Development of New Robust Optimal Score Function for the Weibull Distributed Error Term in Multilevel Models

A popular robust estimation technique for linear models is the rank-based method as an alternative to the ordinary least square (OLS) and restricted maximum likelihood (REML) in the presence of extreme observations. This method is applied in machine reliability analysis and quantum engineering, especially in artificial intelligence and optimization problems where outliers are commonly observed. This technique is also extended for the multilevel model, where the shape of error distribution contributes a significant role in more efficient estimation. In this study, we proposed the Weibull score function for the Weibull distributed error terms in the multilevel model. The efficiency of the proposed score function is compared with the existing Wilcoxon score function and the traditional method REML via Monte Carlo simulations after adding simulated extreme observations. For small values of shape parameter in Weibull distribution of error term showing the presence of outliers, the Weibull score function was found to be efficient as compared to the Wilcoxon and REML methods. However, for a large value of shape parameter, Wilcoxon score appeared either equally efficient than the Weibull score function. REML is observed least precise in all situations. These findings are verified through a real application on test scores data, with a small value of shape parameter, and the Weibull score function turned out the most efficient.