Sensitivity testing for quantile estimation under constraints

Abstract

Sensitivity testing involves assessing how a stimulus applied to test specimens affects the probability of a binary response. While various methods for conducting such tests efficiently have been proposed, there has been limited research on testing under constraints on the stimulus levels that can be applied. These constraints can lead to failure conditions in widely used methods. We consider two types of constraints commonly encountered in practical experiments. We propose a new sensitivity testing procedure, called Constrained Three Phase Optimal Design (C-3POD), that is robust to failure due to these constraints. We give proofs regarding the convergence, bias, and error of the Robbins-Monro and Robbins-Monro-Joseph (RMJ) procedures under these constraints to justify their use both separately and within the C-3POD procedure. The performance of C-3POD is compared with the Three Phase Optimal Design (3POD) method under various constraints in Monte Carlo simulations. The Robbins-Monro and RMJ procedures are shown to converge with bias, with the resulting error being bounded by the severity of the constraint. The C-3POD method eliminates failures where testing cannot be completed according to the procedure, whereas 3POD exhibited varying failure rates, from minor to significant. In terms of accuracy, the methods demonstrated similar mean-square error at small sample sizes, irrespective of constraints, but the new method showed superior accuracy with larger samples. Our findings are relevant for applications where testing conditions are constrained, such as in the sensitivity testing of explosive materials, offering an efficient and reliable alternative to traditional methods.