A family of likelihood functions to make inferences about the reliability parameter for many stress-strength distributions

Many research papers in statistical literature address the estimation of the reliability parameter in stress-strength models, considering different types of distributions for stress and for strength. We have found that for many of these distributions, their corresponding profile likelihood functions of the reliability parameter can be grouped in a family of likelihood functions, with a simple algebraic structure that facilitates making inferences about this parameter. The novel family of likelihood functions, proposed here, maximum likelihood estimation procedures and suitable reparameterizations, were used to obtain a simple closed-form expression for the likelihood confidence interval of the reliability parameter. This new approach is particularly useful when small and/or unequal sample sizes are involved. Simulation studies for some distributions were carried out to illustrate the performance of the likelihood confidence intervals for the reliability parameter, and adequate coverage frequencies were obtained. The simplicity of our unifying proposal is shown here using three stress-strength distributions that have been analysed individually in statistical literature. However, there are many distributions for which inferences about the reliability parameter could be easily obtained using the proposed family.