From Gaussian Distribution to Weibull Distribution

Abstract

The Gaussian distribution is one of the most widely used statistical distributions, but there are a lot of data that do not conform to Gaussian distribution. For example, structural fatigue life is mostly in accordance with the Weibull distribution rather than the Gaussian distribution, and the Weibull distribution is in a sense a more general full state distribution than the Gaussian distribution. However, the biggest obstacle affecting the application of the Weibull distribution is the complexity of the Weibull distribution, especially the estimation of its three parameters is relatively difficult. In order to avoid this difficulty, people used to solve this problem by taking the logarithm to make the data appear to be more consistent with the Gaussian distribution. But in fact, this approach is problematic, because from the physical point of view, the structure of the data has changed and the physical meaning has changed, so it is not appropriate to use logarithmic Gaussian distribution to fit the original data after logarithm. The author thinks that Z.T. Gao method can give the estimation of three parameters of Weibull distribution conveniently, which lays a solid mathematical foundation for Weibull distribution to directly fit the original data.