A New Flexible Integrated Linear–Weibull Lifetime Model: Analytical Characterization and Real-Data Studies

In this work, we introduce a new four-parameter distribution, called the integrated linear–Weibull (ILW) model, constructed by embedding a dynamic linear component within the Weibull framework. The ILW distribution is capable of capturing a wide variety of symmetric and asymmetric density shapes and accommodates diverse failure-rate behaviors. We derive several of its key mathematical and statistical properties, including moments, extropy, cumulative residual entropy, order statistics, and their asymptotic forms. The mean residual life function and its reciprocal relationship with the failure rate are also obtained. An algorithm for generating random samples from the ILW distribution is provided, and model identifiability is examined numerically through the Kullback–Leibler divergence. Parameter estimation is carried out via maximum likelihood, and a simulation study is conducted to assess the accuracy of the estimators; the results show improvements in estimator performance as sample size increases. Finally, three real datasets involving failure-time observations and one describing hydrological and epidemiological data are analyzed to demonstrate the practical usefulness of the ILW model. In these applications, the proposed model exhibits competitive or superior performance relative to several existing lifetime distributions based on standard model selection criteria and goodness-of-fit measures.