I'm working on fitting an exponential decay curve to a data set. While searching for techniques I found Wolfram's page describing how to easily accomplish it by taking the natural log of the predictor equation. http://mathworld.wolfram.com/LeastSquaresFittingExponential.html
I'm confused by equations (5) and beyond, as there is no justification to the claim that the prediction gives greater weight to small y values. In order to test the claim, I generated a sample data set (took an exponential decay function and added random noise) and compared the predictive model generated by the standard linear regression vs. the one suggested by equations (5) and beyond. I found that the sum of the residuals squared was significantly smaller (6 million vs. 11 million) for the predictive model generated by using the parameters calculated from (9) and (10).
Can someone give me a reason for this bias? Is multiplying by the response (y) just a convenient choice, or can it be shown that it is the optimal correction?
Also, if this is indeed a known bias, why doesn't software designed to generate exponential fits take it into account? I generated the fit using Oracle (REGR_SLOPE, REGR_INTERCEPT) and Excel (Trendline) and both gave me the same fit, which wasn't as good as the one I generated by equations (9) and (10) on Wolfram's site, based on my analysis of the residuals.