A Mixing Distribution Approach to Estimating Particle Size Distributions
Spherical particles are dispersed randomly in a three-dimensional body. The centers of the spheres are distributed according to a dilute Poisson process. The radii of such spheres have a distribution G independent of everything else. A random probe (line, plane or thin slice) is cut through the volumes. Taking the viewpoint of nonparametric estimation of mixing distributions, we propose a new procedure that deals with the shortcomings of the classical procedures. We consider linear, planar and thin slice data. In all three cases, our approach performs better than the classical procedure. In addition, we prove consistency results. In the random plane case, we discuss the right way and the wrong way to bootstrap the distribution of a stereological estimate, corresponding to whether we have taken the structure of the problem into account or not. In the thin slice case, when G is mixed or discrete, the formulas involve a decomposition of H into its continuous and discrete component. This makes the estimation problem more complicated but also more interesting especially in the discrete case. We propose a few procedures which involve a decomposition of the data corresponding to that of H.
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0.9 quantiles 1000 independent samples admissible range algorithm Anderssen and Jakeman's asymptotic variance BOOTSTRAP ESTIMATES resample Bulgren's estimate C&B MLE Choi and Bulgren's classical estimator classical procedures consistent estimate Deely and Kruse's dense in 0,A density dH(y distribution function Doctor of Philosophy equations estimate H estimate of G estimates over 100 ESTIMATES resample 1000 estimating G(9 fe(y fQ(y G is discrete Imply inversion formula Jakeman's estimate Kruse's estimate large numbers law of large lemma let Q linear linear programming m.s.e. shown maximum likelihood estimate METHOD M.S.E. minimum distance method mixing distribution nonparametric MLE planar data Poisson process probability probability mass functions product integration estimate puts mass 0.2 Q e Q quadratic programming random plane random probe sample sequence sample size Simulation results spectral derivative spectral differentiation stationary distribution sufficiently large support of G terms of m.s.e. trials TRUE VALUES based variance