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Curing a Summing Error That Occurs Automatically When Fitting a Function to Binomial or Poisson Distributed Data

Abstract

Without special precautions a sum-rule error occurs automatically when a chi-squared procedure is used to fit a funtion to binomial or Poisson distributed histogram data if the function has at least one linear parameter. Since the square of the variance per channel is equal to the mean population, errors are usually approximated using (G2~=yi>0)}; this choice for approximating the variance gives a per-channel error weighting of 1/yi that automatically results in a sum-rule error. This sum-rule error consistently and systematically underestimates the total sum of the data points by an amount equal to the value of %*, resulting in Zjyj-Zjfj= J& where %i = £j(vi - QVyi an^ f'i= f(Xj,{parameters}). In contrast, using {o'-f=(¦>()} gives the error weighting per channel of 1/fj that automatically results in a less well known sum rule error. This sum-rule error which is only half as large but opposite insign consistently and systematically overestimates the total sum ofthe data X? points by an amount equal to half the value of %?, that is, itresults in - Zjf- =- L y ,where Xr = (vi " 'i)"'/'i- The good news is a combination of error weightings may be constructed which completely eliminates the otherwise automatically cocuring sum-rule error by taking advantage of cancellations occuring between the two sum-rule errors implicit in the two above-mentioned approaches to error-weighting per channel. This fortunitous linear combination ofsum-rule error swill combine and cancel ifthe fitting funtion is a sufficiently viable choice so that Xr= Xy = v (number ofdegrees of freedom); 1 2 consequently a weighted linear combination of these two definitions may be used, X 2 = 3 Xy + !%?• This choice for X" = is 1 1 2 equivalent to choosing an error weighting of „'_' = 3yj +:'>(; ,and it essentially eliminates summing errors so that Xjyi- Zjf-. An alternate method is presented and proven for {jLt; = f-}infitting a function using Maximum Likelihood.

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