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arXiv · 2012.08294

Maximum $\log_q$ Likelihood Estimation for Parameters of Weibull Distribution and Properties: Monte Carlo Simulation

Mehmet Niyazi Çankaya·Roberto Vila
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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 $\rho(x;\boldsymbol{\theta})=\log_q\big[f(x;\boldsymbol{\theta})\big]$, 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 $\rho(x;\boldsymbol{\theta})$ 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.

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Mehmet Niyazi Çankaya, Roberto Vila. 2020-12-15. Maximum $\log_q$ Likelihood Estimation for Parameters of Weibull Distribution and Properties: Monte Carlo Simulation. https://arxiv.org/abs/2012.08294

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