: 10.1093/biomet/17.1-2.84 Google Scholar: Lookup Link · JFM 51.0392.02 · doi:10.1093/biomet/17.1-2.84
[25] ZOLOTAREV, V. M. (1957). Mellin-Stieltjes transforms in probability theory. Theory Probab. Appl. 2 433-460. Digital Object Identifier: 10.1137/1102031 Google Scholar: Lookup Link MathSciNet: MR0108843 · doi:10.1137/1102031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.