Why fitting a logistic is nearly impossible from early data

This explains why estimating the asymptote of a logistic curve from early (one-sided) data is effectively impossible: when observations lie on one tail of the S‑curve the logistic is nearly exponential there, and small measurement errors produce huge changes in the fitted limiting value. For three equally spaced times there is a closed-form expression for the logistic limit computed from the three observed y's, and taking partial derivatives shows that all sensitivities share the same denominator, y1^2 − y0·y2. That quantity is exactly zero for a pure exponential and only small when the points are in a tail, so the derivatives (and hence the estimate of the limit) blow up.

A concrete numeric example makes the effect stark: with true parameters set so the limit equals 1 and observations at t = −2, −1.5, −1 the three y values give a perfect fit L = 1, but an extremely tiny change in the earliest y produces an infinite estimated limit; a small interval of perturbations of that point yields predicted limits ranging from 1 to infinity. The conclusion is practical: to predict the logistic asymptote reliably you need data that span both sides of the inflection point, not just the early exponential‑like growth.