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Maximum likelihood estimators based on discrete component lifetimes of a k-out-of-n system

Author

Listed:
  • Anna Dembińska

    (Warsaw University of Technology)

  • Krzysztof Jasiński

    (Nicolaus Copernicus University)

Abstract

This paper deals with parametric inference about the independent and identically distributed discrete lifetimes of components of a k-out-of-n system. We consider the maximum likelihood estimation assuming that the available data consists of component failure times observed up to and including the moment of the breakdown of the system. First, we provide general conditions for the almost sure existence of a strongly consistent sequence of maximum likelihood estimators (MLE’s). Then, we focus on three typical discrete failure distributions—the Poisson, binomial and negative binomial distributions—and prove that in these cases the MLE’s are unique, provided they exist, and that they are strongly consistent. Finally, we complete our results by Monte Carlo simulation study. Interestingly, the inference considered in the paper can be viewed as equivalent to one based on Type-II right censored discrete data. Therefore, our results can as well be applied to the case when Type-II right censored sample from a discrete distribution is observed.

Suggested Citation

  • Anna Dembińska & Krzysztof Jasiński, 2021. "Maximum likelihood estimators based on discrete component lifetimes of a k-out-of-n system," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 407-428, June.
    • Handle: RePEc:spr:testjl:v:30:y:2021:i:2:d:10.1007_s11749-020-00724-0
    DOI: 10.1007/s11749-020-00724-0
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    References listed on IDEAS

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    1. Fatih Tank & Serkan Eryilmaz, 2015. "The distributions of sum, minima and maxima of generalized geometric random variables," Statistical Papers, Springer, vol. 56(4), pages 1191-1203, November.
    2. Jorge Navarro & Hon Keung Tony Ng & Narayanaswamy Balakrishnan, 2012. "Parametric inference for component distributions from lifetimes of systems with dependent components," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(7), pages 487-496, October.
    3. M. Hermanns & E. Cramer, 2018. "Inference with progressively censored k-out-of-n system lifetime data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(4), pages 787-810, December.
    4. Chien-Tai Lin & N. Balakrishnan, 2011. "Asymptotic properties of maximum likelihood estimators based on progressive Type-II censoring," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 349-360, November.
    5. Fanhui Kong & Heliang Fei, 1996. "Limit theorems for the maximum likelihood estimate under general multiply Type II censoring," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(4), pages 731-755, December.
    6. Hon Ng & Jorge Navarro & Narayanaswamy Balakrishnan, 2012. "Parametric inference from system lifetime data under a proportional hazard rate model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(3), pages 367-388, April.
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