Abstract
The Weibull distribution is a very applicable model for the lifetime data. Therefore, comparing the parameters of two Weibull distributions is very important. However, there is not an appropriate method for comparing the shape or scale parameters in the literatures based on record values. In this paper, we have proposed a simple exact method for testing and constructing confidence interval for the ratio of shape parameters in two Weibull distributions. In addition, a simple exact method is proposed for inference about the common shape parameter. For comparing the scale parameters, we use the concepts of generalized confidence interval and generalized p value, and derive approaches when the shape parameters are equal or unequal. Also, we give a generalized approach for inference about the common scale parameter. At the end, we investigate inference about stress–strength reliability. Simulation results show that proposed approaches are satisfactory. All approaches are illustrated using a real example.
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References
Abd-Elfattah A, Mohamed MO (2009) Estimating system reliability of competing Weibull failures with censored sampling. Selcuk J Appl Math 10(2):53–65
Ahsanullah M (1995) Record statistics. Nova Science Publishers Commack, New York
Amiri N, Azimi R, Yaghmaei F, Babanezhad M (2013) Estimation of stress-strength parameter for two-parameter Weibull distribution. Int J Adv Stat Probab 1(1):4–8
Arnold B, Balakrishnan N, Nagaraja H (1998) Records. Wiley, New York
Asgharzadeh A, Abdi M (2011) Joint confidence regions for the parameters of the Weibull distribution based on record. ProbStat Forum 4:12–24
Asgharzadeh A, Valiollahi R, Raqab MZ (2011) Stress-strength reliability of Weibull distribution based on progressively censored samples. Stat Oper Res Trans 35(2):103–124
Baklizi A (2012) Inference on \(Pr(X < Y)\) in the two-parameter Weibull model based on records. ISRN Probab Stat 1–11
Baklizi A (2013) Interval estimation of the stress-strength reliability in the two-parameter exponential distribution based on records. J Stat Comput Simul 1–10
Balakrishnan N, Chan PS (1993) Record values from Rayleigh and Weibull distributions and associated inference. In: International conference on extremes and applications, MD
Balakrishnan N, Chan PS (1994) Record values from Rayleigh and Weibull distributions and associated inference. In: NIST Special Publication 866, proceedings of the conference on extreme value theory and applications, vol 3, pp 41–41
Chandler KN (1952) The distribution and frequency of record values. J R Stat Soc Ser B (Methodological) 14(2):220–228
Dallas AC (1982) Some results on record values from the exponential and Weibull law. Acta Math Acad Sci Hung 40(3):307–311
Friedrich T, Knapp G (2013) Generalised interval estimation in the random effects meta regression model. Comput Stat Data Anal 64:165–179
Hoinkes LA, Padgett WJ (1994) Maximum likelihood estimation from record-breaking data for the Weibull distribution. Qual Reliab Eng Int 10(1):5–13
Jafari AA (2012) Inferences on the ratio of two generalized variances: independent and correlated cases. Stat Methods Appl 21(3):297–314
Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations: theory and applications. World Scientific, Singapore
Krishnamoorthy K, Lin Y (2010) Confidence limits for stress-strength reliability involving Weibull models. J Stat Plan Inference 140(7):1754–1764
Krishnamoorthy K, Lu Y (2003) Inferences on the common mean of several normal populations based on the generalized variable method. Biometrics 59(2):237–247
Krishnamoorthy K, Lin Y, Xia Y (2009) Confidence limits and prediction limits for a Weibull distribution based on the generalized variable approach. J Stat Plan Inference 139(8):2675–2684
Kundu D, Gupta R (2006) Estimation of \(P(Y < X)\) for Weibull distribution. IEEE Trans Reliab 55(2):270–280
Lawless JF (2003) Statistical models and methods for lifetime data. Wiley-Interscience, New York
Nelson W (1982) Applied life data analysis. Wiley, New York
Soliman AA, Abd Ellah AH (2006) Comparison of estimates using record statistics from Weibull model: Bayesian and non-Bayesian approaches. Comput Stat Data Anal 51(3):2065–2077
Teimouri M, Gupta AK (2012) On the Weibull record statistics and associated inferences. Statistica 72(2):145–162
Teimouri M, Nadarajah S (2013) Bias corrected MLEs for the Weibull distribution based on records. Stat Methodol 13:12–24
Tsui KW, Weerahandi S (1989) Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. J Am Stat Assoc 84(406):602–607
Weerahandi S (1993) Generalized confidence intervals. J Am Stat Assoc 88(423):899–905
Weerahandi S (1995) Exact statistical methods for data analysis. Springer, New York
Wu JW, Tseng HC (2006) Statistical inference about the shape parameter of the Weibull distribution by upper record values. Stat Pap 48(1):95–129
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The authors would like to thank the Editor-in-Chief and two referees for their comments and suggestions which have contributed to improving the manuscript.
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Zakerzadeh, H., Jafari, A.A. Inference on the parameters of two Weibull distributions based on record values. Stat Methods Appl 24, 25–40 (2015). https://doi.org/10.1007/s10260-014-0278-3
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DOI: https://doi.org/10.1007/s10260-014-0278-3