The estimation of the Weibull parameters seems to be a serious problem in certain situations. Maybe this article is a good starting point:

Boudt, K., Caliskan, D., Croux, C. Robust and explicit estimators of Weibull parameters. Metrika, 2009; 73(2):187-209. doi:10.1007/s00184-009-0272-1

I also know a Diploma thesis from 2011 on the problem. But this may be of limited help because it is written in German.