Skip to main content
SpringerLink
Log in

Hypothesis testing for reliability with a three-parameter Weibull distribution using minimum weighted relative entropy norm and bootstrap

  • Published:
Journal of Zhejiang University SCIENCE C Aims and scope Submit manuscript

Abstract

With the help of relative entropy theory, norm theory, and bootstrap methodology, a new hypothesis testing method is proposed to verify reliability with a three-parameter Weibull distribution. Based on the relative difference information of the experimental value vector to the theoretical value vector of reliability, six criteria of the minimum weighted relative entropy norm are established to extract the optimal information vector of the Weibull parameters in the reliability experiment of product lifetime. The rejection region used in the hypothesis testing is deduced via the area of intersection set of the estimated truth-value function and its confidence interval function of the three-parameter Weibull distribution. The case studies of simulation lifetime, helicopter component failure, and ceramic material failure indicate that the proposed method is able to reflect the practical situation of the reliability experiment.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abbasi, B., Niaki, S.T.A., Khalife, M.A., Faize, Y., 2011. A hybrid variable neighborhood search and simulated annealing algorithm to estimate the three parameters of the Weibull distribution. Exp. Syst. Appl., 38(1):700–708. [doi:10.1016/j.eswa.2010.07.022]

    Article  Google Scholar 

  • Bartkute-Norkuniene, V., Sakalauskas, L., 2011. Consistent estimator of the shape parameter of three-dimensional Weibull distribution. Commun. Stat. Theory Meth., 40(16): 2985–2996. [doi:10.1080/03610926.2011.562785]

    Article  MathSciNet  MATH  Google Scholar 

  • Basu, A., Manda, A., Pardo, L., 2010. Hypothesis testing for two discrete populations based on the Hellinger distance. Stat. Prob. Lett., 80(3–4):206–214. [doi:10.1016/j.spl.2009.10.008]

    Article  MATH  Google Scholar 

  • Chatterjee, S., Bandopadhyay, S., 2012. Reliability estimation using a genetic algorithm-based artificial neural network: an application to a load-haul-dump machine. Exp. Syst. Appl., 39(12):10943–10951. [doi:10.1016/j.eswa.2012.03.030]

    Article  Google Scholar 

  • Deng, J., Gu, D.S., Li, X.B., 2004. Parameters and quantile estimation for fatigue life distribution using probability weighted moments. Chin. J. Comput. Mech., 21(5): 609–613 (in Chinese).

    Google Scholar 

  • de Santis, F., 2004. Statistical evidence and sample size determination for Bayesian hypothesis testing. J. Stat. Plan. Infer., 124(1):121–144. [doi:10.1016/S0378-3758(03)00198-8]

    Article  MATH  Google Scholar 

  • Duffy, S.F., Palko, J.L., Gyekenyesi, J.P., 1993. Structural reliability analysis of laminated CMC components. J. Eng. Gas Turb. Power, 115(1):103–108. [doi:10.1115/1.2906663]

    Article  Google Scholar 

  • Efron, B., 1979. Bootstrap methods. Ann. Stat., 7(1):1–36. [doi:10.1214/aos/1176344552]

    Article  MathSciNet  MATH  Google Scholar 

  • El-Adll, M.E., 2011. Predicting future lifetime based on random number of three parameters Weibull distribution. Math. Comput. Simul., 81(9):1842–1854. [doi:10.1016/j.matcom.2011.02.003]

    Article  MathSciNet  MATH  Google Scholar 

  • Ennis, D.M., Ennis, J.M., 2010. Equivalence hypothesis testing. Food Qual. Pref. 21(3):253–256. [doi:10.1016/j.foodqual.2009.06.005]

    Article  Google Scholar 

  • Feng, D.C., Chen, F., Xu, W.L., 2012. Learning robust principal components from L1-norm maximization. J. Zhejiang Univ.-Sci. C (Comput. & Electron.), 13(12):901–908. [doi: 10.1631/jzus.C1200180]

    Article  Google Scholar 

  • Harris, T.A., 1991. Rolling Bearing Analysis (3rd Ed.). John Wiley & Sons, New York.

    Google Scholar 

  • Jiang, R.Y., Zuo, M.J., 1999. Reliability Model and Application. China Machine Press, Beijing (in Chinese).

    Google Scholar 

  • Johnson, L.G., 1970. Theory and Technique of Variation Research. Elsevier, New York.

    Google Scholar 

  • Kaplan, E.L., Meier, P., 1958. Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc., 53(282): 457–481. [doi:10.1080/01621459.1958.10501452]

    Article  MathSciNet  MATH  Google Scholar 

  • Konishi, Y., Nishiyama, Y., 2009. Hypothesis testing in rank-size rule regression. Math. Comput. Simul., 79(9): 2869–2878. [doi:10.1016/j.matcom.2008.10.012]

    Article  MathSciNet  MATH  Google Scholar 

  • Kula, K.S., Apaydin, A., 2009. Hypotheses testing for fuzzy robust regression parameters. Chaos Solit. Fract., 42(4): 2129–2134. [doi:10.1016/j.chaos.2009.03.140]

    Article  MATH  Google Scholar 

  • Lee, J., Pan, R., 2012. Bayesian analysis of step-stress accelerated life test with exponential distribution. Qual. Rel. Eng. Int., 28(3):353–361. [doi:10.1002/qre.1251]

    Article  Google Scholar 

  • Li, Q., Liu, S.L., Pan, X.H., Zheng, S.Y., 2012. A new method for studying the 3D transient flow of misaligned journal bearings in flexible rotor-bearing systems. J. Zhejiang Univ.-Sci. A (Appl. Phys. & Eng.), 13(4):293–310. [doi: 10.1631/jzus.A1100228]

    Article  Google Scholar 

  • Luxhoj, T.J., Shyur, H.J., 1995. Reliability curve fitting for aging helicopter components. Rel. Eng. Syst. Safety, 48(3): 229–234. [doi:10.1016/0951-8320(95)00018-W]

    Article  Google Scholar 

  • Martin, M.A., 2007. Bootstrap hypothesis testing for some common statistical problems: a critical evaluation of size and power properties. Comput. Stat. Data Anal., 51(12): 6321–6342. [doi:10.1016/j.csda.2007.01.020]

    Article  MATH  Google Scholar 

  • Nelson, W., 1990. Accelerated Testing: Statistical Models, Test Plans and Data Analysis. John Wiley & Sons, New York.

    Google Scholar 

  • NiesŁony, A., Ruzicka, M., Papuga, J., Hodr, A., Balda, M., Svoboda, J., 2012. Fatigue life prediction for broad-band multiaxial loading with various PSD curve shapes. Int. J. Fatigue, 44:74–88. [doi:10.1016/j.ijfatigue.2012.05.014]

    Article  Google Scholar 

  • Pierce, D.M., Zeyen, B., Huigens, B.M., Fitzgerald, A.M., 2011. Predicting the failure probability of device features in MEMS. IEEE Trans. Dev. Mat. Rel., 11(3):433–441. [doi:10.1109/TDMR.2011.2159117]

    Article  Google Scholar 

  • Prawoto, Y., Dillon, B., 2012. Failure analysis and life assessment of coating: the use of mixed mode stress intensity factors in coating and other surface engineering life assessment. J. Fail. Anal. Prev., 12(2):190–197. [doi:10.1007/s11668-011-9525-1]

    Article  Google Scholar 

  • Qian, L.F., 2012. The Fisher information matrix for a three-parameter exponentiated Weibull distribution under type II censoring. Stat. Methodol., 9(3):320–329. [doi:10.1016/j.stamet.2011.08.007]

    Article  MathSciNet  Google Scholar 

  • Shafieezadeh, A., Ellingwood, B.R., 2012. Confidence intervals for reliability indices using likelihood ratio statistics. Struct. Safety, 38:48–55. [doi:10.1016/j.strusafe.2012.04.002]

    Article  Google Scholar 

  • Tan, P., He, W.T., Lin, J., Zhao, H.M., Chu, J., 2011. Design and reliability, availability, maintainability, and safety analysis of a high availability quadruple vital computer system. J. Zhejiang Univ.-Sci. A (Appl. Phys. & Eng.), 12(12):926–935. [doi: 10.1631/jzus.A11GT003]

    Article  Google Scholar 

  • Toasa Caiza, P.D., Ummenhofer, T., 2011. General probability weighted moments for the three-parameter Weibull distribution and their application in S-N curves modeling. Int. J. Fatigue, 33(12):1533–1538. [doi:10.1016/j.ijfatigue.2011.06.009]

    Article  Google Scholar 

  • Woodbury, A.D., 2004. A FORTRAN program to produce minimum relative entropy distributions. Comput. Geosci., 30(1):131–138. [doi:10.1016/j.cageo.2003.09.001]

    Article  Google Scholar 

  • Woodbury, A.D., Ulrych, T.J., 1998. Minimum relative entropy and probabilistic inversion in groundwater hydrology. Stoch. Hydrol. Hydraul., 12(5):317–358. [doi:10.1007/s004770050024]

    Article  MATH  Google Scholar 

  • Xia, X.T., 2012. Forecasting method for product reliability along with performance data. J. Fail. Anal. Prev., 12(5): 532–540. [doi:10.1007/s11668-012-9592-y]

    Article  Google Scholar 

  • Xia, X.T., Chen, J.F., 2011. Fuzzy hypothesis testing and time series analysis of rolling bearing quality. J. Test. Eval., 39(6):1144–1151. [doi:10.1520/JTE103371]

    MathSciNet  Google Scholar 

  • Zhang, Y., Zheng, Y.M., Yang, M., Li, H., Jin, Z.H., 2012. Design and implementation of the highly-reliable, lowcost housekeeping system in the ZDPS-1A pico-satellite. J. Zhejiang Univ.-Sci. C (Comput. & Electron.), 13(2): 83–89. [doi: 10.1631/jzus.C1100079]

    Google Scholar 

  • Zhao, X.P., Lu, Z.Z., Wang, W.H., 2010. A practical and effective method for estimating confidence limits of population percentile and reliability of three-parameter Weibull distribution on probability weighted moment. J. Northw. Polytechn. Univ., 38(3):470–475 (in Chinese).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechatronical Engineering, Henan University of Science and Technology, Luoyang, 471003, China

    Xin-tao Xia, Yin-ping Jin, Yan-tao Shang & Long Chen

  2. School of Mechatronical Engineering, Northwestern Polytechnical University, Xi’an, 710072, China

    Yong-zhi Xu

Authors
  1. Xin-tao Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yin-ping Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong-zhi Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yan-tao Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Long Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin-tao Xia.

Additional information

Project (Nos. 51075123 and 50675011) supported by the National Natural Science Foundation of China

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xia, Xt., Jin, Yp., Xu, Yz. et al. Hypothesis testing for reliability with a three-parameter Weibull distribution using minimum weighted relative entropy norm and bootstrap. J. Zhejiang Univ. - Sci. C 14, 143–154 (2013). https://doi.org/10.1631/jzus.C12a0241

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1631/jzus.C12a0241

Key words

CLC number

Search

Navigation