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Say I have some data, I assume a fit of the form: $\alpha_1 e^{\beta_1 k}$, and I extract some values $\alpha_1>0$ and $\beta_1<0$ for $k=[1,\ldots]$. Say this appears to be a "near perfect" fit for my data, i.e. my data points fall within some $\epsilon$ of $f = \alpha_1 e^{\beta_1 k}$ .

Now, say later I learn that the "true" form of the fit should be: $k \alpha e^{\beta k}$. No longer having access to my data, can I say something about $\alpha_2$ and $\beta_2$ (besides the fact that $|\beta_2|>|\beta_1|$)? I am specifically interested in being able to say something about the ratio: $\frac{\beta_1}{\beta_2}$

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I would create some artificial data resembling the original data as much as possible (whatever I can remember) and then just run two fits and compare the coefficients. If you say that you had a near perfect fit, then an artificial signal will be easy to create (exponential plus random noise?). Looking at the confidence intervals of the coefficients from both fits might also help. Intuitively, the two fitting functions are quite different so if you had a good fit with one I wouldn't expect a good fit with the other. – Fixed Point Jan 5 '13 at 7:11
@FixedPoint Yup, I've certainly tried that. But I'm curious how one would proceed without doing that "experiment"? – user55257 Jan 5 '13 at 8:57
Time to dust off some old books. Just out of curiosity, do you remember the correlation coefficient from the first fit, just to get an idea of how good the fit actually was? And also, when you tried the numerical experiment, what was the result? – Fixed Point Jan 5 '13 at 9:01
@FixedPoint I didn't actually perform the original fit, and the correlation coefficient isn't provided. This is from an old whitepaper in my lab group. – user55257 Jan 5 '13 at 9:18

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