Revisiting Maximum Log-Likelihood Parameter Estimation for Two-Parameter Weibull Distributions: Theory and Applications

Abstract

In this article, we reexamine properties of maximum log-likelihood parameter estimation for two-parameter Weibull distributions which have been applied in many different sciences. Finding reasons for this popularity is a key question. Our main contribution is a thorough existence and uniqueness proof for a global maximizer with respect to the parameter space. We first provide existence and uniqueness of local maximizers by Schauder’s first fixed point theorem, monotony arguments and local concavity of its Hessian matrix. Thus, we can prove our main result of existence and uniqueness of a global maximizer by considering all limiting cases with respect to the parameter space. We finally strengthen our theoretical findings on four data sets. On the one hand, two synthetic data sets underline our need for our data assumptions while, on the other hand, we choose two data sets from wind engineering and reliability engineering to demonstrate usefulness in real-world applications.