Agnes Kovacs

The estimation of the tensile strength of certain brittle fibers at extremely short lengths from tensile strength measurements of these fibers at longer lengths is of fundamental importance in the theory of modern fiber composite materials. We examine a competing risks model in which the fiber strength arises as the minimum of two (independent) Weibull random variables having distinct Weibull exponents. In the case of silicon carbide fibers, for example, the fracture of the fiber is determined by competition between the surface and inner defects, the strength distribution of each defect type being of Weibull form. In practice, it is often the case that the cause of failure is unknown for any particular specimen. We show how the EM (expectation maximization) algorithm can be used to compute maximum likelihood estimates of the four model parameters. The advantage of the EM technique is that the maximization step is reduced to two separate single variable numerical maximizations, thus avoiding the need to estimate an inverse Hessian matrix. We also derive an explicit formula for the observed information matrix, which allows us to derive asymptotic confidence intervals for the parameters and for the mean fiber strength. Several goodness-of-fit techniques are also considered for testing the validity of the model and its assumptions. This model also appears in the life-testing of a series system having two components that each follow a simple Weibull model.