Nonparametric estimation of weighted proportions with application to particle size measurement

Weighted proportions arise naturally in particle size measurement where one is interested in estimating the proportion of the total volume or surface corresponding to specific particle diameters. The particles are assumed to be spherical, and observations consist of the particle diameters. We consid er the weighted proportion as a nonparametric functional of the diameter di stribution, and investigate how well this functional can be estimated. It i s shown that the commonly used estimator is asymptotically efficient, but i t is very sensitive to sample fluctuations. We propose a modified estimator which is also asymptotically efficient and has better small-sample propert ies. Numerical experiments with real datasets demonstrate the improved prac tical behaviour of the proposed estimators.