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Alpha-Power Transformed Lindley Distribution: Properties and Associated Inference with Application to Earthquake Data

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Abstract

The Lindley distribution has been generalized by many authors in recent years. A new two-parameter distribution with decreasing failure rate is introduced, called Alpha Power Transformed Lindley (APTL, in short, henceforth) distribution that provides better fits than the Lindley distribution and some of its known generalizations. The new model includes the Lindley distribution as a special case. Various properties of the proposed distribution, including explicit expressions for the ordinary moments, incomplete and conditional moments, mean residual lifetime, mean deviations, L-moments, moment generating function, cumulant generating function, characteristic function, Bonferroni and Lorenz curves, entropies, stress-strength reliability, stochastic ordering, statistics and distribution of sums, differences, ratios and products are derived. The new distribution can have decreasing increasing, and upside-down bathtub failure rates function depending on its parameters. The model parameters are obtained by the method of maximum likelihood estimation. Also, we obtain the confidence intervals of the model parameters. A simulation study is carried out to examine the bias and mean squared error of the maximum likelihood estimators of the parameters. Finally, two data sets have been analyzed to show how the proposed models work in practice.

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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. 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 ex- tended lindley distribution. J Korean Stat Soc 41(1):75–85

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Barreto-Souza W, Cordeiro GM, Simas AB (2011) Some results for beta Frechet distribution. Commun Stat Theory Methods 40(5):798–811

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  8. Bourguignon M, Silva RB, Cordeiro GM (2014) The Weibull-G family of probability distributions. J Data Sci 12:53–68

    Google Scholar 

  9. Cordeiro GM, Castro M (2011) A new family of generalized distributions. J Stat Comput Simul 81:883–898

    Article  Google Scholar 

  10. Dey S, Alzaatreh A, Zhang C, Kumar D (2017a) A new extension of generalized exponential distribution with application to ozone data, OZONE: Sci Eng. https://doi.org/10.1080/01919512.2017.1308817

  11. Dey S, Sharma VK, Mesfioui M (2017b) A new extension of weibull distribution with application to lifetime data. Ann Data Sci https://doi.org/10.1007/s40745-016-0094-8

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

    Article  Google Scholar 

  13. 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 

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

    Article  Google Scholar 

  15. 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 

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

    Article  Google Scholar 

  17. Ghitany M, Al-Mutairi D, Nadarajah S (2008) Zero-truncated Poisson-Lindley distribution and its application. Math Comput Simul 79(3):279–287

    Article  Google Scholar 

  18. 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 

  19. 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 

  20. Glaser RE (1980) Bathtub and related failure rate characterization. J Am Stat Assoc 75:667–672

    Article  Google Scholar 

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

    Google Scholar 

  22. Groves-Kirkby CJ, Denman AR, Phillips PS (2009) Lorenz Curve and Gini Coefficien: novel tools for analysing seasonal variation of environmental radon gas. J Environ Manage 90:2480–2487

    Article  Google Scholar 

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

    Google Scholar 

  24. Hoskings JRM (1990) L-moments: analysis and estimation of distribution using linear combinations of order statistics. J R Stat Soc B 52:105–124

    Google Scholar 

  25. Jelinek HF, Pham P, Struzik ZR, Spence I (2007) Short term ECG recording for the identifiction of cardiac autonomic neuropathy in people with diabetes mellitus. In: Proceedings of the 19th international conference on noise and fluctuations, Tokyo, Japan

  26. Jones MC (2015) On families of distributions with shape parameters. Int Stat Rev 83(2):175–192

    Article  Google Scholar 

  27. Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New York

    Google Scholar 

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

    Book  Google Scholar 

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

    Article  Google Scholar 

  30. Kus C (2007) A new lifetime distribution. Comput Stat Data Anal 51:4497–4509

    Article  Google Scholar 

  31. 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 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  35. Mahdavi A, Kundu D (2017) A new method for generating distributions with an application to exponential distribution. Commun Stat Theory Methods 46(13):6543–6557

    Article  Google Scholar 

  36. 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 

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

    Google Scholar 

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

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  41. Rao M, Chen Y, Vemuri BC, Wang F (2004) Cumulative residual entropy: a new measure of information. IEEE Trans Inf Theory 50:1220–1228

    Article  Google Scholar 

  42. Rényi A (1961) On measures of entropy and information. Procedings of fourth berkeley symposium mathematics statistics and probability, vol 1. University of California Press, Berkeley, pp 547–561

    Google Scholar 

  43. Sankaran M (1970) The discrete poisson-lindley distribution. Biometrics 26(1):145–149

    Article  Google Scholar 

  44. 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 

  45. Sharma VK, Singh SK, Singh U, Merovci F (2016) The generalized inverse lindley distribution: a new inverse statistical model for the study of upside-down bathtub data. Commun Stat Theory Methods 45(19):5709–5729

    Article  Google Scholar 

  46. Shaked M, Shanthikumar JG (1994) Stochastic orders and their applications. Academic Press, Boston

    Google Scholar 

  47. 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 

  48. Tahmasbi R, Rezaei S (2008) A two-parameter lifetime distribution with decreasing failure rate. Comput Stat Data Anal 52:3889–3901

    Article  Google Scholar 

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

    Google Scholar 

  50. Zografos K, Balakrishnan N (2009) On families of beta- and generalized gamma-generated distributions and associated inference. Stat Methodol 6:344–362

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the Editor, Associate Editor and anonymous Referee for careful reading and for comments which greatly improved the paper.

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Authors and Affiliations

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

    Sanku Dey

  2. Department of Mathematics and Statistics, University of North Carolina, Wilmington, NC, USA

    Indranil Ghosh

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

    Devendra Kumar

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  1. Sanku Dey
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  2. Indranil Ghosh
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  3. Devendra Kumar
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Correspondence to Devendra Kumar.

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Dey, S., Ghosh, I. & Kumar, D. Alpha-Power Transformed Lindley Distribution: Properties and Associated Inference with Application to Earthquake Data. Ann. Data. Sci. 6, 623–650 (2019). https://doi.org/10.1007/s40745-018-0163-2

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  • DOI: https://doi.org/10.1007/s40745-018-0163-2

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