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A New Modified Alpha Power Weibull Distribution: Properties, Parameter Estimation and Application

Abstract

A new lifetime distribution called alpha power modified Weibull (APMW) distribution based on the alpha power transformation method has been studied. Various statistical properties of the newly developed distribution including quantiles, moments, stress-strength parameter, Bonferroni and Lorentz curve, residual and reversed residual lifetime function, stress-strength parameter, entropy and order statistics have been obtained. Percentage points of the APMW distribution for different values of the parameters have been obtained. The method of maximum likelihood estimation (MLE) has been used for estimating the parameters. A simulation study has been performed to evaluate the behaviour of the MLEs in terms of the sample size n and revealed that as the value of the sample size increases the value of the mean square error decreases showing the reliability of the estimators. The efficiency and flexibility of the new distribution are illustrated by analysing three real-life data sets. In each case, the APMW distribution provides a better fit indicating that the APMW distribution is a justifiable choice for fitting the considered data sets.

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Availability of data and materials

The study materials and the related data set mentioned in this article are from secondary sources. The authors confirm that the data supporting the findings of this study are available in Aldeni et al. (2017), Proschan (1963) and Kosznik-Biernacka (2018).

Code availability

R Programming Software.

References

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

    Article  Google Scholar 

  • Abdullah M, Ibrahim NA (2017) New extension of exponentiated Weibull distribution with properties and application to survival data. In: Proceeding second ISI regional statistics conference (Session CPS34), Indonesia, 20–24 March

  • Aldeni M, Lee C, Famoye F (2017) Families of distributions arising from the quantile of generalized lambda distribution. J Stat Distrib Appl 4(25):1–18

    MATH  Google Scholar 

  • Almalki SJ (2018) A reduced new modified Weibull distribution. Commun Stat Theory Methods 47(10):2297–2313

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  • Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing: probability models. Holt, Rinehart and Winston, New York

    MATH  Google Scholar 

  • Bebbington M, Lai CD, Zitikis R (2007) A flexible Weibull extension. Reliab Eng Syst Saf 92:719–726

    Article  Google Scholar 

  • Butler RJ, McDonald JB (1989) Using incomplete moments to measure inequality. J Econometr 42(1):109–119

    MathSciNet  Article  Google Scholar 

  • Cordeiro GM, Silva RB (2014) The complementary extended Weibull power series class of distributions. Ciência e Natura, Santa Maria 36:1–13

    Google Scholar 

  • Cordeiro GM, Silva GO, Ortega EMM (2013) The beta-Weibull geometric distribution. Stat A J Theor Appl Stat 47(4):817–834

    MathSciNet  MATH  Google Scholar 

  • Dey S, Sharma VK, Mesfioui M (2017) A new extension of Weibull distribution with application to lifetime data. Ann Data Sci 4(1):31–61

    Article  Google Scholar 

  • Doostmoradi A, Zadkarami MR, Sheykhabad AR (2014) A new modified Weibull distribution and its application. J Stat Res Iran 11:97–118

    Article  Google Scholar 

  • El-Morshedy M, El-Bassiouny AH, El-Gohary A (2017) Exponentiated inverse flexible Weibull extension distribution. J Stat Appl Probab 6(1):169–183

    Article  Google Scholar 

  • Ijaz M, Asim SM, Alamgir MF, Khan SA, Manzoor S (2021) A Gull Alpha Power Weibull distribution with applications to real and simulated data. PLoS ONE 15(6):1–19

    Google Scholar 

  • Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, Vol I, 2nd edn. John Wiley, New York

    MATH  Google Scholar 

  • Kosznik-Biernacka S (2007) Makehams generalised distribution. Comput Methods Sci Technol 13(2):113–120

    Article  Google Scholar 

  • Lai CD, Xie M (2003) A modified Weibull distribution. IEEE Trans Reliab 52(1):33–37

    Article  Google Scholar 

  • Lai CD, Murthy DNP, Xie M (2011) Weibull distributions. John Wiley and Sons, pp. 282–287

  • Mahdavi A, Kundu D (2016) A new method for generating distributions with an application to exponential distribution. Commun Stat Theory Methods 46(13):6543–6557

    MathSciNet  Article  Google Scholar 

  • 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. Biometrica 84(3):641–652

    MathSciNet  Article  Google Scholar 

  • Morais AL, Souza WB (2011) A compound class of Weibull and power series distributions. Comput Stat Data Anal 55(3):1410–1425

    MathSciNet  Article  Google Scholar 

  • Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure rate data. IEEE Trans Reliab 42(2):299–302

    Article  Google Scholar 

  • Murthy DNP, Xie M, Jiang R (2004) Weibull Models. John Wiley & Sons Inc, Publication. https://doi.org/10.1186/s40488-017-0081-4

    Book  MATH  Google Scholar 

  • Nassar M, Alzaatreh A, Mead M, Abo-Kasem O (2017) Alpha power Weibull distribution: properties and applications”. Communication in Statistics-Theory and Methods 46(20):10236–10252

    MathSciNet  Article  Google Scholar 

  • Nassar M, Alzaatreh A, Abo-Kasem O, Mead M, Mansoor M (2018) A new method of generalized distributions based on alpha power transformation with application to cancer data. Ann Data Sci. https://doi.org/10.1007/s40745-018-0144-5

    Article  Google Scholar 

  • Pal M, Ali MM, Woo J (2003) Exponentiated Weibull distribution. Statistica (bologna) 66(2):141–147

    MathSciNet  MATH  Google Scholar 

  • Park S, Park J (2018) A general class of flexible Weibull distributions. Commun Stat Theory Methods 47(4):767–778

    MathSciNet  Article  Google Scholar 

  • Pham H, Lai CD (2007) On recent generalization of the Weibull distribution. IEEE Trans Reliab 56(3):454–458

    Article  Google Scholar 

  • Proschan F (1963) Theoretical explanation of observed decreasing failure rate. Technometrics 5(3):375–383

    Article  Google Scholar 

  • Ramadan DA, Magdy WA (2018) On the alpha-power inverse Weibull distribution. Int J Comput Appl 181(11)

  • Salem AM, Abo-Kasem OE (2011) Estimation for the parameters of the exponentiated Weibull distribution based on progressive hybrid censored samples. Int J Contemporary Math Sci 6(35):1713–1724

    MathSciNet  MATH  Google Scholar 

  • Sarhan AM, Zaindin M (2009) Modified Weibull distribution. Appl Sci 11:123–136

    MathSciNet  MATH  Google Scholar 

  • Silva GO, Ortega EMM, Cordeiro GM (2010) The beta modified Weibull distribution. Lifetime Data Anal 16:409–430

    MathSciNet  Article  Google Scholar 

  • Weibull W (1951) A statistical distribution functions of wide applicability. J Appl Mech 51:293–297

    Article  Google Scholar 

Download references

Acknowledgements

We would like to extend our sincere gratitude to the reviewers for their valuable comments and suggestions. It has greatly improved the manuscript.

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No funds, grants, or other support was received for this study.

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Contributions

This work was carried out in collaboration among the authors. SC, designed the study, performed the statistical analysis and wrote the first draft of the manuscript. BD, managed the literature searches. All authors have contributed to the studies, conception and design which immensely help in the development of this article in all stages of the article formation.

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Correspondence to Bhanita Das.

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Chettri, S., Das, B. & Chakraborty, S. A New Modified Alpha Power Weibull Distribution: Properties, Parameter Estimation and Application. J Indian Soc Probab Stat 22, 417–449 (2021). https://doi.org/10.1007/s41096-021-00111-4

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Keywords

  • Alpha power modified Weibull distribution
  • Moment
  • Quantile
  • Stress-strength reliability
  • Entropy
  • Parameter estimation
  • Simulation