Estimation of the scale parameter of the half-logistic distribution under progressively type II censored sample

Based on a progressively type II censored sample, the maximum likelihood and Bayes estimators of the scale parameter of the half-logistic distribution are derived. However, since the maximum likelihood estimator (MLE) and Bayes estimator do not exist in an explicit form for the scale parameter, we consider a simple method of deriving an explicit estimator by approximating the likelihood function and derive the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney and Kadane in J Am Stat Assoc 81:82-86, 1986) and importance sampling methods are used for obtaining the Bayes estimator. In order to compare the performance of the MLE, approximate MLE and Bayes estimates of the scale parameter, we use Monte Carlo simulation.