Weibull Distribution with Linear Shape Function

Abstract

Featured Application: The proposed model, a generalization of Weibull-type distributions, finds wide application in analyzing data concerning the lifetime of technical and biological components. It may be particularly useful in reliability engineering, for example, for modeling the failure rate of mechanical devices, electronic systems, or critical infrastructure components, as well as in medicine for analyzing patient survival. Thanks to the flexibility of shaping the hazard function, the model enables a more accurate fit to data compared to classical approaches, which can support maintenance planning, risk forecasting, and resource optimization. Potential applications also include the analysis of large datasets from IoT sensors, which opens the possibility of its use in predictive maintenance systems. The paper is intended to put forward a modified Weibull-type lifetime model. Modification consists of replacing the shape parameter of the original Weibull model with the shape function. It is self-evidently a novelty among lifetime models. The model in question will further be named the Weibull-sf model. To present the Weibull-sf, we need appropriate background. The background comes from an extensive review performed on 165 Weibull-type lifetime models we found in the source literature. Performing this review, we focused on two properties of the models: modality of failure density functions, as well as shape of the hazard rate functions. It does not matter that these are strongly interrelated, incidentally. The Weibull-sf lifetime model has the valuable property of flexibility. It may have a bathtub-like hazard rate function and bimodal density function. This is exactly what reliability analysts want to have. Foreseeing the huge numerical problems one will face when trying the maximum-likelihood method, we promote the method of the least absolute values that is a “close relative” to the method of least squares. Examples of fitting the Weibull-sf to real data are given. The cumulative failure functions of bimodal models with a bathtub-like hazard rate function and R codes are given.