Accelerated life tests have become increasingly important because of higher customer expectations for better reliability, more complicated products with more components, rapidly changing technologies advances, and the clear need for rapid product development. Hence, accelerated life tests have been widely used in manufacturing industries, particularly to obtain timely information on the reliability. Maximum likelihood estimation is the starting point when it comes to estimating the parameters of the model. In this paper, besides the method of maximum likelihood, nine other frequentist estimation methods are proposed to obtain the estimates of the exponentiated exponential distribution parameters under constant stress accelerated life testing. We consider two parametric bootstrap confidence intervals based on different methods of estimation. Furthermore, we use the different estimates to predict the shape parameter and the reliability function of the distribution under the usual conditions. The performance of the ten proposed estimation methods is evaluated via an extensive simulation study. As an empirical illustration, the proposed estimation methods are applied to an accelerated life test data set.