Parametric Hazard Functions

Hazard models are used only moderately in the transportation field compared with their levels of application in other areas, such as medicine, political science, and economics. Many specifications of the hazard models are not well known, whereas the primary application of hazard models is limited to a basic proportional continuous hazard formulation with continuous failure time and Weibull baseline hazard function. Besides the recent advances in the methodology of hazard-based models, the basic proportional hazard formulation remains the proffered hazard-based modeling approach. This paper aims to present some popular specifications of hazard-based models and also discusses some formulations that are less frequently used in practical models. In this study, the Weibull baseline hazard, which is a monotonic function, is replaced by the nonmonotonic log logistic function, and the modified formulation is presented. The log logistic and Weibull formulations are also presented, not only in continuous form but also in discrete hazard formulations for applications in which the failures are observed in discrete time intervals. Finally, all four combinations of discrete and continuous formulations with Weibull and log logistic baselines are discussed with the option of unobserved heterogeneity of type of gamma distribution. The accelerated failure time model, which is an alternative to the proportional hazard models, is discussed. This paper attempts to present a comprehensive overview of the applications of the duration model in the field of transportation.

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