Bayesian and Maximum Likelihood Estimation of the Scale Parameter of a New Weighted Weibull Distribution

Abstract

The newly proposed weighted Weibull distribution is a three-parameter lifetime model characterised by a high degree of flexibility for modelling real-world data. It comprises one scale parameter and two shape parameters, which together govern its adaptability and shape characteristics. Despite the established importance of parameter estimation in model fitting and practical applications, no consensus has yet been reached regarding a universally superior estimation approach for the parameters of this distribution. Accordingly, the present study develops Bayesian estimators for the scale parameter of the weighted Weibull distribution using two non-informative priors (Uniform and Jeffreys) and one informative prior (Gamma). The estimation is carried out under three different loss functions, namely the squared error loss function (SELF), quadratic loss function (QLF), and precautionary loss function (PLF). The resulting Bayesian estimates are compared with the maximum likelihood estimation (MLE) approach through Monte Carlo simulation studies. The mean squared error (MSE) is employed as the primary criterion for evaluating and comparing estimator efficiency. The findings indicate that estimators derived under the quadratic loss function consistently exhibit the lowest MSE values across all prior distributions considered. In particular, the Bayesian estimator based on the Gamma prior combined with the quadratic loss function demonstrates superior performance compared to both the maximum likelihood estimator and Bayesian estimators obtained under SELF and PLF with Uniform and Jeffreys priors. Furthermore, variations in the shape parameters are found to have no substantial effect on the performance of the scale parameter estimators. The study concludes by recommending that future research should extend the analysis to the estimation of the shape parameters, which are critical for broader applications of the weighted Weibull distribution.