# Fit a function whose asymptotics is known

Consider the experimental blue curve in the figure that follows

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?

• Have you tried least squares? Oct 8, 2020 at 16:58
• 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?
– Eli
Oct 8, 2020 at 18:02
• @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 Oct 8, 2020 at 19:44
• What is your research question? Oct 8, 2020 at 19:46
• 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.
– whuber
Oct 8, 2020 at 20:02

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