Interval Estimation of the Stress-Strength Reliability with Independent Normal Random Variables

Abstract

This article develops a procedure to obtain highly accurate confidence interval estimates for the stress-strength reliability R = P(X > Y) where X and Y are data from independent normal distributions of unknown means and variances. Our method is based on third-order likelihood analysis and is compared to the conventional first-order likelihood ratio procedure as well as the approximate methods of Reiser and Guttman (1986) and Guo and Krishnamoorthy (2004). The use of our proposed method is illustrated by an empirical example and its superior accuracy in terms of coverage probability and error rate are examined through Monte Carlo simulation studies.

Keywords:

• Reliability
• Likelihood analysis
• Interval estimation
• Normal distribution.

Mathematics Subject Classification:

• 62F03
• 62N05

Notes

Note that the Reiser and Guttman (1986) and the Guo and Krishnamoorthy (2004) methods use s²_x and s²_y in their estimation of σ²_x and σ²_y instead of and like the MLE.

In contrast to the Reiser & Guttman (1986) and Guo & Krishnamoorthy (2004) methods, whose rates of convergence are unknown.

Practically, the confidence interval I^{1 − α}_ψ is obtained by numerically solving for ψ in the inequality |r*(ψ)| ⩽ z_α/2 using a sufficient number of grid points of ψ chosen in an appropriate range.