<p>In certain situations, it is not feasible to maintain a constant sample size due to various reasons. Though there is extensive research on prediction problems involving a random sample size, the estimation problem has received significantly less attention from authors in this respect. This paper proposes a novel approach to address this gap. In addition, it seems that the application of the E-Bayesian procedure for inferential problems based on random sample sizes has remained unexplored in the literature. This paper aims to study the maximum likelihood, Bayesian, and E-Bayesian methods to estimate the parameter of a general family of distributions, namely the proportional hazard rate family of distributions, based on progressively Type II censored data with fixed and random sizes. For the Bayesian and E-Bayesian estimations, we employ the balanced squared error loss function. We derive the E-posterior risks of the E-Bayesian estimators under this loss function, considering three different priors for the hyperparameters. To assess our theoretical results, we provide a simulation study, along with graphical comparisons. We also present two real data applications for illustration. The simulation results reveal that the estimators with truncated Poisson distributed sizes may yield superior performance compared to those with fixed and truncated geometric distributed sizes.</p>