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Inference on q-Weibull parameters

Author

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  • Xiang Jia

    (National University of Defense Technology)

  • Saralees Nadarajah

    (University of Manchester)

  • Bo Guo

    (National University of Defense Technology)

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.

Suggested Citation

  • Xiang Jia & Saralees Nadarajah & Bo Guo, 2020. "Inference on q-Weibull parameters," Statistical Papers, Springer, vol. 61(2), pages 575-593, April.
    • Handle: RePEc:spr:stpapr:v:61:y:2020:i:2:d:10.1007_s00362-017-0951-3
    DOI: 10.1007/s00362-017-0951-3
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    References listed on IDEAS

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    1. Almalki, Saad J. & Nadarajah, Saralees, 2014. "Modifications of the Weibull distribution: A review," Reliability Engineering and System Safety, Elsevier, vol. 124(C), pages 32-55.
    2. Costa, U.M.S. & Freire, V.N. & Malacarne, L.C. & Mendes, R.S. & Picoli Jr., S. & de Vasconcelos, E.A. & da Silva Jr., E.F., 2006. "An improved description of the dielectric breakdown in oxides based on a generalized Weibull distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(1), pages 209-215.
    3. Joarder, Avijit & Krishna, Hare & Kundu, Debasis, 2011. "Inferences on Weibull parameters with conventional type-I censoring," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 1-11, January.
    4. Saralees Nadarajah & Xiang Jia, 2017. "Estimation of $$P(Y > X)$$ P ( Y > X ) for the Weibull distribution," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 8(2), pages 1762-1774, November.
    5. Nadarajah, Saralees & Kotz, Samuel, 2007. "On the q-type distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 377(2), pages 465-468.
    6. Picoli, S. & Mendes, R.S. & Malacarne, L.C., 2003. "q-exponential, Weibull, and q-Weibull distributions: an empirical analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(3), pages 678-688.
    7. Xu, Meng & Droguett, Enrique López & Lins, Isis Didier & das Chagas Moura, Márcio, 2017. "On the q-Weibull distribution for reliability applications: An adaptive hybrid artificial bee colony algorithm for parameter estimation," Reliability Engineering and System Safety, Elsevier, vol. 158(C), pages 93-105.
    8. Zhang, L.F. & Xie, M. & Tang, L.C., 2007. "A study of two estimation approaches for parameters of Weibull distribution based on WPP," Reliability Engineering and System Safety, Elsevier, vol. 92(3), pages 360-368.
    9. Jia, Xiang & Wang, Dong & Jiang, Ping & Guo, Bo, 2016. "Inference on the reliability of Weibull distribution with multiply Type-I censored data," Reliability Engineering and System Safety, Elsevier, vol. 150(C), pages 171-181.
    10. K. Jose & Shanoja Naik & Miroslav Ristić, 2010. "Marshall–Olkin q-Weibull distribution and max–min processes," Statistical Papers, Springer, vol. 51(4), pages 837-851, December.
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