Maximum-Likelihood Fits to Histograms for Improved Parameter Estimation

Abstract

Straightforward methods for adapting the familiar $\chi ^2$ statistic to histograms of discrete events and other Poisson distributed data generally yield biased estimates of the parameters of a model. The bias can be important even when the total number of events is large. For the case of estimating a microcalorimeter’s energy resolution at 6 keV from the observed shape of the Mn K$\alpha$ fluorescence spectrum, a poor choice of $\chi ^2$ can lead to biases of at least 10 % in the estimated resolution when up to thousands of photons are observed. The best remedy is a Poisson maximum-likelihood fit, through a simple modification of the standard Levenberg-Marquardt algorithm for $\chi ^2$ minimization. Where the modification is not possible, another approach allows iterative approximation of the maximum-likelihood fit.