Accurate higher-order likelihood inference on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$P(Y>X)$$</EquationSource> </InlineEquation>

The stress-strength reliability <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$R=P(Y>X)$$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$X$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$Y$$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$R$$</EquationSource> </InlineEquation> has been widely examined under various assumptions on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$X$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$Y$$</EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$R$$</EquationSource> </InlineEquation>. The proposed procedures are illustrated under the assumptions of Gaussian and exponential models for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$(X,Y)$$</EquationSource> </InlineEquation>. 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 <Emphasis FontCategory="NonProportional">R software is provided. Copyright Springer-Verlag 2013