Optimal experimental design for systems with bivariate failures under a bivariate Weibull function

Summary In manufacturing industry, it may be important to study the relationship between machine component failures under stress. Examples include the failures of integrated circuits and memory chips in electronic merchandise given various levels of electronic shock. Such studies are important for the development of new products and for the improvement of existing products. We assume two‐component systems for simplicity and we assume that the joint probability of failures increases with stress as a cumulative bivariate Weibull function. Optimal designs have been developed for two correlated binary responses by using the Gumbel model, the bivariate binary Cox model and the bivariate probit model. In all these models, the amount of damage ranges from −∞ to ∞. In the Weibull model, the amount of damage is positive, which is natural for experimental factors such as voltage, tension or pressure. We describe locally optimal designs under bivariate Weibull assumptions. Since locally optimal designs with non‐linear models depend on predetermined parameter values, misspecified parameter values may lead to inefficient designs. However, we find that optimal designs under the Weibull model are surprisingly efficient over a wide range of misspecified parameter values. To improve the efficiency, we recommend a multistage procedure. We show how using a two‐stage procedure can provide a substantial improvement over a design that was optimal for misspecified parameters.