This paper addresses the estimation of the unknown parameters of the alpha power exponential distribution (Mahdavi and Kundu, 2017) using nine frequentist estimation methods. We discuss the finite sample properties of the parameter estimates of the alpha power exponential distribution via Monte Carlo simulations. The potentiality of the distribution is analyzed by means of two real data sets from the fields of engineering and medicine. Finally, we use the maximum likelihood method to derive the estimates of the distribution parameters under competing risks data and analyze one real data set.