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A note on multiple roots of a likelihood equation for Weibull sequential order statistics

Author

Listed:
  • Marcus Johnen

    (RWTH Aachen University)

  • Stefan Bedbur

    (RWTH Aachen University)

  • Udo Kamps

    (RWTH Aachen University)

Abstract

A multi-sample set-up of sequential order statistics from Weibull distribution functions with known scale parameters and a common unknown shape parameter is considered. The respective likelihood equation may have multiple roots even in the single-sample case, which is demonstrated by a simple example and illustrated with a simulation study. Uniqueness of the root of the likelihood equation and of the maximum likelihood estimator is examined with respect to different models of ordered data, sufficient conditions for uniqueness are shown, and the distribution of the number of roots of the likelihood equation is seen to be independent of the unknown shape parameter.

Suggested Citation

  • Marcus Johnen & Stefan Bedbur & Udo Kamps, 2020. "A note on multiple roots of a likelihood equation for Weibull sequential order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(4), pages 519-525, May.
    • Handle: RePEc:spr:metrik:v:83:y:2020:i:4:d:10.1007_s00184-019-00743-4
    DOI: 10.1007/s00184-019-00743-4
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    References listed on IDEAS

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    1. Vuong, Q.N. & Bedbur, S. & Kamps, U., 2013. "Distances between models of generalized order statistics," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 24-36.
    2. Balakrishnan, N. & Kateri, M., 2008. "On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 2971-2975, December.
    3. Erhard Cramer & Udo Kamps, 1996. "Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(3), pages 535-549, September.
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