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A new family of the continuous distributions: the extended Weibull-G family

Year 2019, Volume: 68 Issue: 1, 248 - 270, 01.02.2019
https://doi.org/10.31801/cfsuasmas.451602

Abstract

In this study, we present a new family of continuous distributions via an extended form of the Weibull distribution. Some special members of the newly defined family are discussed and the new univariate continuous distributions are introduced. The mathematical properties are obtained for any members of the family such as expansions of the density, hazard rate function, quantile function, moments and order statistics. We obtain the distribution parameters by maximum likelihood method. The simulation study to evaluate the performance of the estimated parameters based on the selected member of the this new family is also given. The lifetime data example is discussed to illustrate the applicability of the distribution.

References

  • Abouammoh, A.M., Abdulghani, S.A. and Qamber, I.S. On partial orderings and testing of new better than renewal used classes, Reliab. Eng. Syst. Safety (1994), 43(1), 37-41.
  • Alexander, C., Cordeiro, G.M., Ortega, E.M.M. Sarabia, J.M. Generalized beta-generated distributions, Computational Statistics and Data Analysis (2012), 56(6), 1880-1897.
  • Alizadeh, M., Cordeiro, G.M., de Brito, E. and Demetrio, C.L.B. The beta Marshall-Olkin family of distributions, Journal of Statistical Distributions and Applications (2015), 2(4), DOI: 10.1186/s40488-015-0027-7.
  • Alizadeh, M., Tahir, M.H., Cordeiro, G.M., Zubair, M. and Hamedani, G.G. The Kumaraswamy Marshall-Olkin family of distributions, Journal of the Egyptian Mathematical Society (2015), 23(3), 546-557.
  • Alizadeh, M., Emadi, M., Doostparast, M., Cordeiro, G.M., Ortega, E.M.M. and Pescim, R.R. A new family of distributions: the Kumaraswamy odd log-logistic, properties and applications, Hacettepe Journal of Mathematics and Statistics (2015), 44(6), 1491-1512.
  • Aljarrah, M.A., Lee, C. and Famoye, F. On generating T-X family of distributions using quantile functions, Journal of Statistical Distributions and Applications (2014), 1(2), DOI: 10.1186/2195-5832-1-2.
  • Almalki, S.J. and Nadarajah, S. Modifications of the Weibull distribution: A review, Reliability Engineering and System Safety (2014), 124, 32-55.
  • Alzaatreh, A., Lee, C. and Famoye, F. A new method for generating families of continuous distributions, Metron (2013), 71(1), 63-79.
  • Alzaatreh, A., Lee, C. and Famoye, F. Family of generalized gamma distributions: Properties and applications, Hacettepe Journal of Mathematics and Statistics (2016), 45(3), 869-886.
  • Alzaatreh, A., Lee, C. and Famoye, F. T-normal family of distributions: A new approach to generalize the normal distribution, Journal of Statistical Distributions and Applications (2014), 1(16), DOI: 10.1186/2195-5832-1-16.
  • Alzaghal, A., Famoye, F. and Lee, C. Exponentiated T-X family of distributions with some applications, International Journal of Probability and Statistics (2013), 2(3), 31-49.
  • Batsidis, A. and Lemonte, A.J. On the Harris extended family of distributions, Statistics (2015), 49(6), 1400-1421.
  • Bourguignon, M., Silva, R.B. and Cordeiro, G.M. The Weibull-G family of probability distributions, Journal of Data Science (2014), 12(1), 53-68.
  • Chen, Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics and Probability Letters (2000), 49(2), 155-161.
  • Cordeiro, G.M., Alizadeh, M., Tahir, M.H., Mansoor, M., Bourguignon, M. And Hamedani, G.G. The beta odd log-logistic generalized family of distributions, Hacettepe Journal of Mathematics and Statistics (2016), 45(3), 1175-1202. Cordeiro, G.M. and de Castro, A new family of generalized distributions, Journal of Statistical Computation and Simulation (2011), 81(7), 883-898.
  • Cordeiro, G.M., Ortega, E.M.M. and da Cunha, D.C.C. The exponentiated generalized class of distributions, Journal of Data Science (2013), 11(1), 1-27.
  • Cordeiro, G.M., Ortega, E.M.M., Popovic, B.V. and Pescim, R.R. The Lomax generator of distributions: Properties, minification process and regression model, Applied Mathematics and Computation (2014), 247, 465-486.
  • Cordeiro, G.M., Ortega, E.M.M. and Ramires, T.G. A new generalized Weibull family of distributions: mathematical properties and applications, Journal of Statistical Distributions and Applications (2015), 2(13), DOI 10.1186/s40488-015-0036-6.
  • Cordeiro, G.M., Saboor, A., Khan, M.N, Ozel, G. and Pascoa, M.A.R. The Kumaraswamy Exponential-Weibull Distribution: Theory and Applications, Hacettepe Journal of Mathematics and Statistics (2016), 45(4), 1203-1229. Eugene, N., Lee, C. and Famoye, F. Beta-normal distribution and its applications, Commun. Stat. Theory Methods (1997), 31(4), 497-512.
  • Gupta, R. and Kundu, D. Generalized exponential distribution, Aust. N. Z. J. Statist. (1999), 41(2), 172-188. Hassan, A.S. and Hemeda, S.E. A New Family of Additive Weibull-Generated Distributions, International Journal of Mathematics And its Applications (2016), 4(2), 151-164.
  • Kenney, J.F. Mathematics of Statistics, Chapman and Hall, 1939.
  • Korkmaz, M.Ç. and Genç, A.I. A New Generalized Two-Sided Class of Distributions with an Emphasis on Two-Sided Generalized Normal Distribution, Communications in Statistics Simulation and Computation (2017), 46(2), 1441-1460.
  • Lai, C.C., Murthy, D.N.P. and Xie, M. Weibull distributions, Wiley Interdisciplinary Reviews: Computational Statistics (2011), 3(3), 282-287.
  • <label>MO</label> Marshall, A.N. and Olkin, I. A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families, Biometrika (1997), 84(3), 641-652.
  • Moors, J.J.A. A quantile alternative for kurtosis, Statistician (1998), 37(1), 25-32.
  • Mudholkar, G.S. and Srivastava, D.K. Exponentiated Weibull family for analyzing bathtub failure rate data, IEEE Transactions on Reliability (1993), 42(2), 299-302.
  • Mudholkar, G.S., Srivastava, D.K. and Freimer, M. The exponentiated Weibull family: A reanalysis of the bus-motor failure data, Technometrics (1995), 37(4), 436-445.
  • Nadarajah, S. and Kotz, S. The exponentiated type distributions, Acta Applicandae Mathematica (2006), 92(2), 97-111.
  • Peng, X. and Yan, Z. Estimation and application for a new extended Weibull distribution, Reliability Engineering and System Safety (2014), 121, 34-42.
  • Phani, K.K.A new modified Weibull distribution function, Communications of the American Ceramic Society (1987), 70(8), 182-184.
  • R Development Core Team R: A Language and Environment for Statistical Computing, Vienna, Austria, 2012.
  • Sarhan, A.M., Ahmad, A.A. and Ibtesam, A. Exponentiated generalized linear exponential distribution, Applied Mathematical Modelling (2013), 37(5), 2838-2849.
  • Smith, R.M. and Bain, L.J. An exponential power life-testing distribution, Communications in Statistics - Theory and Methods (1975), 4(5), 469-481.
  • Tahir, M.H., Cordeiro, G.M., Alzaatreh, A., Mansoor, M. and Zubair, M. The Logistic-X family of distributions and its applications,, Communications in Statistics-Theory and Methods (2016), 45(24), 7326-7349.
  • Tahir, M.H. and Nadarajah, S. Parameter induction in continuous univariate distribution: Well-established G families, Ann. Braz. Acad. Sci., (2015), 87, 539-568.
  • Tahir, M.H., Zubair, M., Mansoor, M., Cordeiro, G.M., Alizadeh, M. and Hamedani, G.G. A new Weibull-G family of distributions, Hacettepe Journal of Mathematics and Statistics (2016), 45(2), 629-647.
  • Van dorp, J. R. and Kotz, S. The standard two-sided power distribution and its properties: with applications in financial engineering, The American Statistician (2002), 56(2), 90-99.
  • Zografos, K. and Balakrishnan, N. On families of beta- and generalized gamma-generated distributions and associated inference, Statistical Methodology (2009), 6(4), 344-362.
Year 2019, Volume: 68 Issue: 1, 248 - 270, 01.02.2019
https://doi.org/10.31801/cfsuasmas.451602

Abstract

References

  • Abouammoh, A.M., Abdulghani, S.A. and Qamber, I.S. On partial orderings and testing of new better than renewal used classes, Reliab. Eng. Syst. Safety (1994), 43(1), 37-41.
  • Alexander, C., Cordeiro, G.M., Ortega, E.M.M. Sarabia, J.M. Generalized beta-generated distributions, Computational Statistics and Data Analysis (2012), 56(6), 1880-1897.
  • Alizadeh, M., Cordeiro, G.M., de Brito, E. and Demetrio, C.L.B. The beta Marshall-Olkin family of distributions, Journal of Statistical Distributions and Applications (2015), 2(4), DOI: 10.1186/s40488-015-0027-7.
  • Alizadeh, M., Tahir, M.H., Cordeiro, G.M., Zubair, M. and Hamedani, G.G. The Kumaraswamy Marshall-Olkin family of distributions, Journal of the Egyptian Mathematical Society (2015), 23(3), 546-557.
  • Alizadeh, M., Emadi, M., Doostparast, M., Cordeiro, G.M., Ortega, E.M.M. and Pescim, R.R. A new family of distributions: the Kumaraswamy odd log-logistic, properties and applications, Hacettepe Journal of Mathematics and Statistics (2015), 44(6), 1491-1512.
  • Aljarrah, M.A., Lee, C. and Famoye, F. On generating T-X family of distributions using quantile functions, Journal of Statistical Distributions and Applications (2014), 1(2), DOI: 10.1186/2195-5832-1-2.
  • Almalki, S.J. and Nadarajah, S. Modifications of the Weibull distribution: A review, Reliability Engineering and System Safety (2014), 124, 32-55.
  • Alzaatreh, A., Lee, C. and Famoye, F. A new method for generating families of continuous distributions, Metron (2013), 71(1), 63-79.
  • Alzaatreh, A., Lee, C. and Famoye, F. Family of generalized gamma distributions: Properties and applications, Hacettepe Journal of Mathematics and Statistics (2016), 45(3), 869-886.
  • Alzaatreh, A., Lee, C. and Famoye, F. T-normal family of distributions: A new approach to generalize the normal distribution, Journal of Statistical Distributions and Applications (2014), 1(16), DOI: 10.1186/2195-5832-1-16.
  • Alzaghal, A., Famoye, F. and Lee, C. Exponentiated T-X family of distributions with some applications, International Journal of Probability and Statistics (2013), 2(3), 31-49.
  • Batsidis, A. and Lemonte, A.J. On the Harris extended family of distributions, Statistics (2015), 49(6), 1400-1421.
  • Bourguignon, M., Silva, R.B. and Cordeiro, G.M. The Weibull-G family of probability distributions, Journal of Data Science (2014), 12(1), 53-68.
  • Chen, Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics and Probability Letters (2000), 49(2), 155-161.
  • Cordeiro, G.M., Alizadeh, M., Tahir, M.H., Mansoor, M., Bourguignon, M. And Hamedani, G.G. The beta odd log-logistic generalized family of distributions, Hacettepe Journal of Mathematics and Statistics (2016), 45(3), 1175-1202. Cordeiro, G.M. and de Castro, A new family of generalized distributions, Journal of Statistical Computation and Simulation (2011), 81(7), 883-898.
  • Cordeiro, G.M., Ortega, E.M.M. and da Cunha, D.C.C. The exponentiated generalized class of distributions, Journal of Data Science (2013), 11(1), 1-27.
  • Cordeiro, G.M., Ortega, E.M.M., Popovic, B.V. and Pescim, R.R. The Lomax generator of distributions: Properties, minification process and regression model, Applied Mathematics and Computation (2014), 247, 465-486.
  • Cordeiro, G.M., Ortega, E.M.M. and Ramires, T.G. A new generalized Weibull family of distributions: mathematical properties and applications, Journal of Statistical Distributions and Applications (2015), 2(13), DOI 10.1186/s40488-015-0036-6.
  • Cordeiro, G.M., Saboor, A., Khan, M.N, Ozel, G. and Pascoa, M.A.R. The Kumaraswamy Exponential-Weibull Distribution: Theory and Applications, Hacettepe Journal of Mathematics and Statistics (2016), 45(4), 1203-1229. Eugene, N., Lee, C. and Famoye, F. Beta-normal distribution and its applications, Commun. Stat. Theory Methods (1997), 31(4), 497-512.
  • Gupta, R. and Kundu, D. Generalized exponential distribution, Aust. N. Z. J. Statist. (1999), 41(2), 172-188. Hassan, A.S. and Hemeda, S.E. A New Family of Additive Weibull-Generated Distributions, International Journal of Mathematics And its Applications (2016), 4(2), 151-164.
  • Kenney, J.F. Mathematics of Statistics, Chapman and Hall, 1939.
  • Korkmaz, M.Ç. and Genç, A.I. A New Generalized Two-Sided Class of Distributions with an Emphasis on Two-Sided Generalized Normal Distribution, Communications in Statistics Simulation and Computation (2017), 46(2), 1441-1460.
  • Lai, C.C., Murthy, D.N.P. and Xie, M. Weibull distributions, Wiley Interdisciplinary Reviews: Computational Statistics (2011), 3(3), 282-287.
  • <label>MO</label> Marshall, A.N. and Olkin, I. A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families, Biometrika (1997), 84(3), 641-652.
  • Moors, J.J.A. A quantile alternative for kurtosis, Statistician (1998), 37(1), 25-32.
  • Mudholkar, G.S. and Srivastava, D.K. Exponentiated Weibull family for analyzing bathtub failure rate data, IEEE Transactions on Reliability (1993), 42(2), 299-302.
  • Mudholkar, G.S., Srivastava, D.K. and Freimer, M. The exponentiated Weibull family: A reanalysis of the bus-motor failure data, Technometrics (1995), 37(4), 436-445.
  • Nadarajah, S. and Kotz, S. The exponentiated type distributions, Acta Applicandae Mathematica (2006), 92(2), 97-111.
  • Peng, X. and Yan, Z. Estimation and application for a new extended Weibull distribution, Reliability Engineering and System Safety (2014), 121, 34-42.
  • Phani, K.K.A new modified Weibull distribution function, Communications of the American Ceramic Society (1987), 70(8), 182-184.
  • R Development Core Team R: A Language and Environment for Statistical Computing, Vienna, Austria, 2012.
  • Sarhan, A.M., Ahmad, A.A. and Ibtesam, A. Exponentiated generalized linear exponential distribution, Applied Mathematical Modelling (2013), 37(5), 2838-2849.
  • Smith, R.M. and Bain, L.J. An exponential power life-testing distribution, Communications in Statistics - Theory and Methods (1975), 4(5), 469-481.
  • Tahir, M.H., Cordeiro, G.M., Alzaatreh, A., Mansoor, M. and Zubair, M. The Logistic-X family of distributions and its applications,, Communications in Statistics-Theory and Methods (2016), 45(24), 7326-7349.
  • Tahir, M.H. and Nadarajah, S. Parameter induction in continuous univariate distribution: Well-established G families, Ann. Braz. Acad. Sci., (2015), 87, 539-568.
  • Tahir, M.H., Zubair, M., Mansoor, M., Cordeiro, G.M., Alizadeh, M. and Hamedani, G.G. A new Weibull-G family of distributions, Hacettepe Journal of Mathematics and Statistics (2016), 45(2), 629-647.
  • Van dorp, J. R. and Kotz, S. The standard two-sided power distribution and its properties: with applications in financial engineering, The American Statistician (2002), 56(2), 90-99.
  • Zografos, K. and Balakrishnan, N. On families of beta- and generalized gamma-generated distributions and associated inference, Statistical Methodology (2009), 6(4), 344-362.
There are 38 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Mustafa Çağatay Korkmaz 0000-0003-3302-0705

Publication Date February 1, 2019
Submission Date April 5, 2017
Acceptance Date August 7, 2018
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Korkmaz, M. Ç. (2019). A new family of the continuous distributions: the extended Weibull-G family. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 248-270. https://doi.org/10.31801/cfsuasmas.451602
AMA Korkmaz MÇ. A new family of the continuous distributions: the extended Weibull-G family. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):248-270. doi:10.31801/cfsuasmas.451602
Chicago Korkmaz, Mustafa Çağatay. “A New Family of the Continuous Distributions: The Extended Weibull-G Family”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 248-70. https://doi.org/10.31801/cfsuasmas.451602.
EndNote Korkmaz MÇ (February 1, 2019) A new family of the continuous distributions: the extended Weibull-G family. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 248–270.
IEEE M. Ç. Korkmaz, “A new family of the continuous distributions: the extended Weibull-G family”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 248–270, 2019, doi: 10.31801/cfsuasmas.451602.
ISNAD Korkmaz, Mustafa Çağatay. “A New Family of the Continuous Distributions: The Extended Weibull-G Family”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 248-270. https://doi.org/10.31801/cfsuasmas.451602.
JAMA Korkmaz MÇ. A new family of the continuous distributions: the extended Weibull-G family. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:248–270.
MLA Korkmaz, Mustafa Çağatay. “A New Family of the Continuous Distributions: The Extended Weibull-G Family”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 248-70, doi:10.31801/cfsuasmas.451602.
Vancouver Korkmaz MÇ. A new family of the continuous distributions: the extended Weibull-G family. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):248-70.

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