Estimation for two Gompertz populations under a balanced joint progressive Type-II censoring scheme

Comparative lifetime experiments are remarkable when the study is to ascertain the relative merits of two competing products regarding the duration of their service life. This paper considers the comparative lifetime experiments of two Gompertz populations under a balanced joint progressive Type-II censoring scheme. The lifetime distributions of the units are assumed to follow the Gompertz distribution with a common shape but different scale parameters. The maximum likelihood estimates of the unknown parameters are derived. The existence of the maximum likelihood estimates is proved. Expectation-maximization and stochastic expectation-maximization algorithms are provided to calculate the estimates. The bootstrap-p, bootstrap-t, and approximate confidence intervals are established. To obtain the Bayesian estimates, it is assumed that the prior of scale parameters is a Beta-Gamma distribution and the prior of the common shape parameter is an independent Gamma distribution. Under squared error loss and LINEX loss functions, the Metropolis-Hastings algorithm is provided to compute the Bayes estimates and the credible intervals. Further, the statistical inferences with order restriction are studied when it is known a priori that the expectation of the lifespan of one population is shorter than that of the other population. A wide range of simulation experiments is conducted to evaluate the performance of the proposed methods. Finally, the lifetimes of white organic light-emitting diodes and the breaking strengths of jute fiber of gauge lengths are analyzed to illustrate the practical application of the proposed model and methods.