@Glen_b OK, let me simplify it like this:

Suppose there's a system (in physics, if you like) that follows the law y = log x - a x or similar. I can derive the -a x part with some approximation for large x, but I don't have the log x part. I'm using log x only as an example here, but this could be something completely different and much more complicated. The point is that for large x the linear law holds, according to theory.

Now I have some experimental data, which might look like this:

I actually generated this from x - \ln a, with a little bit of added noise.

Looking at the figure it seems that the end of the curve is linear. Theory confirms that it should be linear. Now I need to find the slope of the line, a.

Are there principled methods that help me decide which part of the data to throw away before fitting a line on it?

@Glen_b If I put it this way, is it specific enough? This is now restricted for linear data.

It's not my actual problem, but I don't want to bring up the details of that (it would be a distraction), and I believe that this one is close enough to it.