Clarification of the use of CHIsquare and likelihood functions in fits to histograms
Abstract
We consider the problem of fitting curves to histograms in which the data obey multinomial or Poisson statistics. Techniques commonly used by physicists are examined in light of standard results found in the statistics literature. We review the relationship between multinomial and Poisson distributions, and clarify a sufficient condition for equality of the area under the fitted curve and the number of events on the histogram. Following the statisticians, we use the likelihood ratio test to construct a general χ^{2} statistic, χ_{λ}^{2}, which yields parameter and error estimates identical to those of the method of maximum likelihood. The χ_{λ}^{2} statist further useful for testing goodnessoffit since the value of its minimum asymptotically obeys a classical chisquare distribution. One should be aware, however, of the potential for statistical bias, especially when the number of events is small.
 Publication:

Nuclear Instruments and Methods in Physics Research
 Pub Date:
 April 1984
 DOI:
 10.1016/01675087(84)900164
 Bibcode:
 1984NIMPR.221..437B