The Iterated Weighted Least-Squares Fit
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Least-squares methods are popular in statistical inference, but are widely believed to produce unbiased results only for normally distributed data. We demonstrate that commonly used variants of the least-squares method indeed produce biased estimates when applied to Poisson-distributed data, but that an iterated weighted least-squares method produces the same unbiased result as the maximum-likelihood method. For linear models, the iterated weighted least-squares method converges faster than the equivalent maximum-likelihood method and does not require problem-specific starting values, which is a practical advantage. The same holds for binomially distributed data. We further show that the unbinned maximum-likelihood method can be derived as a limiting case of the iterated least-squares fit when the bin width goes to zero.
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