Robust Parameter Estimation for Dual-population Weibull Models under Fuzzy Data Conditions

Estimating parameters accurately in the presence of uncertain and imprecise data is a key challenge in statistical analysis, particularly for complex models involving two populations. Fuzzy data provides a structured way to handle such uncertainties by effectively representing real-world ambiguity. While extensive research has been conducted on parameter estimation for single-population models using fuzzy data, extending these methods to dual populations remains a difficult task. This study addresses the issue by developing estimation techniques for two Weibull distributions that share a common scale parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>σ</mml:mi> </mml:math> but have different shape parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , under fuzzy data conditions. We apply the expectation-maximization (EM) algorithm for Maximum Likelihood estimation and utilize a Bayesian approach with TK approximation for parameter estimation. To further refine Bayesian estimates, Gibbs sampling is employed to derive posterior distributions. Through Monte Carlo simulations on both simulated and real-world datasets, we evaluate the accuracy and robustness of our estimators, demonstrating their effectiveness in handling imprecise data. Additionally, asymptotic and HPD confidence intervals are also obtained. This research highlights the importance of reliable statistical methods for dual-population Weibull models, contributing to improved analytical precision across various domains.