Parameter estimation for Weibull probability distribution function of initial fatigue quality

Abstract

This paper considers the problem of determining the parameters of a Weibull distribution when a limited set of experimental data of such a distribution is available. The problem is of particular interest in the probabilistic approach to durability analysis of flight structures, in which the distribution of time to crack initiation for a given crack length as well as the equivalent initial flaw size distribution are usually described through a three-parameter Weibull distribution. The aims of this paper are the determination of a high-quality estimation procedure and the solution of the problem of the determination of the number of experimental data that are necessary to estimate, with a given confidence level, the parameters of the distribution and some related statistics. The maximum likelihood and the least square methods are compared with regard to the estimation quality. It is shown, by Monte Carlo simulations, that the maximum likelihood approach gives better results than the least square approach. The problem of obtaining confidence intervals for the number of cracks with greater length than the reference one for a fixed service time on a single sample basis is addressed and solved through the bootstrap method.