Estimation of parameters and quantiles of the Weibull distribution

We propose three new estimators of the Weibull distribution parameters which lead to three new plug-in estimators of quantiles. One of them is a modification of the maximum likelihood estimator and two of them are based on nonparametric estimators of the Gini coefficient. We also make some review of estimators of the Weibull distribution parameters and quantiles. We compare the small sample performance (in terms of bias and mean squared error) of the known and new estimators and extreme quantiles. Based on simulations, we obtain, among others, that the proposed modification of the maximum likelihood estimator of the shape parameter has a smaller bias and mean squared error than the maximum likelihood estimator, and is better or as good as known estimators when the sample size is not very small. Moreover, one of the proposed estimator, based on the nonparametric estimator of the Gini coefficient, leads to good extreme quantiles estimates (better than the maximum likelihood estimator) in the case of small sample sizes.