I was wondering why maximum likelihood estimation is not mentioned much with regard to quantile matching estimation even though it's possible since you can estimate quantile sample variance using the formula given above p(1-p) / (n f(ξp)2) and figure it out from there assuming you also know the sample size. Then I did some testing and came to the conclusion that weighted least square is just as, if not more, good as maximum likelihood estimation in figuring out the true distribution parameter, perhaps due to some precision loss and numerical instability associated with this specific form of MLE (this was done using Python)