Maximum likelihood estimation of parameters of spherical particle size distributions from profile size measurements and its application for small samples

Microscopy research often requires recovering particle-size distributions in three dimensions from only a few (10 - 200) profile measurements in the section. This problem is especially relevant for petrographic and mineralogical studies, where parametric assumptions are reasonable and finding distribution parameters from the microscopic study of small sections is essential. This paper deals with the specific case where particles are approximately spherical (i.e. Wicksell's problem). The paper presents a novel approximation of the probability density of spherical particle profile sizes. This approximation uses the actual non-smoothness of mineral particles rather than perfect spheres. The new approximation facilitates the numerically efficient use of the maximum likelihood method, a generally powerful method that provides the distribution parameter estimates of the minimal variance in most practical cases. The variance and bias of the estimates by the maximum likelihood method were compared numerically for several typical particle-size distributions with those by alternative parametric methods (method of moments and minimum distance estimation), and the maximum likelihood estimation was found to be preferable for both small and large samples. The maximum likelihood method, along with the suggested approximation, may also be used for selecting a model, for constructing narrow confidence intervals for distribution parameters using all the profiles without random sampling and for including the measurements of the profiles intersected by section boundaries. The utility of the approach is illustrated using an example from glacier ice petrography.