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In this paper, we consider a nonlinear inverse problem occurring in nuclearscience. Gamma rays randomly hit a semiconductor detector which produces animpulse response of electric current. Because the sampling period of themeasured current is larger than the mean inter arrival time of photons, theimpulse responses associated to different gamma rays can overlap: thisphenomenon is known as pileup. In this work, it is assumed that the impulseresponse is an exponentially decaying function. We propose a novel method toinfer the distribution of gamma photon energies from the indirect measurementsobtained from the detector. This technique is based on a formula linking thecharacteristic function of the photon density to a function involving thecharacteristic function and its derivative of the observations. We establishthat our estimator converges to the mark density in uniform norm at alogarithmic rate. A limited Monte-Carlo experiment is provided to support ourfindings.

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