We propose a novel lifetime model by extending the new exponential-gamma distribution to the exponentiated new exponential-gamma distribution. This extension allows for the derivation of a more flexible density function that combines the characteristics of the exponential and gamma distributions. We present various statistical properties of the newly proposed method, including the cumulative function, probability density function, moment-generating function, and moments. Additionally, we discuss the estimation of parameters using maximum likelihood. To compare the performance of our newly developed model with existing probability distributions (gamma, exponential, Lindley, generalized gamma, generalization of the generalized gamma, and new exponential-gamma distribution), we employ model selection criteria such as the Akaike Information Criterion (AIC), the corrected Akaike Information Criterion (AICC), and the Bayesian Information Criterion (BIC). The application of these criteria to different models demonstrates that our proposed model outperforms the other six models across various datasets. For instance, in the first dataset, the AIC, AICC, and BIC values for our model are 366.975, 373.805, and 373.805, respectively, whereas the values for the other six models (exponential, Lindley, generalized gamma, generalization of the generalized gamma) range from 503.012 to 834.327. We conduct simulation studies to assess the efficiency of our proposed model. Furthermore, we apply the proposed method to three real data applications to further examine its effectiveness. It is important to note that the quantile function of the proposed model does not have a closed-form solution, requiring the computation of the quantile function through the Newton-Raphson iterative approach.
