# 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.

## Keywords

Lindley distribution Moments Stress-strength reliability Maximum likelihood estimation## Mathematics Subject Classification

60E05 62F10## Notes

### 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.

## References

- 1.Alzaatreh A, Lee C, Famoye F (2013) A new method for generating families of continuous distributions. Metron 71:63–79CrossRefGoogle Scholar
- 2.Asgharzadeh A, Bakouch HS, Nadarajah S, Sharafi F (2016) A new weighted Lindley distribution with application. Braz J Probab Stat 30(1):1–27CrossRefGoogle 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):75–85CrossRefGoogle Scholar
- 4.Barreto-Souza W, Bakouch HS (2013) A new lifetime model with decreasing failure rate. Statistics 47:465–476CrossRefGoogle Scholar
- 5.Barreto-Souza W, Cordeiro GM, Simas AB (2011) Some results for beta Frechet distribution. Commun Stat Theory Methods 40(5):798–811CrossRefGoogle Scholar
- 6.Bennette S (1983) Log-logistic regression models for survival data. Appl Stat 32:165–171CrossRefGoogle Scholar
- 7.Bonferroni CE (1930) Elmenti di statistica generale. Libreria Seber, FirenzeGoogle Scholar
- 8.Bourguignon M, Silva RB, Cordeiro GM (2014) The Weibull-G family of probability distributions. J Data Sci 12:53–68Google Scholar
- 9.Cordeiro GM, Castro M (2011) A new family of generalized distributions. J Stat Comput Simul 81:883–898CrossRefGoogle 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 83:414–425CrossRefGoogle 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 36:587–602CrossRefGoogle Scholar
- 14.Eugene N, Lee C, Famoye F (2002) Beta-normal distribution and its applications. Commun Stat Theory Methods 31:497–512CrossRefGoogle Scholar
- 15.Gail MH (2009) Applying the Lorenz curve to disease risk to optimize health benefits under cost constraints. Stat Interface 2:117–121CrossRefGoogle Scholar
- 16.Ghitany M, Atieh B, Nadarajah S (2008) Lindley distribution and its application. Math Comput Simul 78:493–506CrossRefGoogle Scholar
- 17.Ghitany M, Al-Mutairi D, Nadarajah S (2008) Zero-truncated Poisson-Lindley distribution and its application. Math Comput Simul 79(3):279–287CrossRefGoogle 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 81:1190–1201CrossRefGoogle Scholar
- 19.Ghitany M, Al-Mutairi D, Balakrishnan N, Al-Enezi L (2013) Power Lindley distribution and associated inference. Comput Stat Data Anal 64:20–33CrossRefGoogle Scholar
- 20.Glaser RE (1980) Bathtub and related failure rate characterization. J Am Stat Assoc 75:667–672CrossRefGoogle Scholar
- 21.Gradshteyn IS, Ryzhik IM (2000) Table of integrals, series, and products, 6th edn. Academic Press, San DiegoGoogle 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 90:2480–2487CrossRefGoogle Scholar
- 23.Han L, Min X, Haijie G, Anupam G, John L, Larry W (2011) Forest density estimation. J Mach Learn Res 12:907–951Google Scholar
- 24.Hoskings JRM (1990) L-moments: analysis and estimation of distribution using linear combinations of order statistics. J R Stat Soc B 52:105–124Google 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, JapanGoogle Scholar
- 26.Jones MC (2015) On families of distributions with shape parameters. Int Stat Rev 83(2):175–192CrossRefGoogle Scholar
- 27.Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New YorkGoogle Scholar
- 28.Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. Wiley, Hoboken, NJCrossRefGoogle Scholar
- 29.Kreitmeier W, Linder T (2011) High-resolution scalar quantization with Rényi entropy constraint. IEEE Trans Inf Theory 57:6837–6859CrossRefGoogle Scholar
- 30.Kus C (2007) A new lifetime distribution. Comput Stat Data Anal 51:4497–4509CrossRefGoogle 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 2:1247–1251CrossRefGoogle 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 5:219–238CrossRefGoogle Scholar
- 33.Lindley D (1958) Fiducial distributions and bayes theorem. J R Stat Soc Ser B 20:102–107Google Scholar
- 34.Lorenz MO (1905) Methods of measuring the concentration of wealth. Publ Am Stat Assoc 9:209–219Google 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):6543–6557CrossRefGoogle 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 7:21–39Google Scholar
- 37.Nadarajah S, Bakouch HS, Tahmasbi R (2011) A generalized lindley distribution. Sankhya B 73(2):331–359Google 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, FranceGoogle Scholar
- 39.Proschan F (1963) Theoretical explanation of observed decreasing failure rate. Technometrics 5:375–383CrossRefGoogle Scholar
- 40.Radice A (2009) Use of the Lorenz curve to quantify statistical non uniformity of sediment transport rate. J Hydraul Eng 10:320–326CrossRefGoogle Scholar
- 41.Rao M, Chen Y, Vemuri BC, Wang F (2004) Cumulative residual entropy: a new measure of information. IEEE Trans Inf Theory 50:1220–1228CrossRefGoogle 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 547–561Google Scholar
- 43.Sankaran M (1970) The discrete poisson-lindley distribution. Biometrics 26(1):145–149CrossRefGoogle Scholar
- 44.Shanker R, Sharma S, Shanker R (2013) A two-parameter lindley distribution for modeling waiting and survival times data. Appl Math 4:363–368CrossRefGoogle 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):5709–5729CrossRefGoogle Scholar
- 46.Shaked M, Shanthikumar JG (1994) Stochastic orders and their applications. Academic Press, BostonGoogle 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 2011:125CrossRefGoogle Scholar
- 48.Tahmasbi R, Rezaei S (2008) A two-parameter lifetime distribution with decreasing failure rate. Comput Stat Data Anal 52:3889–3901CrossRefGoogle Scholar
- 49.Zakerzadeh H, Dolati A (2009) Generalized lindley distribution. J Math Ext 3(2):13–25Google Scholar
- 50.Zografos K, Balakrishnan N (2009) On families of beta- and generalized gamma-generated distributions and associated inference. Stat Methodol 6:344–362CrossRefGoogle Scholar