We test different fit methods of weighted binned data. The tested weight distributions are normal, exponential, and uniform. The toy case is a common HEP fit of a gaussian peak above a smooth background (here exponential).
Fits of weighted data are discussed in F. James, “Statistical Methods in Experimental Physics” (2nd edition), World Scientific, in section 8.5. Two approximate least-squares methods for binned data are presented in section 8.5.4. Here we test the method \(Q_1'\).
Bohm and Zech (2014) derived an approximate maximum-likelihood method for weighted histograms with on average positive weights. The authors show that their approximation yields less biased results than the aforementioned least-squares method.
Still, a notable bias is present even when the SPD method is used. The largest bias is found for exponentially distributed weights with deviations of 4-5% in the fitted parameters.
The remaining bias of the SPD method originates from the fact that the true weight variance in each bin is unknown and is replaced by the sample variance. This is inaccurate in bins with small number of samples and fails for empty bins, which must be skipped. The same is true for the alternative least-squares fit, however, and the biases are even larger there.
The study suggest that biases are larger if the weight distribution deviates more from the normal distribution.
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