Skip to main content
SpringerLink
Log in

\(\alpha \) Logarithmic Transformed Family of Distributions with Application

  • Published:
Annals of Data Science Aims and scope Submit manuscript

Abstract

In this paper, a new three-parameter distribution, called \(\alpha \) logarithmic transformed generalized exponential distribution (\(\alpha LTGE\)) is proposed. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, moment generating function, mean deviation about the mean and median, mean residual life, Bonferroni curve, Lorenz curve, Gini index, Rényi entropy, stochastic ordering and order statistics are derived. It appears to be a distribution capable of allowing monotonically increasing, decreasing, bathtub and upside-down bathtub shaped hazard rates depending on its parameters. The maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non-linear equations only. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance covariance matrix. Finally, two empirical applications of the new model to real data are presented for illustrative purposes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Alzaatreh A, Lee C, Famoye F (2013) A new method for generating families of continuous distributions. Metron 71:63–79

    Article  Google Scholar 

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

    Google Scholar 

  3. Azzalini A (1986) Further results on a class of distributions which includes the normal ones. Statistica 46:199–208

    Google Scholar 

  4. Barreto-Souza WM, Cordeiro GM, Simas AB (2011) Some results for beta Fréchet distribution. Commun Stat Theory Methods 40:798–811

    Article  Google Scholar 

  5. Barreto-Souza W, Santos AHS, Cordeiro GM (2010) The beta generalized exponential distribution. J Stat Comput Simul 80(2):159–172

    Article  Google Scholar 

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

    Google Scholar 

  7. Casella G, Berger RL (1990) Statistical inference. Brooks/Cole Publishing Company, California

    Google Scholar 

  8. Cooray K, Ananda MMA (2005) Modeling actuarial data with a composite lognormal-Pareto model. Scand Actuar J 5:321–334

    Article  Google Scholar 

  9. Dey S, Alzaatreh A, Zhang C, Kumar D (2017) A new extension of generalized exponential distribution with application to Ozone data. Ozone Sci Eng. doi:10.1080/01919512.2017.1308817

    Google Scholar 

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

    Article  Google Scholar 

  11. Egghe L (2002) Development of hierarchy theory for digraphs using concentration theory based on a new type of Lorenz curve. Math Comput Modell 36:587–602

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Gail MH (2009) Applying the Lorenz curve to disease risk to optimize health benefits under cost constraints. Stat Interface 2:117–121

    Article  Google Scholar 

  14. Gradshteyn IS, Ryzhik IM (2014) Tables of integrals, series and products, 8th edn. Academic Press, New York

    Google Scholar 

  15. Gupta RD, Kundu D (1999) Generalized exponential distribution. Aust N Z J Stat 41:173–188

    Article  Google Scholar 

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

    Article  Google Scholar 

  17. Groves-Kirkby CJ, Denman AR, Phillips PS (2009) Lorenz curve and Gini coefficient: novel tools for analysing seasonal variation of environmental radon gas. J Environ Manag 90:24802487

    Article  Google Scholar 

  18. Han L, Min X, Haijie G, Anupam G, John L, Larry W (2011) Forest density estimation. J Mach Lear Res 12:907–951

    Google Scholar 

  19. Jelinek HF, Tarvainen MP, Cornforth DJ (2012) Rényi entropy in identification of cardiac autonomic neuropathy in diabetes. In: Proceedings of the 39th conference on computing in cardiology, pp 909–911

  20. Kenney JF, Keeping E (1962) Mathematics of statistics. D. Van Nostrand Company, Princeton

    Google Scholar 

  21. Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. Wiley, Hoboken

    Book  Google Scholar 

  22. Kotz S, Nadarajah S (2000) Extreme value distributions: theory and applications. Imperial College Press, London

    Book  Google Scholar 

  23. Kreitmeier W, Linder T (2011) High-resolution scalar quantization with Rényi entropy constraint. IEEE Trans Inf Theory 57:6837–6859

    Article  Google Scholar 

  24. Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New York

    Google Scholar 

  25. Lee C, Famoye F, Alzaatreh A (2013) Methods for generating families of univariate continuous distributions in the recent decades. Wiley Interdiscip Rev Comput Stat 5:219–238

    Article  Google Scholar 

  26. Lorenz MO (1905) Methods of measuring the concentration of wealth. Q Publ Am Stat Assoc 9:209–219

    Google Scholar 

  27. Mahdavi A, Kundu D (2016) A new method for generating distributions with an application to exponential distribution. Commun Stat Theory Methods. doi:10.1080/03610926.2015.1130839

    Google Scholar 

  28. Maldonado A, Ocón RP, Herrera A (2007) Depression and cognition: new insights from the lorenz curve and the gini index. Int J Clin Health Psychol 7:21–39

    Google Scholar 

  29. Maurya SK, Kaushik A, Singh RK, Singh SK, Singh U (2016) A new method of proposing distribution and its application to real data. Imp J Interdiscip Res 2(6):1331–1338

    Google Scholar 

  30. 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 

  31. Mead M (2014a) An extended Pareto distribution. Pak J Stat Oper Res 10(3):313–329

    Article  Google Scholar 

  32. Mead M (2014b) A new generalization of Burr XII distribution. J Stat Adv Theory Appl 12(2):53–73

    Google Scholar 

  33. Merovci F (2013) Transmuted exponentiated exponential distribution. Math Sci Appl E Notes 1(2):11–122

    Google Scholar 

  34. Moors JJA (1988) A quantile alternative for kurtosis. J R Stat Soc Ser D Stat 37:25–32

    Google Scholar 

  35. Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans Reliab 42:299–302

    Article  Google Scholar 

  36. Nassar MM, Nada NK (2011) The beta generalized Pareto distribution. J Stat Adv Theory Appl 6:1–17

    Google Scholar 

  37. Pappas V, Adamidis K, Loukas S (2012) A family of lifetime distributions. Int J Qual Stat Reliab. doi:10.1155/2012/760687

    Google Scholar 

  38. Popescu TD, Aiordachioaie D (2013) Signal segmentation in time-frequency plane using Renyi entropy—application in seismic signal processing. In: 2013 conference on control and fault-tolerant systems (SysTol), Oct 9–11, 2013. Nice, France

  39. Radice A (2009) Use of the Lorenz curve to quantify statistical nonuniformity of sediment transport rate. J Hydraul Eng 10:320–326

    Article  Google Scholar 

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

  41. Ristić MM, Kundu D (2015) Marshall–Olkin generalized exponential distribution. Metron 73(3):317–333

    Article  Google Scholar 

  42. Sucic V, Saulig N, Boashash B (2011) Estimating the number of components of a multicomponent nonstationary signal using the short-term time-frequency Rényi entropy. J Adv Signal Process 2011:125

    Article  Google Scholar 

  43. Tahir MH, Cordeiro GM, Alizadeh M, Muhammad Mansoor M, Zubair M, Gholamhossein GH (2015) The odd generalized exponential family of distributions with applications. J Stat Distrib Appl 2:1–28

    Article  Google Scholar 

  44. Tahir M, Hussain M, Cordeiro G, Hamedani G, Mansoor M, Zubair M (2016) The Gumbel-Lomax distribution: properties and applications. J Stat Theory Appl 15(1):61–79

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, 793001, India

    Sanku Dey

  2. Department of Statistics, Faculty of Commerce, Zagazig University, Zagazig, Egypt

    Mazen Nassar

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

    Devendra Kumar

Authors
  1. Sanku Dey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mazen Nassar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Devendra Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Devendra Kumar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dey, S., Nassar, M. & Kumar, D. \(\alpha \) Logarithmic Transformed Family of Distributions with Application. Ann. Data. Sci. 4, 457–482 (2017). https://doi.org/10.1007/s40745-017-0115-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40745-017-0115-2

Keywords

Search

Navigation