A New One-Parameter Model Supports an Upside-Down Bathtub Failure Rate: Theory, Inference, and Real-World Applications

Researchers often develop ordinal hazard distributions, whether increasing or decreasing, into multi-parameter distributions to derive various forms of the hazard function. This process necessitates the formulation of a multi-parameter hazard function, which involves a more complex mathematical expression. In contrast, this study introduces a new one-parameter lifetime model, termed the Inverted Z–Lindley (IZL) distribution, which is capable of capturing an upside-down bathtub-shaped failure rate without sacrificing analytical simplicity. Fundamental distributional properties of the IZL model are rigorously established, including closed-form expressions for the probability density, cumulative distribution, reliability, and hazard rate functions. Theoretical analysis shows that the density is strictly positive, unimodal, positively skewed, and heavy-tailed, while the hazard rate is unimodal with vanishing limits at both extremes. Fractional moments are obtained, and the non-existence of classical moments is formally justified, motivating the use of quantile-based and inactivity-time reliability measures. Besides the quantile function, several key reliability measures, including the mean inactivity time and strong mean inactivity time functions, and order statistics, are also developed. Inferential procedures are constructed under Type-II censoring using both likelihood-based and Bayesian frameworks. The existence and uniqueness of the frequentist estimator are established, while Bayesian estimation is implemented via Markov chain Monte Carlo methods under informative gamma priors. Several interval estimation techniques—including asymptotic, bootstrap, Bayesian credible, and highest posterior density intervals—are developed and compared through extensive Monte Carlo simulations. The practical relevance of the proposed model is demonstrated using real datasets from environmental health and communication engineering, where the IZL distribution consistently outperforms fifteen well-established inverted lifetime models according to likelihood-based criteria, information measures, and goodness-of-fit diagnostics. Overall, the IZL model offers a powerful, interpretable, and computationally efficient alternative for modeling heavy-tailed lifetime data with non-monotone failure behavior, contributing meaningfully to modern distribution theory and applied reliability analysis.