Weibull Distribution with Linear Shape Function

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.