Statistical Inference of Weighted Extropy Under Outlier Influence: An MCMC Approach with Data Applications

Extropy has recently become a focal point of study as a measure of uncertainty in probability distributions and serves as the dual complement to entropy. This paper suggests estimating extropy and weighted extropy for the power function distribution in the presence of outliers. Both Bayesian and conventional estimating methods are recommended. The Bayesian estimators for the extropy measures are produced for both symmetric and asymmetric loss functions using partially informative and non-informative priors. Bayesian estimates are calculated using a method called the Gibbs sampler, which is part of the Markov chain Monte Carlo approach. Extensive simulations evaluated with certain precision metrics display the results of empirical Bayesian and non-Bayesian extropy estimates when k outliers are presented. Under a symmetric loss function, Bayesian estimates of both extropy measures performed better than those under the linear exponential and minimal expected loss functions in the majority of simulation study situations. It can be concluded that Bayesian estimates based on the minimum expected loss function performed the poorest in both homogeneous (non-outlier) and outlier situations. According to the simulation research, larger sample sizes led to appreciable improvements in key accuracy metrics for all extropy estimates. Across all outlier and homogeneous case scenarios, the accuracy measures of the Bayesian estimates have the lowest values in the case of a partially informative prior compared to the others in the non-informative prior case. The effectiveness of the recommended methods is shown by applications to real datasets, such as failure times of air conditioning systems and lifetimes of electronic tubes. The aforementioned examples demonstrate the versatility and use of extropy and weighted extropy measures in simulating uncertainty and reliability in the field of reliability.