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Consider the experimental blue curve in the figure that followsenter image description here

It turns out that I know the expression for the tails (-they are power laws- red and green curves) and I want to fit the curve using this information. What is the best way to proceed?

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    $\begingroup$ Have you tried least squares? $\endgroup$ Oct 8, 2020 at 16:58
  • $\begingroup$ Can you provide more information? It's not clear what you want to do. Do you want to describe your data with p1 for negative values and p2 for positive values? $\endgroup$
    – Eli
    Oct 8, 2020 at 18:02
  • $\begingroup$ @Eli, I realized that I had a misspelling in the most important part. I know that the tails are power laws and I want to fit the whole blue curve. The problem is that I don't know what model to use in order to fit it by using, for instance, least squares. Also I edited the figure to make it more clear $\endgroup$
    – dapias
    Oct 8, 2020 at 19:44
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    $\begingroup$ What is your research question? $\endgroup$ Oct 8, 2020 at 19:46
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    $\begingroup$ With so much data it isn't going to matter much how you fit the curves if all you want is a reasonable graphical approximation. Just plot the two tails separately on log-log axes against $|\omega|$ and estimate the slopes. But if the fit has scientific meaning or you need to provide a standard error, then it is crucial to supply more information about the nature of the random-looking variation around the expected values. $\endgroup$
    – whuber
    Oct 8, 2020 at 20:02

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A rather good fit is obtained with the function below. The numerical values of parameters are shown on the figure where the red curve represents the fitted function.

enter image description here

This leads to asymptotics : $\quad\sim(-\omega)^{-0.633}$ at $\omega \to -\infty\quad$ and $\quad\sim(\omega)^{-0.746}$ at $\omega \to +\infty$

The above values of parameters are not accurate because the data comes from scanning the figure published in the question. This is not a good way to get the data.

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  • $\begingroup$ An analysis of the residuals will reveal potentially interesting failures in the fit, especially in the left tail. This is no surprise, because your functions do not have the asymptotic behavior specified in the question. $\endgroup$
    – whuber
    Oct 22, 2020 at 13:21
  • $\begingroup$ @wuber. The blue curve given in the question doesn't have the asymptotic behaviour specified in the question. $\endgroup$
    – JJacquelin
    Oct 28, 2020 at 6:42

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