In this article, we make mathematical and practical contributions to the Bell-X family of absolutely continuous distributions. As a main member of this family, a special distribution extending the modeling perspectives of the famous Burr XII (BXII) distribution is discussed in detail. It is called the Bell-Burr XII (BBXII) distribution. It stands apart from the other extended BXII distributions because of its flexibility in terms of functional shapes. On the theoretical side, a linear representation of the probability density function and the ordinary and incomplete moments are among the key properties studied in depth. Some commonly used entropy measures, namely Rényi, Havrda and Charvat, Arimoto, and Tsallis entropy, are derived. On the practical (inferential) side, the associated parameters are estimated using seven different frequentist estimation methods, namely the methods of maximum likelihood estimation, percentile estimation, least squares estimation, weighted least squares estimation, Cramér von-Mises estimation, Anderson-Darling estimation, and right-tail Anderson-Darling estimation. A simulation study utilizing all these methods is offered to highlight their effectiveness. Subsequently, the BBXII model is successfully used in comparisons with other comparable models to analyze data on patients with acute bone cancer and arthritis pain. A group acceptance sampling plan for truncated life tests is also proposed when an item's lifetime follows a BBXII distribution. Convincing results are obtained.

In this paper, we propose a novel family of distributions called the odd Muth-G distributions by using Transformed-Transformer methodology and study their essential properties. The distinctive feature of the proposed family is that it can provide numerous special models with significant applications in reliability analysis. The density of the new model is expressible in terms of linear combinations of generalized exponentials, a useful feature to extract most properties of the proposed family. Some of the structural properties are derived in the form of explicit expressions such as quantile function, moments, probability weighted moments and entropy. The model parameters are estimated following the method of maximum likelihood principle. Weibull is selected as a baseline to propose an odd Muth-Weibull distribution with some useful properties. In order to confirm that our results converge with minimized mean squared error and biases, a simulation study has been performed. Additionally, a plan acceptance sampling design is proposed in which the lifetime of an item follows an odd Muth-Weibull model by taking median lifetime as a quality parameter. Two real-life data applications are presented to establish practical usefulness of the proposed model with conclusive evidence that the model has enough flexibility to fit a wide panel of lifetime data sets.