Advertisement

Order Statistics from the Power Lindley Distribution and Associated Inference with Application

  • Devendra KumarEmail author
  • Anju Goyal
Article
  • 6 Downloads

Abstract

Power Lindley distribution has been proposed recently by Ghitany et al. (Comput Stat Data Anal 64:20–33, 2013) as a simple and useful reliability model for analysing lifetime data. This model provides more flexibility than the Lindley distribution in terms of the shape of the density and hazard rate functions as well as its skewness and kurtosis. For this distribution, exact explicit expressions for single moments, product moments, marginal moment generating functions and joint moment generating functions of each of these order statistics are derived. By using these relations, we have tabulated the expected values, second moments, variances and covariances of order statistics from samples of sizes up to 10 for various values of the parameters. In addition, we use these moments to obtain the best linear unbiased estimates of the location and scale parameters based on Type-II right-censored samples. In addition, we carry out some numerical illustrations through Monte Carlo simulations to show the usefulness of the findings. Finally, we apply the findings of the paper to some real data set.

Keywords

Power Lindley distribution Order statistics Single moment Double moment Type-II right censoring Best linear unbiased estimator 

Notes

Acknowledgements

The authors would like to thank the the editor and the referee for their comments which helped improve the paper.

References

  1. 1.
    Arnold BC, Balakrishnan N, Nagaraja HN (2003) A first course in order statistics. Wiley, New YorkGoogle Scholar
  2. 2.
    Ashour SK, Eltehiwy MA (2015) Exponentiated power Lindley distribution. J Adv Res 6:895–905CrossRefGoogle Scholar
  3. 3.
    Bakouch H, Al-Zahrani B, Al-Shomrani A, Marchi V, Louzada F (2012) An extended lindley distribution. J Korean Stat Soc 41(1):75–85CrossRefGoogle Scholar
  4. 4.
    Balakrishnan N, Chandramouleeswaran MP (1996) BLUEs of location and scale parameters of Laplace distribution based on Type-II censored samples and associated. Microelectron Reliab 36:371374Google Scholar
  5. 5.
    Barreto-Souza W, Bakouch HS (2013) A new lifetime model with decreasing failure rate. Statistics 47:465–476CrossRefGoogle Scholar
  6. 6.
    Bhaumik DK, Kapur K, Gibbons RD (2009) Testing parameters of a gamma distribution for small samples. Technometrics 51:326–334CrossRefGoogle Scholar
  7. 7.
    Burkschat M (2010) Linear estimators and predictors based on generalized order statistics from generalized Pareto distributions. Commun Stat-Theory Methods 39:311–326CrossRefGoogle Scholar
  8. 8.
    Ghitany M, Atieh B, Nadarajah S (2008) Lindley distribution and its application. Math Comput Simul 78(4):493–506CrossRefGoogle Scholar
  9. 9.
    Ghitany M, Alqallaf F, Al-Mutairi D, Husain HA (2011) A two-parameter weighted Lindley distribution and its applications to survival data. Math Comput Simul 81:1190–1201CrossRefGoogle Scholar
  10. 10.
    Ghitany M, Al-Mutairi D, Balakrishnan N, Al-Enezi L (2013) Power lindley distribution and associated inference. Comput Stat Data Anal 64:20–33CrossRefGoogle Scholar
  11. 11.
    Jaheen ZF (2005) Estimation based on generalized order statistics from the Burr model. Commun Stat Theory Methods 34:785–794CrossRefGoogle Scholar
  12. 12.
    Jabeen R, Ahmad A, Feroze N, Gilani GM (2013) Estimation of location and scale parameters of Weibull distribution using generalized order statistics under Type II singly and doubly censored data. Int J Adv Sci Technol 55:6780Google Scholar
  13. 13.
    Kumar D, Dey S, Nadarajah S (2017) Extended exponential distribution based on order statistics. Commun Stat-Theory Methods 46(18):9166–9184CrossRefGoogle Scholar
  14. 14.
    Kumar D, Dey S (2017a) Power generalized Weibull distribution based on order statistics. J Stat Res 51:61–78Google Scholar
  15. 15.
    Kumar D, Dey S (2017b) Relations for moments of generalized order statistics from extended exponential distribution. Am J Math Manag Sci 36:378–400Google Scholar
  16. 16.
    Lindley D (1958) Fiducial distributions and bayes theorem. J R Stat Soc Ser B 20:102–107Google Scholar
  17. 17.
    Meyer J (1987) Two-moment decision models and expected utility maximization. Am Econ Rev 77:421430Google Scholar
  18. 18.
    Mahmoud MAW, Sultan KS, Amer SM (2003) Order statistics from inverse weibull distribution and associated inference. Comput Stat Data Anal 42:149–163CrossRefGoogle Scholar
  19. 19.
    Nadarajah S, Bakouch HS, Tahmasbi R (2011) A generalized lindley distribution. Sankhya B 73(2):331–359CrossRefGoogle Scholar
  20. 20.
    Sanmel P, Thomas PY (1997) Estimation of location and scale parameters of U-shaped distribution. J Int Soc Agric Stat 50:7594Google Scholar
  21. 21.
    Sankaran M (1970) The discrete Poisson–Lindley distribution. Biometrics 26(1):145–149CrossRefGoogle Scholar
  22. 22.
    Shanker R, Sharma S, Shanker R (2013) A two-parameter lindley distribution for modeling waiting and survival times data. Appl Math 4:363–368CrossRefGoogle Scholar
  23. 23.
    Sultan KS, Childs A, Balakrishnan N (2000) Higher order moments of order statistics from the power function distribution and Edgeworth approximate inference. In: Balakrishnan N, Melas VB, Ermakove S (eds) Advances in stochastic simulation and methods. Birkhauser, Boston, pp 245–282CrossRefGoogle Scholar
  24. 24.
    Sultan KS, Al-Thubyani WS (2016) Higher order moments of order statistics from the Lindley distribution and associated inference. J Stat Comput Simul 86(17):3432–3445CrossRefGoogle Scholar
  25. 25.
    Wasserman GS (2003) Reliability verfication, testing and analysis in engineering design. Marcel Dekker, NewYorkGoogle Scholar
  26. 26.
    Wu SJ, Chen YJ, Chang CT (2007) Statistical inference based on progressively censored samples with random removals from the Burr type XII distribution. J Stat Comput Simul 77:19–27CrossRefGoogle Scholar
  27. 27.
    Zakerzadeh H, Dolati A (2009) Generalized lindley distribution. J Math Ext 3(2):13–25Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of StatisticsCentral University of HaryanaMahendragarhIndia
  2. 2.Department of StatisticsPanjab UniversityChandigarhIndia

Personalised recommendations