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When doing a weighted least squares fit of a model to data, I want to examine the residuals to see if their histogram matches the expected probability distribution. Since I am minimizing

[tex]

\chi^2 = \sum_i w_i \left[ y_i - Y(x_i) \right]^2

[/tex]

would I define my (normalized/studentized) residuals as

[tex]

r_i = \frac{\sqrt{w_i} ( y_i - Y(x_i) )}{\sigma \sqrt{1 - h_i}}

[/tex]

or in the usual way as

[tex]

r_i = \frac{y_i - Y(x_i)}{\sigma \sqrt{1 - h_i}}

[/tex]

where [itex]h_i[/itex] are the statistical leverages and [itex]\sigma[/itex] is the residual standard deviation. I am using a robust (iterative) least squares procedure to determine the weights [itex]w_i[/itex] in order to detect and eliminate outliers, so the residual histogram should match the expected distribution of the M-estimator function I'm using (Huber in my case).

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# Weighted least squares residuals

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