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Generalized Lindley Distribution Based on Order Statistics and Associated Inference with Application

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Abstract

The generalized Lindley distribution is an important distribution for analyzing the stress–strength reliability models and lifetime data, which is quite flexible and can be used effectively in modeling survival data. It can have increasing, decreasing, upside-down bathtub and bathtub shaped failure rate. In this paper, we derive the exact explicit expressions for the single, double (product), triple and quadruple moments of order statistics from the generalized Lindley distribution. 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. Also, 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.

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References

  1. Arnold BC, Balakrishnan N, Nagaraja HN (2003) A first course in order statistics. Wiley, New York

    Google Scholar 

  2. Asgharzadeh A, Bakouch HS, Nadarajah S, Sharafi F (2016) A new weighted Lindley distribution with application. Braz J Probab Stat 30(1):1–27

    Article  Google Scholar 

  3. Bakouch H, Al-Zahrani B, Al-Shomrani A, Marchi V, Louzada F (2012) An extended lindley distribution. J Korean Stat Soc 41(1):75–85

    Article  Google Scholar 

  4. Balakrishnan N, Zhu X, Al-Zaharani B (2015) Recursive computation of the single and product moments of order statistics for the complementary exponential-geometric distribution. J Stat Comput Simul 85:2187–2201

    Article  Google Scholar 

  5. Barreto-Souza W, Bakouch HS (2013) A new lifetime model with decreasing failure rate. Statistics 47:465–476

    Article  Google Scholar 

  6. Bennette S (1983) Log-logistic regression models for survival data. Appl Stat 32:165–171

    Article  Google Scholar 

  7. Bhaumik DK, Kapur K, Gibbons RD (2009) Testing parameters of a gamma distribution for small samples. Technometrics 51:326–334

    Article  Google Scholar 

  8. Efron B (1988) Logistic regression, survival analysis, and the Kaplan–Meier curve. J Am Stat Assoc 83:414–425

    Article  Google Scholar 

  9. Ghitany M, Atieh B, Nadarajah S (2008) Lindley distribution and its application. Math Comput Simul 78:493–506

    Article  Google Scholar 

  10. 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 81:1190–1201

    Article  Google Scholar 

  11. Ghitany M, Al-Mutairi D, Balakrishnan N, Al-Enezi L (2013) Power lindley distribution and associated inference. Comput Stat Data Anal 64:20–33

    Article  Google Scholar 

  12. Kumar D (2013) Relations for marginal and joint moment generating functions of Marshall–Olkin extended logistic distribution based on lower generalized order statistics and characterization. Am J Math Manag Sci 32:19–39

    Google Scholar 

  13. Kumar D (2017) The Singh–Maddala distribution: properties and estimation. Int J Syst Assur Eng Manag 8:1297–1311

    Article  Google Scholar 

  14. Kumar D, Dey S (2017a) Power generalized Weibull distribution based on order statistics. J Stat Res 51(1):61–78

    Google Scholar 

  15. Kumar D, Dey S (2017b) Relations for moments of generalized order statistics from extended exponential distribution. Am J Math Manag Sci 36:378–400

    Google Scholar 

  16. Kumar D, Goyal A (2019) Order statistics from the power Lindley distribution and associated inference with application. Ann Data Sci. https://doi.org/10.1007/s40745-019-00193-9

    Article  Google Scholar 

  17. Kumar D, Dey S, Nadarajah S (2017) Extended exponential distribution based on order statistics. Commun Stat Theory Methods 46(18):9166–9184

    Article  Google Scholar 

  18. Langlands A, Pocock S, Kerr G, Gore S (1997) Long-term survival of patients with breast cancer: a study of the curability of the disease. Br Med J 2:1247–1251

    Article  Google Scholar 

  19. Lindley D (1958) Fiducial distributions and Bayes theorem. J R Stat Soc Ser B 20:102–107

    Google Scholar 

  20. Meyer J (1987) Two-moment decision models and expected utility maximization. Am Econ Rev 77:421430

    Google Scholar 

  21. Nadarajah S, Bakouch HS, Tahmasbi R (2011) A generalized lindley distribution. Sankhya B 73(2):331–359

    Article  Google Scholar 

  22. Sankaran M (1970) The discrete Poisson–Lindley distribution. Biometrics 26(1):145–149

    Article  Google Scholar 

  23. Shanker R, Sharma S, Shanker R (2013) A two-parameter Lindley distribution for modeling waiting and survival times data. Appl Math 4:363–368

    Article  Google Scholar 

  24. Sultan KS, AL-Thubyani WS (2016) Higher order moments of order statistics from the Lindley distribution and associated inference. J Stat Comput Simul 86:3432–3445

    Article  Google Scholar 

  25. Wasserman GS (2003) Reliability verification, testing and analysis in engineering design. Marcel Dekker, New York

    Google Scholar 

  26. Zakerzadeh H, Dolati A (2009) Generalized Lindley distribution. J Math Ext 3(2):13–25

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and the referee for careful reading and comments which greatly improved the article.

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Authors and Affiliations

  1. Department of Statistics, Central University of Haryana, Mahendragarh, India

    Devendra Kumar

  2. Department of Statistics, Panjab University, Chandigarh, India

    Anju Goyal

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  1. Devendra Kumar
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  2. Anju Goyal
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Correspondence to Devendra Kumar.

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Kumar, D., Goyal, A. Generalized Lindley Distribution Based on Order Statistics and Associated Inference with Application. Ann. Data. Sci. 6, 707–736 (2019). https://doi.org/10.1007/s40745-019-00196-6

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  • DOI: https://doi.org/10.1007/s40745-019-00196-6

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