Applications of an extended geometric Brownian motion degradation model

In order to modify the restriction that the path of the geometric Brownian motion can never reach zero, we consider an extended degradation model based on geometric Brownian motion. This model incorporates some important stochastic processes, such as geometric Brownian motion and the sinh-Gaussian process. We determine the convergence behavior of the first passage time as the random effect vanishes. The parameters of the model are estimated using the maximum likelihood method based on the first passage time observations. Both the explicit forms and the asymptotic properties of the estimators are provided. Our model can be applied to measure the brightness of LED lamps and is a reasonable extension of the original brightness model to situations in which the brightness reaches zero. The performance of the proposed model is discussed based on real data analysis and simulations.