A new generalized exponentiated Weibull model called Gumbel-exponentiated Weibull{Logistic}
distribution is introduced and studied. The new distribution extends the
exponentiated Weibull distribution with additional parameters and bimodal
densities. Some new and earlier distributions formed the sub-models of the
proposed distribution. The mathematical properties of the new distribution
including expressions for the hazard function, survival function, moments,
order statistics, mean deviation and absolute mean deviation from the mean, and
entropy were derived. Monte Carlo simulation study was carried out to assess
the finite sample behavior of the parameter estimates by maximum likelihood estimation
approach. The superiority of the new generalized exponentiated Weibull
distribution over some competing distributions was proved empirically using the
fitted results from three real life datasets.
References
[1]
Pearson, K. (1895) Contribution to the Mathematical Theory of Evolution II. Skew Variation in Homogenous Material. Philosophical Transaction of the Royal Society London A, 186, 343-414. https://doi.org/10.1098/rsta.1895.0010
[2]
Johnson, N.L. (1949) Systems of Frequency Curves Generated by Methods of Translation. Biometrika, 36, 149-176. https://doi.org/10.1093/biomet/36.1-2.149
[3]
Hastings, J.C., Mosteller, F., Tukey, J.W. and Windsor, C. (1947) Low Moments for Small Samples: A Comparative Study of Order Statistics. Annals of Statistics, 18, 413-426. https://doi.org/10.1214/aoms/1177730388
[4]
Tukey, J.W. (1960) The Practical Relationship between the Common Transformations of Percentages of Counts and amounts. Technical Report 36. Statistical Techniques Research Group, Princeton University, Princeton.
[5]
Azzalini, A. (1985) A Class of Distributions Which Includes the Normal Ones. Scandinavian Journal of Statistics, 12, 171-178.
[6]
Mudholkar, G.S. and Srivastava, D.K. (1993) Exponentiated Weibull Family for Analyzing Bathtub Failure-Rate Data. IEEE Transactions on Reliability, 42, 299-302.
https://doi.org/10.1109/24.229504
[7]
Adamidis, K. and Loukas, S. (1998) A Lifetime Distribution with Decreasing Failure Rate. Statistics and Probability Letters, 39, 35-42.
https://doi.org/10.1016/S0167-7152(98)00012-1
[8]
Eugene, N., Lee, C. and Famoye, F. (2002) Beta-Normal Distribution and Its Applications. Communications in Statistics: Theory and Methods, 31, 497-512.
https://doi.org/10.1081/STA-120003130
[9]
Cooray, K. and Ananda, M.M.A. (2005) Modeling Actuarial Data with Composite Lognormal-Pareto Model. Scandinavian Actuarial Journal, 2005, 321-334.
https://doi.org/10.1080/03461230510009763
[10]
Alzaatreh, A., Lee, C. and Famoye, F. (2013) A New Method for Generating Families of Continuous Distributions. Metron, 71, 63-79.
https://doi.org/10.1007/s40300-013-0007-y
[11]
Aljarrah, M.A., Lee, C. and Famoye, F. (2014) On Generating T-X Family of Distributions Using Quantile Functions. Journal of Statistical Distributions and Applications, 1, Article No. 2. https://doi.org/10.1186/2195-5832-1-2
[12]
Ibe, G.C., Ekpenyoung, E.J., Anyiam, K. and John, C. (2021) The Weibull-Exponential {Rayleigh} Distribution: Theory and Application. Earthline Journal of Mathematical Sciences, 6, 65-86. https://doi.org/10.34198/ejms.6121.6586
[13]
Mudholkar, G.S., Srivastava, D.K. and Freimer, M. (1995) The Exponentiated Weibull Family: A Reanalysis of Bus-Motor-Failure Data. Technometrics, 37, 436-445.
https://doi.org/10.1080/00401706.1995.10484376
[14]
Madmoudi, E., Meshkat, R.S., Kargar, B. and Kundu, D. (2018) The Extended Exponential Weibull Distribution and Its Applications. Statistica, 78, 363-396.
[15]
Hassan, A. and Elegarhy, M. (2019) Exponentiated Weibull Weibull Distribution: Statistical Properties and Applications. Gazi University Journal of Science, 32, 616-635.
[16]
Shahzad, M.N., Ullah, E. and Hassanan, A. (2019) Beta Exponentiated Modified Weibull Distribution: Properties and Application. Symmetry, 11, Article No. 781.
https://doi.org/10.3390/sym11060781
[17]
Ghnimi, S. and Gasmi, S. (2014) Parameter Estimation for Some Modifications of the Weibull Distribution. Open Journal of Statistics, 4, 597-610.
https://doi.org/10.4236/ojs.2014.48056
[18]
Nader, N., El-Damcese, M. and El-Desouky, B. (2021) The Marshall-Olkin Right Truncated Frechet-Inverted Weibull Distribution: Its Properties and Applications. Open Journal of Modelling and Simulations, 9, 74-89.
https://doi.org/10.4236/ojmsi.2021.91005
[19]
Al-Aqtash, R., Lee, C. and Famoye, F. (2014) Gumbel-Weibull Distribution: Properties and Application. Journal of Modern Applied Statistical Method, 13, Article No. 11. https://doi.org/10.22237/jmasm/1414815000
[20]
Alshawarbeh, E., Famoye, F. and Lee, C. (2013) Beta-Cauchy Distribution: Some Properties and Applications. Journal of Statistical Theory and Applications, 12, 378-391.
https://doi.org/10.2991/jsta.2013.12.4.5
[21]
Nicholas, M.D. and Padgett, W.J. (2006) A Bootstrap Control for Weibull Percentiles. Quality Reliability Engineering International, 22, 141-151.
https://doi.org/10.1002/qre.691
[22]
Weibull, W. (1951) A Statistical Distribution Function of Wide Applicability. Journal of Applied Mechanics, 18, 293-297. https://doi.org/10.1115/1.4010337
[23]
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions. Vol. 1, 2nd Edition, John Wiley and Sons, Inc., New York.
[24]
Nadarajah, S. and Kotz, S. (2006) The Beta Exponential Distribution. Reliability Engineering and System Safety, 91, 689-697. https://doi.org/10.1016/j.ress.2005.05.008
[25]
Barreto-Souza, W., Morais, A.L. and Cordeiro, G.M. (2010) The Weibull-Geometric Distribution. Journal of Statistic al Computation and Simulation, 81, 645-657.
https://doi.org/10.1080/00949650903436554
[26]
Andrews, D.F. and Herzberg, A.M. (1985) Data: A Collection of Problems from Many Fields for the Student and Research Worker (Springer Series in Statistics). Springer, New York.
[27]
Osatohanmwen, P., Oyegue, F.O. and Ogbonmwan, S.M. (2019) A New Member from the T-X Family of Distributions: The Gumbel-Burr XII Distribution and Its Properties. Sankhya A, 81, 298-322. https://doi.org/10.1007/s13171-017-0110-x
