Maximum likelihood is certainly better than the methods of moments.

The following reference presents the methods the well adapted (in particular SEM Stochastic Estimation Maximisation  and BRM Bayesian Restoration Maximisation) for estimating the parameters of a Weibull law, taking into account a small sample of failures (from 1 to 20 and beyond)  in very high censored context. Examples are presented.

Bacha M., Celeux G., Idée E., Lannoy A., Vasseur D. (1998), Estimation de modèles de durées de vie fortement censurées, Collection de la Direction des Etudes et Recherches d'Electricité de France, Editions Eyrolles, Paris, 99, 1998.

I recommend Linear Regression with plotting position formula adjusted for sample size to give unbiased estimates. Please see the attached papers.

Good luck!

Article Unbiased estimates of the Weibull parameters by the linear r...

Article Guidelines for Two-Parameter Weibull Analysis for Flaw-Conta...

Murat Tyryakioglu,

I am developing a paper that compares the performance of the estimation methods. his paper was very good "Unbiased estimates of the Weibull parameters by the linear regression method." I will cite his work in my paper.

Thank you very much

Best regards.

Andre Lannoy,

any public text?

Thx