Abstract
The exponential distribution is widely used in reliability and delay-time analysis owing to its mathematical simplicity and analytical tractability. However, its constant failure rate assumption may be restrictive for systems characterized by heterogeneous failure mechanisms and unobserved variability. To address this limitation, this paper discusses a flexible Frullani–Exponential Delay-Time Distribution (FEDTD) by applying the Frullani construction to the classical exponential model. The proposed distribution provides a flexible probabilistic framework for modelling positive-valued lifetime data while retaining a natural interpretation for defect-to-failure delay-time phenomena. The main statistical and reliability properties of the FEDTD are derived and investigated. Two distinct characterizations of the proposed distribution are derived through the hazard rate function and the mean inactivity time (MIT). Notably, the MIT characterization offers a direct retrospective interpretation of undetected failure duration, providing actionable insights for optimizing inspection intervals and maintenance planning. Parameter estimation is carried out using maximum likelihood and Bayesian approaches, with Bayesian inference implemented through the Metropolis–Hastings algorithm to obtain posterior estimates and credible intervals. The finite-sample performance of the proposed estimators is evaluated through Monte Carlo simulation studies in terms of bias and mean squared error. The practical applicability of the FEDTD is demonstrated using three reliability datasets drawn from mechanical, nuclear engineering, and aviation, and its performance is compared with competing lifetime distributions using standard goodness-of-fit criteria. The empirical results demonstrate the flexibility and competitive performance of the proposed model in diverse lifetime and failure-time data. Thus, the FEDTD provides a tractable and versatile extension of the exponential framework for reliability analysis.