Parameter estimation in extreme-value distributions using the continuation method

An efficient and stable maximum likelihood parameter estimation scheme for the three kinds of extreme-value distributions (the Weibull, the Gumbel and the Fréchet) using the generalized extreme-value distribution and the continuation method is introduced. As the proposed algorithm can almost always obtain the maximum likelihood estimates automatically, it is of considerable practical value. This paper focuses on the Weibull distribution parameter estimation and shows that using the generalized extreme-value distribution is better than the Weibull distribution itself and that the continuation method is more efficient than the grid search method in searching for parameters globally. The paper also shows that when there are no finite local maximum likelihood estimates in the Weibull distribution, it is probable that there are finite local maximum likelihood estimates in the Fréchet distribution; and vice versa. Only complete data sets are considered in this paper, but the algorithm can easily be applied to censored data.