It's well known in FHI and similar circles, that it's impossible to distinguish an exponential (growth going up wildly) from a sigmoid/logistic curve (exponential growth until a turning point - an S shape) - until well after the turning point.
Which means we can't effectively predict that turning point. And so can't distinguish when a sigmoid will have a turning point, even when we know it must have one.
But this doesn't seem to exist in the statistics literature; and it would be very useful to have such a paper or textbook to point to.
We don't have time to write a full paper ourselves, but is there someone on this list with statistical experience who would like to write or co-write such a paper?
Since this result is important and as yet unpublished, it's plausible that such a publication may get an extremely high number of citations.
Cheers!
You may want to look into stiff equations. The logistic equation y'=λy(1-y) is increasingly stiff with increase in λ, meaning, in particular, that the resulting curve depends very sensitively on small relative changes in λ when λ is large. That would give you the exact effect you observe, inability to predict the inflection point from noisy data.
From a numerical methods textbook:
Awesome find! I really like the paper.
I had been looking at Fisher information myself during the weekend, noting that it might be a way of estimating uncertainty in the estimation using the Cramer-Rao bound (but quickly finding that the algebra got the better of me; it *might* be analytically solvable, but messy work).