Count rate estimation of a Poisson process: Unbiased fit vs. central moment analysis of time interval spectra


Stefaan Pommé,, Institute for Reference Materials and Measurements, EC-JRC-IRMM, Retieseweg 111, B-2440 Geel, Belgium and John D. Keightley,, Institute for Reference Materials and Measurements, Retieseweg 111, B-2460 Geel, Belgium.
Least squares fitting procedures and weighted averages imply the assignment of proper weighting factors to the stochastically distributed data involved. In the case of a Poisson distribution, the obvious choice of setting the weighting factor equal to the inverse of the measured value is prone to bias towards low values. Alternatively, using the inverse of the fitted value may turn out to be biased towards higher values, depending on the procedure followed. Additional problems arise when also the possibility of zero counts has to be taken into account. In this work, several least squares statistics are considered and their performance compared with maximum likelihood estimation. The particular case of an exponential time interval distribution of a stationary Poisson process is investigated. Formulas are presented to determine the event rate of the Poisson process from the central moments of the time interval distribution. These count rates are compared to estimated values from the fit procedures.