Bandwidth Selection for Indirect Density Estimation Based on Corrupted Histogram Data

Abstract

Motivated by a number of practical applications, we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts. Due to variability in the process of correction, the histogram cannot be modeled in terms of simple scaled Poisson statistics. This departure necessitates the development of a new methodology for bandwidth selection. A variant of the method of unbiased risk estimation is proposed. The methodology is studied, using numerical simulations and asymptotic analysis tools, in the context of a class of idealized density deconvolution problems. The methodology is adapted for application to the practical reconstruction problem of positron emission tomography (PET). Realistic numerical simulations and physical phantom data are presented to validate the approach in this setting. Some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets are also presented.