标题： Maximum $\log_q$ Likelihood Estimation for Parameters of Weibull Distribution and Properties: Monte Carlo Simulation 显示英文标题

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

摘要： 最大${\log}_q$似然估计方法是已知的最大$\log$似然方法的推广，以克服对非同质观测值（内点和离群点）建模的问题。 参数$q$是一个调参常数，用于管理建模能力。 威布尔分布是一种在工程问题中灵活且流行的分布。 在本研究中，该方法用于在存在非同质观测值时估计威布尔分布的参数。 由于主要思想基于目标函数的建模能力$\rho(x;\boldsymbol{\theta})=\log_q\big[f(x;\boldsymbol{\theta})\big]$，我们观察到得分函数的有限性在内点的稳健估计中不能发挥作用。 检查了威布尔分布的性质。 在数值实验中，当对基础威布尔分布施加不同的污染设计时，通过$\log_q$和其特殊形式$\log$似然方法估计威布尔分布的参数。 优化是通过遗传算法进行的。 通过蒙特卡洛模拟观察了$\rho(x;\boldsymbol{\theta})$的建模能力和对非同质观测值的不敏感性。 $q$的值可以通过模拟中的均方误差进行选择，以及用于评估拟合能力的 Kolmogorov-Smirnov 检验统计量的$p$值。 因此，我们可以解决确定真实数据集的$q$值的问题。 显示英文摘要

摘要： 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.