Likelihood and Bayesian Inference for Exponentiated Lomax Stress-Strength Reliability Under Progressive Type-II Censoring with Binomial Removals

Abstract

This paper develops frequentist and Bayesian inference for the stress–strength (SS) reliability δ = P(W < V), where the strength V and the stress W are independent Exponentiated Lomax (EL) random variables with common shape and scale parameters (β, θ) but distinct exponentiation parameters α₁ and α₂. The lifetime data are progressively Type-II censored (PT-IIC) with binomial removals. It is shown that δ = α₁/(α₁ + α₂). Maximum likelihood estimators (MLEs) of the model parameters and of δ are derived from the analytic score equations, closed-form estimators of the removal probabilities are obtained, and asymptotic confidence intervals are constructed by the delta method. Bayes estimators of δ are developed under the squared error and linear-exponential (LINEX) loss functions, using informative and non-informative gamma priors and a Metropolis–Hastings sampler. In a Monte Carlo study of 128 scenarios with 300 replications each, the root mean squared errors (RMSEs) of all estimators decrease as the sample size n and the number of observed failures m increase. Bayes estimators with informative priors elicited from pilot samples attain the smallest RMSE in all 128 scenarios (average 0.040 versus 0.052 for the MLE), whereas Bayes estimators with non-informative priors perform similarly to the MLE (average RMSE 0.051). Because the informative priors are built from samples generated under the true parameter values, the resulting advantage is an optimistic upper bound on the benefit of reliable prior information. The methods are illustrated with two classical data sets of insulating-fluid breakdown times at 34 kV and 36 kV. From PT-IIC samples with m = 10 per data set, the MLE of δ is 0.644 (95% asymptotic interval 0.462 to 0.792) and the Bayes estimate under squared error loss is 0.632.