I have a situation where I have two samples : Sample A and Sample B of pairs of data $(x,y)$ that I do now know if they belong to the same population. I perform a regression on Sample A and another regression on Sample B, such that $\hat{y}=f_1(x;\theta_1)$ and $\hat{y}=f_2(x;\theta_2)$. After performing regression, I inspect their fits and both curves look quite similar given that $f$ is parametrized by $\theta$. In other words, $\theta_1\sim\theta_2$, or $MSE(\theta_1,\theta_2)<\epsilon$, or any distance metric between $\theta_1$ and $\theta_2$ is `small'.
However: How do I statistically prove that both samples belong to the same population, and that the curves are the same, and that the difference in curve fitting are a product of sampling error and noise? Is there a way to provide confidence intervals or bounds to my assertion? Is this related to gaussian processes? Can I do a multiple rounds of t-testing on each of the parameters in $\theta$ after bootstrapping data points across both samples to get an equivalent of error bars?
Similar questions, but applied to time-series (so perhaps it is still applicable to my problem), although my problem is not a time-series one. In addition, the samples I am collecting data from are not fully overlapping in input $x$: i.e. suppose sample A is sampled from $[0,x^*]$, and sample B is sampled from $[x^*, x']$, s.t. $0\leq x^* \leq x'$:
