CONFIDENCE LIMITS FOR STRESS STRENGTH RELIABILITY INVOLVING WEIBULL MODELS

Abstract The problem of interval estimation of the stress–strength reliability involving two independent Weibull distributions is considered. An interval estimation procedure based on the generalized variable (GV) approach is given when the shape parameters are unknown and arbitrary. The coverage probabilities of the GV approach are evaluated by Monte Carlo simulation. Simulation studies show that the proposed generalized variable approach is very satisfactory even for small samples. For the case of equal shape parameter, it is shown that the generalized confidence limits are exact. Some available asymptotic methods for the case of equal shape parameter are described and their coverage probabilities are evaluated using Monte Carlo simulation. Simulation studies indicate that no asymptotic approach based on the likelihood method is satisfactory even for large samples. Applicability of the GV approach for censored samples is also discussed. The results are illustrated using an example.

[1]  Kam-Wah Tsui,et al.  Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters , 1989 .

[2]  John I. McCool,et al.  Inference on p{y , 1991 .

[3]  A. Cohen,et al.  Maximum Likelihood Estimation in the Weibull Distribution Based On Complete and On Censored Samples , 1965 .

[4]  T. Mathew,et al.  Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals , 2003 .

[5]  Generalized inference for Weibull distributions , 2010 .

[6]  Richard A. Johnson,et al.  Testing reliability in a stress-strength model when X and Y are normally distributed , 1992 .

[7]  K. Krishnamoorthy,et al.  Confidence limits and prediction limits for a Weibull distribution based on the generalized variable approach , 2009 .

[8]  Samaradasa Weerahandi.  Exact Statistical Methods for Data Analysis , 2003 .

[9]  John A. Swets,et al.  Signal Detection Theory and ROC Analysis in Psychology and Diagnostics: Collected Papers , 1996 .

[10]  Irwin Guttman,et al.  Statistical inference for Pr(Y < X): The normal case , 1986 .

[11]  J. Lawless.  Statistical Models and Methods for Lifetime Data , 2002 .

[12]  K. Krishnamoorthy,et al.  New Approximate Inferential Methods for the Reliability Parameter in a Stress–Strength Model: The Normal Case , 2004 .

[13]  Samaradasa Weerahandi,et al.  Generalized Confidence Intervals , 1993 .

[14]  Tsai-Yu Lin,et al.  One- and Two-Sided Tolerance Intervals for General Balanced Mixed Models and Unbalanced One-Way Random Models , 2005, Technometrics.

[15]  I. J. Hall,et al.  Approximate one-sided tolerance limits for the difference or sum of two independent normal variates , 1984 .

[16]  E. Dudewicz,et al.  On Assessing the Precision of Simulation Estimates of Percentile Points , 1984 .

[17]  Samaradasa Weerahandi.  Generalized inference in repeated measures : exact methods in MANOVA and mixed models , 2004 .

[18]  Ray E. Schafer,et al.  On Procedures for Comparing Two Weibull Populations , 1976 .

[19]  K. Krishnamoorthy,et al.  Normal-Based Methods for a Gamma Distribution , 2008, Technometrics.

[20]  K. Krishnamoorthy,et al.  One-Sided Tolerance Limits in Balanced and Unbalanced One-Way Random Models Based on Generalized Confidence Intervals , 2004, Technometrics.

[21]  Sam Weerahandi,et al.  Exact Statistical Methods for Data Analysis , 1998, Journal of the American Statistical Association.

[22]  B Reiser,et al.  Measuring the effectiveness of diagnostic markers in the presence of measurement error through the use of ROC curves. , 2000, Statistics in medicine.

[23]  Thomas Mathew,et al.  Statistical Tolerance Regions: Theory, Applications, and Computation , 2009 .

[24]  Kenneth Constantine,et al.  Confidence interval estimation of P(Y , 1990 .

[25]  Irwin Guttman,et al.  Confidence limits for stress-strength models with explanatory variables , 1988 .

[26]  Mikhail Nikulin,et al.  Statistical planning and inference in accelerated life testing using the CHSS model , 2004 .

[27]  T Hyslop,et al.  Generalized treatment effects for clinical trials. , 2000, Statistics in medicine.

[28]  Kenneth Constantine,et al.  Bootstrapping estimators of P(Y , 1989 .

[29]  Asit P. Basu,et al.  The estimation of P(x < Y) for distributions useful in life testing , 1981 .

[30]  H. Iyer,et al.  Fiducial Generalized Confidence Intervals , 2006 .

[31]  D. Bamber.  The area above the ordinal dominance graph and the area below the receiver operating characteristic graph , 1975 .

[32]  K. Krishnamoorthy,et al.  Inference on Reliability in Two-parameter Exponential Stress–strength Model , 2007 .

[33]  M. V. Menon.  Estimation of the Shape and Scale Parameters of the Weibull Distribution , 1963 .

[34]  Douglas A. Wolfe,et al.  On Constructing Statistics and Reporting Data , 1971 .

[35]  S. Kotz,et al.  The stress-strength model and its generalizations : theory and applications , 2003 .