A new exponential family of distributions with applications to engineering and medical data

This research is centered on exploring a mathematically tractable and versatile family of probability distributions, specifically focusing on one family member. We have used the exponential distribution as a base distribution to create a novel distribution, which we have aptly named the "New Odd-type Exponential Distribution." In this paper, we provide an overview of the essential characteristics inherent to this innovative distribution. This model showcases a variety of hazard functions, including the reverse-J, decreasing, constant, increasing and constant, bathtub, and S-shaped shapes. The estimation of the distribution's parameters is conducted through both classical and Bayesian methods. We validate the accuracy of the classical estimation procedure through simulation studies. These simulations demonstrate a reduction in biases and mean square errors as sample sizes increase, even for smaller samples. To showcase the practicality of the proposed distribution, we apply it to two sets of real-world data, employing both classical and Bayesian approaches. We evaluate the performance of our suggested distribution model using various model selection criteria and goodness-of-fit test statistics. Empirical evidence from these evaluations confirms that our proposed model surpasses some existing models in the literature. Further, the suggested model was analyzed using the Bayesian approach and the Hamiltonian Monte Carlo method. Its predictive capability was also explored by using the posterior predictive checks, and it can predict the data consistently.