Estimating the parameter of a length biased exponential distribution under generalized progressive hybrid censored sample

Length-biased distributions arise naturally in a wide range of biomedical applications, including studies of family disease history, survival and intermediate event analysis, and the investigation of AIDS latency periods following blood transfusion. In addition, biomedical experiments frequently involve situations in which experimental units are withdrawn or lost prior to observation, resulting in censored data. Motivated by these practical considerations, this paper investigates oth maximum likelihood and Bayesian estimation of the unknown parameter, reliability function, and hazard function of the length-biased exponential distribution (LBED) under a generalized progressive hybrid censoring (GPHC) scheme. Bayesian estimators for the unknown parameter, reliability, and hazard functions are developed under flexible loss functions. Since closed-form Bayesian solutions are not available, Lindley’s approximation and Markov chain Monte Carlo (MCMC) methods are employed to obtain point estimates, with the MCMC approach further used to construct credible intervals. The performance of the proposed estimators is evaluated through extensive Monte Carlo simulations.