[indeed30]

I don’t think you can do anything sensible here without making much stronger modelling assumptions. A vanilla non-parametric bootstrap is only valid under a very specific generative story: IID sampling from a population. Many (most?) curve-fitting problems won't satisfy that.

For example, suppose you measure the decay of a radioactive source at fixed times t = 0,1,2,... and fit y = A e^{-kt}. The only randomness is small measurement error with, say, SD = 0.5. The bootstrap sees the huge spread in the y-values that comes from the deterministic decay curve itself, not from noise. It interprets that structural variation as sampling variability and you end up with absurdly wide bootstrap confidence intervals that have nothing to do with the actual uncertainty in the experiment.

[cb321]

These are all big topics, but any "parametric curve fitting" like this tool uses is parameter estimation (the parameters of the various curves). That already makes strong modeling assumptions (usually including IID, Gaussian noise, etc.,) to get the parameter estimates in the first place. I agree it would be even better to have ways to input measurement errors (in both x- & y- !) per your example and have non-bootstrap options (I only said "probably"), residual diagnostics, etc.

Maybe a residuals plot and IID tests of residuals (i.e. tests of some of the strong assumptions!) would be a better next step for the author than error estimates, but I stand by my original feedback. Right now even the simplest case of a straight line fit is reported with only exact slope & intercept (well, not exact, but to an almost surely meaningless 16 decimals!), though I guess he thought to truncate the goodness of fit measures at ~4 digits.

[indeed30]

I think we are just coming at this from different angles. I do understand and agree that we are estimating the parameters of the fit curves.

> That already makes strong modeling assumptions (usually including IID, Gaussian noise, etc.,) to get the parameter estimates in the first place

You lose me here - I don't agree with "usually". I guess you're thinking of examples where you are sampling from a population and estimating features of that population. There's nothing wrong with that, but that is a much smaller domain than curve fitting in general.

If you give me a set of x and y, I can fit a parametric curve that tries to minimises the average squared distance between fit and observed values of y without making any assumptions whatsoever. This is a purely mechanical, non-stochastic procedure.

For example, if you give me the points {(0,0), (1,1), (2,4), (3,9)} and the curve y = a x^b, then I'm going to fit a=1, b=2, and I certainly don't need to assume anything about the data generating process to do so. However there is no concept of a confidence interval in this example - the estimates are the estimates, the residual error is 0, and that is pretty much all that can be said.

If you go further and tell me that each of these pairs (x,y) is randomly sampled, or maybe the x is fixed and the y is sampled, then I can do more. But that is often not the case.

[abdullahkhalids]

What methods can you use the estimate the standard error in this case?

[indeed30]

The radioactive decay example specifically? Fit A and k (e.g. by nonlinear least squares) and then use the Jacobian to obtain the approximate covariance matrix. The diagnonal elements of that matrix give you the standard error estimates.