IDEAS home Printed from https://ideas.repec.org/a/spr/testjl/v30y2021i2d10.1007_s11749-020-00724-0.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11749-020-00724-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11749-020-00724-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Erhard Cramer & Jorge Navarro, 2015. "Progressive Type‐II censoring and coherent systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(6), pages 512-530, September.
    2. Jingjing Qu & Hon Keung Tony Ng & Chul Moon, 2024. "Empirical likelihood ratio tests for homogeneity of component lifetime distributions based on system lifetime data," Computational Statistics, Springer, vol. 39(6), pages 3007-3029, September.
    3. Jian Zhang & Hon Keung Tony Ng & Narayanaswamy Balakrishnan, 2015. "Tests for homogeneity of distributions of component lifetimes from system lifetime data with known system signatures," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(7), pages 550-563, October.
    4. Antonio Di Crescenzo & Patrizia Di Gironimo, 2018. "Stochastic Comparisons and Dynamic Information of Random Lifetimes in a Replacement Model," Mathematics, MDPI, vol. 6(10), pages 1-13, October.
    5. Krzysztof Jasiński, 2021. "The number of failed components in a coherent working system when the lifetimes are discretely distributed," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(7), pages 1081-1094, October.
    6. M. Chahkandi & Jafar Ahmadi & S. Baratpour, 2014. "Non-parametric prediction intervals for the lifetime of coherent systems," Statistical Papers, Springer, vol. 55(4), pages 1019-1034, November.
    7. Zhu, Xiaojun & Balakrishnan, N., 2023. "Non-parametric inference based on reliability life-test of non-identical coherent systems with application to warranty time," Reliability Engineering and System Safety, Elsevier, vol. 232(C).
    8. Paula R. Bouzas & Nuria Ruiz-Fuentes & Carmen Montes-Gijón & Juan Eloy Ruiz-Castro, 2021. "Forecasting counting and time statistics of compound Cox processes: a focus on intensity phase type process, deletions and simultaneous events," Statistical Papers, Springer, vol. 62(1), pages 235-265, February.
    9. Eryilmaz, Serkan, 2016. "A new class of lifetime distributions," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 63-71.
    10. Ryo Miyazaki & Hidetoshi Murakami, 2020. "The non-null limiting distribution of the generalized Baumgartner statistic based on the Fourier series approximation," Statistical Papers, Springer, vol. 61(5), pages 1893-1909, October.
    11. 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.
    12. Eryilmaz, Serkan, 2017. "On compound sums under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 228-234.
    13. 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.
    14. Yandan Yang & Hon Keung Tony Ng & Narayanaswamy Balakrishnan, 2019. "Expectation–maximization algorithm for system-based lifetime data with unknown system structure," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(1), pages 69-98, March.
    15. Krzysztof Jasiński, 2023. "On conditional residual lifetimes of coherent systems consisting of components with discrete lifetimes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(2), pages 205-218, February.
    16. Yunhan Liu & Changchun Gao & Xiaofeng Liu & Ping Luo & Jianguo Ren, 2024. "A Comparison of MLE for Some Index Distributions Based on Censored Samples," Mathematics, MDPI, vol. 12(20), pages 1-15, October.
    17. Louis J. M. Aslett & Frank P. A. Coolen & Simon P. Wilson, 2015. "Bayesian Inference for Reliability of Systems and Networks Using the Survival Signature," Risk Analysis, John Wiley & Sons, vol. 35(9), pages 1640-1651, September.
    18. Krzysztof Jasiński, 2021. "Some conditional reliability properties of k-out-of-n system composed of different types of components with discrete independent lifetimes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(8), pages 1241-1251, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:testjl:v:30:y:2021:i:2:d:10.1007_s11749-020-00724-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.