Higher order inference for stress-strength reliability with independent Burr-type X distributions

In this paper, a small-sample asymptotic method is proposed for higher order inference in the stress-strength reliability model, R=P(Y<X), where X and Y are distributed independently as Burr-type X distributions. In a departure from the current literature, we allow the scale parameters of the two distributions to differ, and the likelihood-based third-order inference procedure is applied to obtain inference for R. The difficulty of the implementation of the method is in obtaining the the constrained maximum likelihood estimates (MLE). A penalized likelihood method is proposed to handle the numerical complications of maximizing the constrained likelihood model. The proposed procedures are illustrated using a sample of carbon fibre strength data. Our results from simulation studies comparing the coverage probabilities of the proposed small-sample asymptotic method with some existing large-sample asymptotic methods show that the proposed method is very accurate even when the sample sizes are small.