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Inference for the unit-Gompertz model based on record values and inter-record times with an application

  • Devendra Kumar
  • Sanku Dey
  • Ehsan Ormoz
  • S. M. T. K. MirMostafaeeEmail author
Article
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Abstract

Mazucheli et al. (Statistica 79:25–43, 2019) introduced a new transformed model called the unit-Gompertz (UG) distribution which exhibits right-skewed (uni-modal) and reversed-J shaped density and its hazard rate function can be increasing and increasing-decreasing-increasing. They worked on the estimation of the model parameters based on complete data sets. In this paper, by using lower record values and inter-record times, we develop inference procedures for the estimation of the parameters and prediction of future record values for the UG distribution. First, we derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariances between two lower record values. Next, we obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Further, we obtain the Bayes estimators under the assumption that the model parameters follow a joint bivariate density function. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential) loss functions with the help of the Tierney–Kadane’s method and Metropolis–Hastings algorithm. Finally, we compute Bayesian point predictors for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.

Keywords

Unit-Gompertz model Maximum likelihood technique Bayesian viewpoint Lower record values 

Mathematics Subject Classification

62F10 62F15 

Notes

Acknowledgements

We are very grateful to the Referee for his/her valuable comments.

References

  1. 1.
    Ahsanullah, M.: Record Statistics. Nova Science Publishers, New York (1995)zbMATHGoogle Scholar
  2. 2.
    Al-Hussaini, E.K., Jaheen, Z.F.: Bayesian estimation of the parameters, reliability and failure rate functions of the Burr type XII failure model. J. Stat. Comput. Simul. 41, 31–40 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Amini, M., MirMostafaee, S.M.T.K.: Interval prediction of order statistics based on records by employing inter-record times: a study under two parameter exponential distribution. Metodološki Zveski 13, 1–15 (2016)Google Scholar
  4. 4.
    Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: Records. John Wiley and Sons, New York (1998)CrossRefGoogle Scholar
  5. 5.
    Balakrishnan, N., Ahsanullah, M.: Recurrence relations for single and product moments of record values from generalized Pareto distribution. Commun. Stat. Theory Methods 23, 2841–2852 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chandler, K.N.: The distribution and frequency of record values. J. R. Stat. Soc. Ser. B (Methodol.) 14, 220–228 (1952)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, M.-H., Shao, Q.-M.: Monte Carlo estimation of Bayesian credible and HPD intervals. J. Comput. Graph. Stat. 8, 69–92 (1999)MathSciNetGoogle Scholar
  8. 8.
    Cook, D.O., Kieschnick, R., McCullough, B.D.: Regression analysis of proportions in finance with self selection. J. Empir. Finance 15, 860–867 (2008)CrossRefGoogle Scholar
  9. 9.
    Dey, S., Dey, T., Salehi, M., Ahmadi, J.: Bayesian inference of generalized exponential distribution based on lower record values. Am. J. Math. Manag. Sci. 32, 1–18 (2013)Google Scholar
  10. 10.
    Dey, S., Pradhan, B.: Generalized inverted exponential distribution under hybrid censoring. Stat. Methodol. 18, 101–114 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fallah, A., Asgharzadeh, A., MirMostafaee, S.M.T.K.: On the Lindley record values and associated inference. J. Stat. Theory Appl. 17, 686–702 (2018)MathSciNetGoogle Scholar
  12. 12.
    Genç, Aİ.: Estimation of \(P(X > Y )\) with Topp–Leone distribution. J. Stat. Comput. Simul. 83, 326–339 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ghitany, M.E., Al-Mutairi, D.K., Balakrishnan, N., Al-Enezi, L.J.: Power Lindley distribution and associated inference. Comput. Stat. Data Anal. 64, 20–33 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Glick, N.: Breaking records and breaking boards. Am. Math. Mon. 85, 2–26 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kızılaslan, F., Nadar, M.: Estimation with the generalized exponential distribution based on record values and inter-record times. J. Stat. Comput. Simul. 85, 978–999 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kızılaslan, F., Nadar, M.: Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times. J. Stat. Comput. Simul. 86, 2471–2493 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kumar, D.: Explicit expressions and statistical inference of generalized Rayleigh distribution based on lower record values. Math. Methods Stat. 24, 225–241 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kumar, D.: \(K\)-th lower record values from Dagum distribution and characterization. Discuss. Math. Probab. Stat. 36, 25–41 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kumar, D., Dey, T., Dey, S.: Statistical inference of exponentiated moment exponential distribution based on lower record values. Commun. Math. Stat. 5, 231–260 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Marinho, P.R.D., Bourguignon, M., Dias, C.R.B.: Adequacy Model: adequacy of probabilistic models and generation of pseudo-random numbers, R package version 1.0.8. https://CRAN.R-project.org/package=AdequacyModel (2013)
  21. 21.
    Mazucheli, J., Menezes, A.F., Dey, S.: Unit-Gompertz distribution with applications. Statistica 79, 25–43 (2019)Google Scholar
  22. 22.
    MirMostafaee, S.M.T.K., Asgharzadeh, A., Fallah, A.: Record values from NH distribution and associated inference. Metron 74, 37–59 (2016a)MathSciNetCrossRefGoogle Scholar
  23. 23.
    MirMostafaee, S.M.T.K., Mahdizadeh, M., Aminzadeh, M.: Bayesian inference for the Topp–Leone distribution based on lower \(k\)-record values. Jpn. J. Ind. Appl. Math. 33, 637–669 (2016b)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nadar, M., Kızılaslan, F.: Estimation and prediction of the Burr type XII distribution based on record values and inter-record times. J. Stat. Comput. Simul. 85, 3297–3321 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nadar, M., Papadopoulos, A., Kızılaslan, F.: Statistical analysis for Kumaraswamy’s distribution based on record data. Stat. Pap. 54, 355–369 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nevzorov, V.B.: Records: Mathematical Theory. American Mathematical Society, Providence (2001)Google Scholar
  27. 27.
    Pak, A., Dey, S.: Statistical inference for the power Lindley model based on record values and inter-record times. J. Comput. Appl. Math. 347, 156–172 (2019)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Papke, L.E., Wooldridge, J.M.: Econometric methods for fractional response variables with an application to 401(k) plan participation rates. J. Appl. Econ. 11, 619–632 (1996)CrossRefGoogle Scholar
  29. 29.
    Plummer, M., Best, N., Cowles, K., Vines, K.: CODA: convergence diagnosis and output analysis for MCMC. R News 6, 7–11 (2006)Google Scholar
  30. 30.
    Plummer, M., Best, N., Cowles, K., Vines, K., Sarkar, D., Bates, D., Almond, R., Magnusson, A.: CODA: output analysis and diagnostics for MCMC, R package version 0.19-2. https://CRAN.R-project.org/package=coda (2018)
  31. 31.
    R Core Team.: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2018)Google Scholar
  32. 32.
    Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)CrossRefGoogle Scholar
  33. 33.
    Ripley, B., Venables, B., Bates, D.M., Hornik, K., Gebhardt, A., Firth, D.: MASS: support functions and datasets for Venables and Ripley’s MASS, R package version 7.3-50. https://CRAN.R-project.org/package=MASS (2018)
  34. 34.
    Samaniego, F.J., Whitaker, L.R.: On estimating population characteristics from record-breaking observations. I. Parametric results. Naval Res. Logist. Q. 33, 531–543 (1986)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Shorrock, R.W.: Record values and inter-record times. J. Appl. Probab. 10, 543–555 (1973)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Tierney, L., Kadane, J.B.: Accurate approximations for posterior moments and marginal densities. J. Am. Stat. Assoc. 81, 82–86 (1986)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Venables, W.N., Ripley, B.D.: Modern Applied Statistics with S, 4th edn. Springer, New York (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  • Devendra Kumar
    • 1
  • Sanku Dey
    • 2
  • Ehsan Ormoz
    • 3
  • S. M. T. K. MirMostafaee
    • 4
    Email author
  1. 1.Department of StatisticsCentral University of HaryanaMahendergarhIndia
  2. 2.Department of StatisticsSt. Anthony’s CollegeShillongIndia
  3. 3.Department of Statistics, Mashhad BranchIslamic Azad UniversityMashhadIran
  4. 4.Department of Statistics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran

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