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 consider the weighted proportion as a nonparametric functional of the diameter distribution, and investigate how well this functional can be estimated. It is shown that the commonly used estimator is asymptotically efficient, but it is very sensitive to sample fluctuations. We propose a modified estimator which is also asymptotically efficient and has better small-sample properties. Numerical experiments with real datasets demonstrate the improved practical behaviour of the proposed estimators.