A novel odd Type-X family of distributions: Model, theory, and applications to medical, insurance, and engineering data sets

The present work investigates a mathematically manageable and versatile family of probability distributions known as the novel odd type-X family of distributions. We utilized the classical Weibull distribution as a base distribution for developing a new heavy-tailed distribution called the novel odd-type Weibull distribution. The hazard function of the novel odd type-Weibull distribution is visually illustrated, showing different behaviors, including reversed-j shape, j-shape, increasing, unimodal, decreasing, and bathtub-shaped curves. We analyzed various key characteristics of the newly derived work, including the quantile function, moments, moment generating function, characteristics function, incomplete moments, mean residual life, and other statistical properties. The maximum likelihood estimation method is used to estimate the distribution parameters, and a Monte Carlo simulation analysis is performed to validate the accuracy of this estimation approach. The numerical and graphical results of the simulation process revealed that as the sample size increases, the biases and mean square errors decrease, even if the samples are small. Additionally, we explore actuarial metrics such as value-at-risk, tailed value-at-risk, tailed-variance, and tailed-variance-premium to assess the tail behavior of the proposed distribution. Furthermore, three sets of real data from diverse fields are analyzed to validate the practical performance (to check the superiority and versatility) of the newly derived distribution. The practical performance of the proposed distribution is evaluated by using model selection criteria, goodness-of-fit measures, and graphical illustrations. The comparison of the proposed distribution is made with several existing distributions, including the Weibull distribution, alpha power transformed Weibull distribution, new exponent Weibull distribution, exponentiated Weibull distribution, flexible-reduced-logarithmic Weibull distribution, Marshal Olkin Weibull distribution, and Kumaraswamy Weibull distribution.