Truncated Weibull Distribution Functions and Moments

Abstract

The Weibull distribution is widely used in natural sciences and engineering, especially in material reliability. For one is often confronted to censored data sample in these fields, the truncated Weibull distribution functions and moments can be of great interest. McEwen and Parresol (1991) became the main reference over the years as they exhibit the moments expressions of the truncated distribution for both the two-parameter and three-parameter Weibull random variables. Unfortunately, a mistake has been made when writing the doubly-truncated three-parameter Weibull probability distribution function and it naturally led to wrong results. In this article, we aim at correcting this error and giving explicit mathematical demonstrations of how moments of truncated Weibull distributions can be computed. In order to be as clear as possible we give the explicit expressions and proofs for the two-parameter Weibull one-sided and doubly-truncated moments as well as for the three-parameter Weibull distribution, although every result can be derived from the doubly-truncated three-parameter Weibull distribution moments. Besides, an R code with a function computing these moments is given in the last section.