New Approximate Inferential Methods for the Reliability Parameter in a Stress–Strength Model: The Normal Case

Abstract

This article considers the problem of testing and interval estimation of the reliability parameter P(Y ₁ > Y ₂), where Y ₁ and Y ₂ are independent normal random variables with unknown means and variances. A new approximate method is proposed, and is compared with two existing approximate methods and a generalized variable method. Simulation studies indicate that the sizes of the existing approximate tests exceed the nominal level considerably in some situations while the new method controls the sizes satisfactorily in all situations considered. The studies also indicated that the generalized variable test is too conservative for small samples. A confidence limit for the reliability parameter based on the new approximate test is also given. The results are extended to the case where Y ₁ and Y ₂ depend on some explanatory variables. The methods are illustrated using three examples, and some recommendations are made regarding what method should be used for practical applications.

Key Words:

• Explanatory variables
• Generalized p-value
• Lognormal distribution
• Noncentral t distribution
• Parametric bootstrap
• Quantile
• Satterthwaite approximation
• Size
• Tolerance limits

Mathematics Subject Classification:

• Primary 62F03
• Secondary 62N05

Acknowledgments