Maximum $$\log _q$$ likelihood estimation for parameters of Weibull distribution and properties: Monte Carlo simulation

Abstract

The maximum log q likelihood estimation method is a generalization of the known maximum log likelihood method to overcome the problem for modeling non-identical observations (inliers and outliers). The parameter q is a tuning constant to manage the modeling capability. Weibull is a flexible and popular distribution for problems in engineering.In this study, this method is used to estimate the parameters of Weibull distribution when non-identical observations exist. Since the main idea is based on modeling capability of objective function ρ(x; θ) = log q f (x; θ) , we observe that the finiteness of score functions cannot play a role in the robust estimation for inliers. The properties of Weibull distribution are examined. In the numerical experiment, the parameters of Weibull distribution are estimated by log q and its special form, log, likelihood methods if the different designs of contamination into underlying Weibull distribution are applied. The optimization is performed via genetic algorithm. The modeling competence of ρ(x; θ) and insensitiveness to non-identical observations are observed by Monte Carlo simulation. The value of q can be chosen by use of the mean squared error in simulation and the p-value of Kolmogorov-Smirnov test statistic used for evaluation of fitting competence. Thus, we can overcome the problem about determining of the value of q for real data sets.