Accurate higher-order likelihood inference on \(P(Y<X)\)
The stress-strength reliability \(R=P(Y<X)\), where \(X\) and \(Y\) are independent continuous random variables, has obtained wide attention in many areas of application, such as in engineering statistics and biostatistics. Classical likelihood-based inference about \(R\) has been widely examined under various assumptions on \(X\) and \(Y\). However, it is well-known that first order inference can be inaccurate, in particular when the sample size is small or in presence of unknown parameters. The aim of this paper is to illustrate higher-order likelihood-based procedures for parametric inference in small samples, which provide accurate point estimators and confidence intervals for \(R\). The proposed procedures are illustrated under the assumptions of Gaussian and exponential models for \((X,Y)\). Moreover, simulation studies are performed in order to study the accuracy of the proposed methodology, and an application to real data is discussed. An implementation of the proposed method in the R software is provided.
higher-order likelihood inference confidence interval stress-strength model modified signed log-likelihood small sample size area under the roc curve