This thesis examines different classical and Bayesian methods for estimating parameters of the two-parameter exponential distribution, a flexible and valuable model for analyzing randomly censored data in survival and reliability studies. Unlike the standard one-parameter exponential model, which can produce biased results when data have a significant minimum threshold, the two-parameter model includes a location parameter representing a minimum guarantee period, alongside the usual scale parameter. This is particularly relevant in random censoring scenarios, where subjects or items exit studies unpredictably. For computational simplicity, we assume that both failure and censoring times follow a two-parameter exponential distribution with a shared location parameter but different scale parameters, balancing mathematical convenience with realistic assumptions. We first explore established estimation techniques to estimate the unknown parameters, including the maximum likelihood, method of moments, L-moments, least squares, weighted least squares, and Bayesian estimators using the generalized entropy loss function with flexible priors that vary in informativeness. Additionally, we propose new estimators, such as the percentile-based method and its modified version, maximum product of spacings, and Bayesian estimators under the linear-exponential loss function, using the same priors. Through extensive Monte Carlo simulations, we evaluate these estimators’ performance and some key reliability metrics to assess the system performance. Finally, we apply the two-parameter model to real-world datasets from medical science and reliability engineering and compare the results with the standard one-parameter model to highlight the value of incorporating a location parameter for accurate estimation in randomly censored data studies.