Statistical Inference for the Entropy of the Transmuted Weibull Distribution Under Progressive Type-II Censored Samples

This paper investigates statistical inference for the Shannon entropy of the Transmuted Weibull Distribution under progressively Type-II censored samples. The Transmuted Weibull Distribution is obtained by applying the quadratic rank transmutation map to the cumulative distribution function of the two-parameter Weibull distribution, thereby substantially enhancing its modeling flexibility while preserving the analytical tractability of the baseline distribution. Consequently, it provides greater flexibility for modeling lifetime data exhibiting pronounced skewness and complex hazard rate behaviors. First, a closed-form expression for the Shannon entropy of the Transmuted Weibull Distribution is derived. From a frequentist perspective, the maximum likelihood estimators of the model parameters are obtained numerically using the Newton–Raphson algorithm, and the corresponding maximum likelihood estimator of Shannon entropy is derived through the invariance property of maximum likelihood estimation. To quantify estimation uncertainty, asymptotic confidence intervals are constructed using the Delta method together with the observed Fisher information matrix, while Bootstrap confidence intervals are also developed to improve finite-sample inference. From a Bayesian perspective, posterior inference is conducted using a hybrid Gibbs sampling algorithm within the Markov chain Monte Carlo framework. Bayesian point estimators of Shannon entropy are obtained under the squared error loss function, the absolute error loss function, and the 0–1 loss function, corresponding to the posterior mean, posterior median, and posterior mode, respectively. In addition, highest posterior density credible intervals are constructed for the Shannon entropy. The proposed methods are evaluated through an extensive Monte Carlo simulation study under three representative progressively Type-II censoring schemes. Estimation performance is assessed in terms of bias, mean squared error, interval coverage probability, and average interval length. The simulation results demonstrate that the Bayesian estimators consistently outperform the maximum likelihood estimator, particularly for small sample sizes and heavy censoring, while the highest posterior density credible intervals achieve more accurate coverage probabilities and shorter interval lengths. Finally, the proposed inferential procedures are illustrated using a real dataset consisting of remission times from 128 bladder cancer patients, demonstrating their practical applicability and robustness.