  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.
