Defining and Applying a New Approximation for the Parametric Probability Densities of Spherical Particle Profile Sizes

Ekaterina Poliakova

doi:10.5566/ias.2539

Authors

Ekaterina Poliakova Arctic and Antarctic Research Institute, Norwegian University of Science and Technology

DOI:

https://doi.org/10.5566/ias.2539

Keywords:

distribution parameters, maximum likelihood, probability density function, small samples, spherical particles, statistics

Abstract

A novel approximation for the parametric probability densities of spherical particle profile diameters is suggested. The spheres are approximated from solids obtained by rotating regular polygons around their axes of symmetry. This approximation facilitates using the maximum likelihood method in samples that have too few profiles for applying classical stereological approaches.

References

Akaike H (1974). A new look at the statistical model identification. IEEE T Automat Contr 19:716–23.

Baddeley A, Jensen EBV (2004). Stereology for statisticians. CRC Press.

Casella G, Berger RL (2002). Statistical inference. Duxbury Pacific Grove, CA.

Cruz-Orive LM (2017). Stereology: a historical survey. Image Anal Stereol 36:153–77,DOI: 10.5566/ias.1767.

Depriester D, Kubler R (2019). Resolution of the Wicksell’s equation by minimum distance estimation. Image Anal Stereol 38:213–26, DOI: 10.5566/ias.2133.

Durand G, Gagliardini O, Thorsteinsson T, Svensson A, Kipfstuhl S, Dahl-Jensen D (2006). Ice microstructure and fabric: an up-to-date approach for measuring textures. J Glaciol 52:619–30, doi:10.3189/172756506781828377.

Goldsmith P (1967). The calculation of true particle size distributions from the sizes observed in a thin slice. Brit J Appl Phys 18:813.

Gulbin Y (2008). On estimation and hypothesis testing of the grain size distribution by the Saltykov method. Image Anal Stereol 27:163–74, DOI: 10.5566/ias.v27.p163–74.

Hobolth A, Jensen EBV (2002). A note on design-based versus model-based variance estimation in stereology. Adv Appl Probab 34:484–90,https://www.jstor.org/stable/1428242.

Keiding N, Jensen ST (1972). Maximum likelihood estimation of the size distribution of liver cell nuclei from the observed distribution in a plane section. Biometrics 28:813–29.

Larsen RJ, Marx ML (2012). Introduction to Mathematical Statistics and Its Applications: Pearson New International Edition. Pearson Prentice Hall.

Lehmann EL, Romano JP (2006). Testing statistical hypotheses. Springer Science &Business Media.

Lipenkov VY, Ekaykin AA, Polyakova EV, Raynaud D (2016). Characterization of subglacial Lake Vostok as seen from physical and isotope properties of accreted ice. Philos T Roy Soc A 374:1 – 15, https://doi.org/10.1098/rsta.2014.0303.

Lopez-Sanchez MA, Llana-F ́unez S (2016). An extension of the Saltykov method to quantify 3D grain size distributions in mylonites. J Struct Geol 93:149–61,https://doi.org/10.1016/j.jsg.2016.10.008.

Poliakova E (2020). Maximum likelihood estimation of parameters of spherical particle size distributions from profile size measurements and its application for small samples. ArXiv preprint : https://arxiv.org/abs/2008.09091.

Schweder T, Hjort NL (2016). Confidence, likelihood, probability, vol. 41. Cambridge University Press.

Shao J (2003). Mathematical Statistics. Springer-Verlag New York.