In stata, robust regression (rreg) uses weights proportional to the size of the residuals. Is this conceptually the same as weighted OLS (weight by 1/variance)? And both can be applied, for example, in scenarios with heteroskedasticity?
They are not the same, because in weighted OLS, weights are given apriori, they are not a function of size of residuals, as in some types of robust regression. However, there is a connecteion, as robust regression sometimes can be calculated via IRLS, iteratively reweighted least squares. For the last, see can you give a simple intuitive explanation of IRLS method to find the MLE of a GLM?