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A New Generalization of the Extended Exponential Distribution with an Application

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Abstract

We introduce a new lifetime distribution namely, transmuted extended exponential distribution which generalizes the extended exponential distribution proposed by Nadarajah and Haghighi (Statistics 45:543–558, 2011) with an additional parameter using the quadratic rank transmutation map which was studied by Shaw and Buckley (The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map, 2009. arXiv:0901.0434) to provide greater flexibility in modeling data from a practical point of view. In this paper, our main focus is on estimation from frequentist point of view, yet, some statistical and reliability characteristics for the model are derived. We briefly describe different estimation procedures namely, the method of maximum likelihood estimation, maximum product of spacings estimation and least square estimation. Monte Carlo simulations are performed to compare the performance of the proposed methods of estimations for both small and large samples. Finally, the potentiality of the model is analyzed by means of one real data set.

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References

  1. Aarset MV (1987) How to identify bathtub of hazard rate. IEEE Trans Reliab R–36:106–108

    Article  Google Scholar 

  2. Akram M, Hayat A (2014) Comparison of estimators of the Weibull distribution. J Stat Theory Pract 8(2):238–259

    Article  Google Scholar 

  3. Alkasasbeh MR, Raqab MZ (2009) Estimation of the generalized logistic distribution parameters: comparative study. Stat Methodol 6(3):262–279

    Article  Google Scholar 

  4. Aryal GR, Tsokos CP (2009) On the transmuted extreme value distribution with application. Nonlinear Anal Theory Methods Appl 71:1401–1407

    Article  Google Scholar 

  5. Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178

    Google Scholar 

  6. Barakat HM (2015) A new method for adding two parameters to a family of distributions with application to the normal and exponential families. Stat Methods Appl 24(3):359–372

    Article  Google Scholar 

  7. Bonferroni CE (1930) Elmenti di statistica generale. Libreria Seber, Firenze

    Google Scholar 

  8. Cheng RCH, Amin NAK (1979) Maximum product of spacings estimation with applications to the lognormal distribution. Technical report, Department of Mathematics, University of Wales

  9. Cheng RCH, Amin NAK (1983) Estimating parameters in continuous univariate distributions with a shifted origin. J R Stat Soc B 45(3):394–403

    Google Scholar 

  10. Cheng RCH, Iles TC (1987) Corrected maximum likelihood in non-regular problems. J R Stat Soc B 49(1):95–101

    Google Scholar 

  11. Cheng RCH, Traylor L (1995) Non-regular maximum likelihood problems. J R Stat Soc B 57(1):3–44

    Google Scholar 

  12. Dey S, Dey T, Kundu D (2014) Two-parameter Rayleigh distribution: different methods of estimation. Am J Math Manag Sci 33:55–74

    Google Scholar 

  13. Dey S, Ali S, Park C (2015) Weighted exponential distribution: properties and different methods of estimation. J Stat Comput Simul. https://doi.org/10.1080/00949655.2014.992346

    Article  Google Scholar 

  14. Dey S, Kumar D, Park C (2016) Transmuted gamma-mixed Rayleigh distribution: properties and estimation with bladder cancer data example. Model Assist Stat Appl 11:293–313

    Google Scholar 

  15. Dey S, Kumar D, Ramos PD, Louzada F (2017) Exponentiated Chen distribution: properties and estimation. Commun Stat Simul Comput 46:8118–8139

    Article  Google Scholar 

  16. Eugene N, Lee C, Famoye F (2002) The beta-normal distribution and its applications. Commun Stat Theory Methods 31:497–512

    Article  Google Scholar 

  17. Ferreira JTAS, Steel MFJ (2006) A constructive representation of univariate skewed distributions. J Am Stat Assoc 101:823–829

    Article  Google Scholar 

  18. Ghosh K, Jammalamadaka SR (2001) A general estimation method using spacings. J Stat Plan Inference 93:71–82

    Article  Google Scholar 

  19. Gupta RD, Kundu D (2001) Generalized exponential distribution: different method of estimations. J Stat Comput Simul 69(4):315–337

    Article  Google Scholar 

  20. Gupta RD, Kundu D (2007) Generalized exponential distribution: existing results and some recent developments. J Stat Plan Inference 137(11):3537–3547

    Article  Google Scholar 

  21. Gradshteyn IS, Ryzhik IM (2000) Table of integrals, series, and products, 6th edn. Academic Press, San Diego

    Google Scholar 

  22. Jones MC (2004) Families of distributions arising from distributions of order statistics. Test 13:1–43

    Article  Google Scholar 

  23. Juarez SF, Schucany WR (2004) Robust and efficient estimation for the generalized Pareto distribution. Extremes 7:237–251

    Article  Google Scholar 

  24. Khan MS, King R, Irene Lena Hudson IL (2016) Transmuted Kumaraswamy distribution. Stat Transit 17:183–210

    Google Scholar 

  25. Kundu D, Raqab MZ (2005) Generalized Rayleigh distribution: different methods of estimations. Comput Stat Data Anal 49(1):187–200

    Article  Google Scholar 

  26. Kumar D, Dey S, Malik MR, Shahbaz MQ (2017a) Recurrence relations for moments and estimation of parameters of extended exponential distribution based on progressive type-II right censored order statistics. J Stat Theory Appl (In press)

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

    Article  Google Scholar 

  28. Lind NC (1994) Information theory and maximum product of spacings estimation. J R Stat Soc Ser B 56:341–343

    Google Scholar 

  29. Louzadaa F, Romana Mari, Canchob VG (2011) The complementary exponential geometric distribution: model, properties, and a comparison with its counterpart. Comput Stat Data Anal 55:2516–2524

    Article  Google Scholar 

  30. Lu W, Shi D (2012) A new compounding life distribution: the Weibull–Poisson distribution. J Appl Stat 39:21–38

    Article  Google Scholar 

  31. Mahmoud M, Mandouh R (2013) On the transmuted Fréchet distribution. J Appl Sci Res 9(10):5553–5561

    Google Scholar 

  32. Marshall AW, Olkin I (1997) A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652

    Article  Google Scholar 

  33. Mazucheli J, Louzada F, Ghitany M (2013) Comparison of estimation methods for the parameters of the weighted Lindley distribution. Appl Math Comput 220:463–471

    Google Scholar 

  34. Merovci F (2013a) Transmuted generalized Rayleigh distribution. J Stat Appl Probab 2(3):1–12

    Google Scholar 

  35. Merovci F (2013b) Transmuted Lindley distribution. Int J Open Probl Comput Sci Math 6(2):63–72

    Article  Google Scholar 

  36. Merovci F (2013c) Transmuted exponentiated exponential distribution. Math Sci Appl E Notes 1(2):112–122

    Google Scholar 

  37. Merovci F, Puka L (2014) Transmuted Pareto distribution. ProbStat Forum 7:1–11

    Google Scholar 

  38. Merovci F, Elbatal I, Ahmed A (2014) The transmuted generalized inverse Weibull distribution. Austrian J Stat 43(2):119–131

    Article  Google Scholar 

  39. Nadarajah S, Haghighi F (2011) An extension of the exponential distribution. Statistics 45:543–558

    Article  Google Scholar 

  40. Nikulin M, Haghighi F (2006) On the power generalized Weibull family: model for cancer censored data. Metron LXVII:75–86

    Google Scholar 

  41. Peng L, Welsh A (2002) Robust estimation of the generalized Pareto distribution. Extremes 4(1):53–65

    Article  Google Scholar 

  42. Pogány TK (2015) The exponentiated exponential Poisson distribution revisited. Statistics 49:918–929

    Article  Google Scholar 

  43. Proschan F (1963) Theoretical explanation of observed decreasing failure rate. Technometrics 5:375–383

    Article  Google Scholar 

  44. Ranneby B (1984) The maximum spacing method: an estimation method related to the maximum likelihood method. Scand J Stat 11:93–112

    Google Scholar 

  45. Rényi A (1961) On measures of entropy and information. In: Proceedings of 4th Berkeley symposium on mathematical statistics and probability, vol 1. University of California Press, Berkeley, pp 547–561

  46. Shaw WT, Buckley IR (2009) The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. arXiv preprint arXiv:0901.0434

  47. Smith RL (1985) Maximum likelihood estimation in a class of non-regular cases. Biometrika 72(1):67–90

    Article  Google Scholar 

  48. Song KS (2001) Rényi information, log likelihood and an intrinsic distribution measure. J Stat Plan Inference 93(1–2):51–69

    Article  Google Scholar 

  49. Teimouri M, Hoseini SM, Nadarajah S (2013) Comparison of estimation methods for the Weibull distribution. Statistics 47(1):93–109

    Article  Google Scholar 

  50. Tian Y, Tian M, Zhu Q (2014) Transmuted linear exponential distribution: a new generalization of the linear exponential distribution. Commun Stat Simul Comput 43(10):2661–2677

    Article  Google Scholar 

Download references

Acknowledgements

The authors express their sincere thanks to the reviewers and the editors for making some useful suggestions on an earlier version of this manuscript which resulted in this improved version.

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Authors and Affiliations

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

    Devendra Kumar & Manoj Kumar

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  1. Devendra Kumar
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  2. Manoj Kumar
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Correspondence to Devendra Kumar.

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Kumar, D., Kumar, M. A New Generalization of the Extended Exponential Distribution with an Application. Ann. Data. Sci. 6, 441–462 (2019). https://doi.org/10.1007/s40745-018-0181-0

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  • DOI: https://doi.org/10.1007/s40745-018-0181-0

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