The Baker Type-I Model: Theory, Comprehensive Inference, and Empirical Evidence from Complex Reliability and Biomedical Data

Recently, two novel extensions of the Weibull distribution have been introduced through Manly’s exponential transformation, offering a flexible mechanism for modeling skewness, tail behavior, and complex hazard rate structures. In this study, we develop a comprehensive theoretical and inferential framework for one of these models, referred to as the Baker–T1 distribution, to establish it as a mature and practically viable lifetime model for reliability and survival analysis. While the Baker–T1 model exhibits remarkable flexibility in capturing skewness, tail behavior, and complex hazard rate shapes, its statistical properties and practical performance have not yet been systematically investigated. To bridge this gap, we derive a wide range of fundamental distributional characteristics, including reliability measures, hazard and reversed-hazard functions, quantiles, moments, skewness, kurtosis, dispersion indices, and order statistics, establishing the model’s analytical tractability and structural richness. An extensive inferential framework is introduced by implementing eight classical estimation techniques, and their finite-sample behavior is rigorously examined through a large-scale Monte Carlo simulation study under diverse parameter configurations. The practical relevance of the Baker–T1 model is further demonstrated using two genuine datasets from biomedical and engineering domains, where it consistently outperforms thirteen competing lifetime distributions according to likelihood-based and information-theoretic criteria.