[cracker_jacks]

> the length of that interval is bound by the ratio of the error in the data over the derivative with respect the parameter

This is interesting! Could you expand on this a bit? Why is the length of the interval bound by the ratio of the data error over the derivative?

[surroundingbox]

The general case require some work and conditions. But to give a hint, the case of only one parameter is an application of the mean value theorem (1). Suppose a model (y = f(p,x) ) with only one parameter p0 and an exact point (x0,y0) (that is y0=f(p0,x)) and a data point (x0,y1) such that y1-y0=error in the data. And that there is a value p1 of the parameter such that f(p1,x0) = y1, then y1 - y0 = f(p1,x0) - f(p0,x0) = f'(sigma) . (p1-p0), so that p1-p0 = (y1-y0)/f'(sigma) that is (error in the parameter) = (error in the data)/(derivative with respect to the parameter) where sigma is between p0 and p1. The general case is a generalization of this idea using the mean value inequality.

(1) https://en.wikipedia.org/wiki/Mean_value_theorem