We consider a density estimation problem arising in nuclear physics. Gamma photons are impinging on a semiconductor detector, producing pulses of current. The integral of this pulse is equal to the total amount of charge created by the photon in the detector, which is linearly related to the photon energy. Because the inter-arrival times of photons can be shorter than the charge collection time, pulses corresponding to different photons may overlap leading to a phenomenon known as pile-up. The distortions on the photon energy spectrum estimate due to pile-up become worse when the photon rate increases, making pile-up correction techniques a must for high counting rate experiments. In this paper, we present a novel technique to correct pile-up, which extends a method introduced by Hall and Park for the estimation of the service time from the busy period in $M/G/∞$ models. It is based on a novel formula linking the joint distribution of the energy and duration of the cluster of pulses and the distribution of the energy of the photons. We then assess the performance of this estimator by providing an expression for its integrated square error. A Monte Carlo experiment is presented to illustrate, with practical examples, the benefits of the pile-up correction.
E. Moulines. F. Roueff. A. Souloumiac. T. Trigano. "Nonparametric inference of photon energy distribution from indirect measurement." Bernoulli 13 (2) 365 - 388, May 2007. https://doi.org/10.3150/07-BEJ5184