The moment exponential distribution has recently been generalized by a number of authors. The two‐parameter alpha power‐transformed moment exponential (APTME) distribution is introduced. In terms of fit, the APTME distribution outperforms the moment exponential distribution. Exact expressions for ordinary moments, incomplete and conditional moments, the moment generating function, the cumulant generating function, and information measures are obtained for some APTME distribution features. To estimate the model parameters, six well‐known frequentist techniques were applied. The behavior of the various estimators was investigated using a simulated exercise. To examine the practical significance of the APTME distribution, real‐world datasets were used. In terms of performance, we show that the APTME distribution beats other models.