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Inference on q-Weibull parameters

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Abstract

The q-Weibull distribution is a generalization of the Weibull distribution and could describe complex systems. We firstly point out how to derive the maximum likelihood estimates (MLEs) and least-squares estimates (LSEs) of the q-Weibull parameters. Next, three confidence intervals (CIs) for the q-Weibull parameters are constructed based on bootstrap methods and asymptotic normality of the MLEs. Explicit expressions for the Fisher information matrix necessary for the asymptotic CIs are derived. A Monte Carlo simulation study is conducted to compare the performances of the MLEs and LSEs as well as the different CIs. The simulation results show that the MLEs are superior to the LSEs in terms of both bias and mean squared error. The bootstrap CIs based on the MLEs are shown to have good coverage probabilities and average interval widths. Finally, a real data example is provided to illustrate the proposed methods.

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References

  • Abe S, Okamoto Y (2001) Nonextensive statistical mechanics and its applications, vol 560. Springer, Berlin

    Book  Google Scholar 

  • Almalki SJ, Nadarajah S (2014) Modifications of the weibull distribution: a review. Reliab Eng Syst Saf 124:32–55

    Article  Google Scholar 

  • Assis EM, Borges EP, Vieira de Melo SA (2013) Generalized q-weibull model and the bathtub curve. Int J Qual Reliab Manag 30(7):720–736

    Article  Google Scholar 

  • Assis EM, Borges EP, Vieira de Melo SA, Schnitman L (2015) Modeling failure rate of a robotic welding station using generalized q-distributions. Int J Quality Reliab Manag 32(2):156–166

    Article  Google Scholar 

  • Cohen AC (1965) Maximum likelihood estimation in the weibull distribution based on complete and on censored samples. Technometrics 7(4):579–588

    Article  MathSciNet  Google Scholar 

  • Costa UMS, Freire VN, Malacarne LC, Mendes RS, Picoli S Jr, De Vasconcelos EA, da Silva Jr EF (2006) An improved description of the dielectric breakdown in oxides based on a generalized weibull distribution. Physica A: 361(1):209–215

    Article  Google Scholar 

  • Gell-Mann M, Tsallis C (2004) Nonextensive entropy: interdisciplinary applications. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Jia X, Guo B (2015) Analysis of non-repairable cold-standby systems in bayes theory. J Stat Comput Simul 86(11):2089–2112

    Article  MathSciNet  Google Scholar 

  • Jia X, Jiang P, Guo B (2015) Reliability evaluation for weibull distribution under multiply type-I censoring. J Central South Univ 22:3506–3511

    Article  Google Scholar 

  • Jia X, Wang D, Jiang P, Guo B (2016) Inference on the reliability of weibull distribution with multiply type-I censored data. Reliab Eng Syst Saf 150:171–181

    Article  Google Scholar 

  • Jia X, Nadarajah S, Guo B (2017) Bayes estimation of \( p (y< x) \) for the weibull distribution with arbitrary parameters. Appl Math Model 47:249–259

    Article  MathSciNet  Google Scholar 

  • Jia X, Guo B (2016) Exact inference for exponential distribution with multiply type-I censored data. Commun Stat-Simul Comput. doi:10.1080/03610918.2016.1235187

  • Joarder A, Krishna H, Kundu D (2011) Inferences on weibull parameters with conventional type-I censoring. Computat Stat Data Anal 55:1–11

    Article  MathSciNet  Google Scholar 

  • Jose KK, Naik SR (2009) On the q-weibull distribution and its applications. Commun Stat-Theor Methods 38(6):912–926

    Article  MathSciNet  Google Scholar 

  • Jose KK, Naik SR, Ristić MM (2010) Marshall-olkin q-weibull distribution and max-min processes. Stat Pap 51(4):837–851

    Article  MathSciNet  Google Scholar 

  • Nadarajah S, Kotz S (2007) On the q-type distributions. Physica A 377(2):465–468

    Article  MathSciNet  Google Scholar 

  • Picoli S Jr, Mendes RS, Malacarne LC (2003) q-exponential, weibull, and q-weibull distributions: an empirical analysis. Physica A 324(3):678–688

    Article  MathSciNet  Google Scholar 

  • Picoli S Jr, Mendes RS, Malacarne LC, Santos RPB (2009) q-distributions in complex systems: a brief review. Braz J Phys 39(2A):468–474

    Article  Google Scholar 

  • Ribeiro EMS, Prataviera GA (2015) Modeling empirical distributions of firm size with q-distributions. Int J Appl Math 28(6):715–725

    Article  Google Scholar 

  • Sartori I, de Assis EM, da Silva AL, Rosana LF, de Melo Vieira, Borges Ernesto P et al (2009) Reliability modeling of a natural gas recovery plant using q-weibull distribution. Comput Aided Chem Eng 27:1797–1802

    Article  Google Scholar 

  • Vuorenmaa T (2006 ) A q-weibull autoregressive conditional duration model and threshold dependence. Technical report, Discussion Paper 117, University of Helsinky

  • Vuorenmaa T (2009) A q-weibull autoregressive conditional duration model with an application to nyse and hse data. SSRN 1952550

  • Xu M, Droguett EL, Lins ID (2017) On the q-weibull distribution for reliability applications: an adaptive hybrid artificial bee colony algorithm for parameter estimation. Reliab Eng Syst Saf 158:93–105

    Article  Google Scholar 

  • Zhang LF, Xie M, Tang LC (2007) A study of two estimation approaches for parameters of weibull distribution based on wpp. Reliab Eng Syst Saf 92(3):360–368

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the two referees and the Editor for careful reading and comments which greatly improved the paper. This work was partially supported by the National Natural Science Foundation of China under Grant Nos.71571188 and 61573370.

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Authors and Affiliations

  1. College of Information System and Management, National University of Defense Technology, Changsha, 410073, Hunan, People’s Republic of China

    Xiang Jia & Bo Guo

  2. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Saralees Nadarajah

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  1. Xiang Jia
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  2. Saralees Nadarajah
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  3. Bo Guo
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Correspondence to Xiang Jia.

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Jia, X., Nadarajah, S. & Guo, B. Inference on q-Weibull parameters. Stat Papers 61, 575–593 (2020). https://doi.org/10.1007/s00362-017-0951-3

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  • DOI: https://doi.org/10.1007/s00362-017-0951-3

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