Nonparametric estimation of mark's distribution of an exponential Shot-noise process

TL;DR

This paper introduces a nonparametric method to estimate the distribution of gamma photon energies from indirect detector measurements affected by pileup, using characteristic functions and their derivatives, with proven convergence properties.

Contribution

It presents a novel nonparametric estimator for the mark distribution in an exponential shot-noise process, addressing the pileup problem in nuclear detection.

Findings

Estimator converges to the true distribution at a logarithmic rate.

Method successfully infers photon energy distribution from overlapping impulse responses.

Monte Carlo experiments support theoretical convergence results.

Abstract

In this paper, we consider a nonlinear inverse problem occurring in nuclear science. Gamma rays randomly hit a semiconductor detector which produces an impulse response of electric current. Because the sampling period of the measured current is larger than the mean inter arrival time of photons, the impulse responses associated to different gamma rays can overlap: this phenomenon is known as pileup. In this work, it is assumed that the impulse response is an exponentially decaying function. We propose a novel method to infer the distribution of gamma photon energies from the indirect measurements obtained from the detector. This technique is based on a formula linking the characteristic function of the photon density to a function involving the characteristic function and its derivative of the observations. We establish that our estimator converges to the mark density in uniform norm at a logarithmic rate. A limited Monte-Carlo experiment is provided to support our findings.