The numpy library will return the Theil-Sen slope and intercept from a set of data. It also provides a confidence interval on the slope. Is it possible to use this confidence interval along with the original data to produce a regression confidence interval similar to those in Simple Linear Regression:
$$ CI = t*\sigma*\sqrt{\frac{1}{n}+\frac{\left(x-x̄\right)^2}{SSx}} $$
Is it still valid to use the Theil-Sen estimator to calculate a regression variance and mean-square-residual, to then calculate the confidence interval?
scipy.stats.mstats.theilslopes
has "median(y) - medslope*median(x)", it's not the only possibility - if that's what you want you'd need to explore the properties of that intercept and the slope together. Figuring out a nonparametric interval for the intercept will be tricky, let alone a simultaneous one for both of them. Bootstrapping would be a possibility, if used in large samples with a suitable choice of procedure $\endgroup$