Robust mean estimation and model selection between Weibull and Inverse Weibull distributions under misspecification

The Weibull and Inverse Weibull distributions are extensively used in reliability engineering, survival analysis, and related fields for modeling lifetime data. Selecting an incorrect model, such as fitting a Weibull distribution to data generated from an Inverse Weibull distribution, or vice versa, can introduce significant bias in parameter estimation, particularly for the mean lifetime, which is a critical reliability measure, and can thereby distort statistical inference. This paper presents an analytical investigation into the effect of such model misspecification on mean estimation. Explicit expressions for the Maximum Likelihood Estimators (MLEs) and Quasi-Maximum Likelihood Estimators (QMLEs) are derived under both correct and incorrect model assumptions, and the resulting errors are quantified using relative bias and relative mean squared error criteria. To enhance robustness against misspecification, a modern divergence-based approach using the Minimum Density Power Divergence Estimator (MDPDE) is incorporated into the analytical framework. Monte Carlo simulations and real data analyses demonstrate that the proposed divergence-based estimators substantially reduce bias and variability under model misspecification while maintaining competitive efficiency under correct specification. Based on these findings, a practical and robust decision framework is proposed for selecting between the Weibull and Inverse Weibull models. The results provide applied researchers with a principled methodology for mitigating the adverse effects of distributional misspecification in lifetime data analysis.