# Bootstrap for nonlinear regression and ranked explanatory variable

I've been wondering how the bootstrap method works in the case of nonlinear regression models where the order of the x-axis (suppose just 1 explanatory variable) matters.

In the case of univariate numerical data, as far as I know, the order of the bootstrapped samples is not important because, usually, a summary statistic is computed, but with bivariate data where one variable represents a rank, I'm not sure how it works or if it's correct to use bootstrap.

For example, consider the following model: $$\mathrm d(\log_{10}k)=\frac{1}{\sigma \sqrt{2\pi}}\int_{-\infty}^{\log_{10}k} e^{-\frac{(y-\mu)^2}{2\sigma^2}} \mathrm dy,\quad k \in\mathbb{N}$$

Here is a plot using real observed data: And using a bootstrapped sample: In both plots, the x-axis is $$\log_{10}k$$. I suppose that in the latter case, trying to fit the model wouldn't work.

Contrary to my intuition, when I use curve_fit from scipy in bootstrapped samples, I get similar estimated parameters to those I get with the original sample.

This plot shows instances of the model using the original estimated (the smooth almost indistinguishable red line) and bootstrapped (in blue) parameters: Is curve_fit implicitly reordering the x-axis before fitting? If that were true, wouldn't it affect the bootstrapped method?