A New Inverse Weibull Distribution: Properties, Classical and Bayesian estimation with Applications


  • Ahmed Z. Afify Department of Statistics, Mathematics and Insurance, Benha University
  • Shawky Ahmed Faculty of Science, Department of Statistics, King Abdulaziz University
  • Mazen Nassar King Abdulaziz University Faculty of Science




Bayesian estimation, Inverse Weibull distribution, Moments, Quantile function


This article proposes a new extension of the inverse Weibull distribution called, loga-
rithmic transformed inverse Weibull distribution which can provide better ts than some
of its well-known extensions. The proposed distribution contains inverse Weibull, inverse
Rayleigh, inverse exponential, logarithmic transformed inverse Rayleigh and logarithmic
transformed inverse exponential distributions as special sub-models. Our main focus is to
derive some of its mathematical properties along with the estimation of its unknown param-
eters using frequentist and Bayesian estimation methods. We compare the performances
of the proposed estimators using extensive numerical simulations for both small and large
samples. The importance and potentiality of this distribution is analyzed via two real data


Abouelmagd, T. H. M., Hamed, M. S., Afy, A. Z., Al-Mo

eh, H. and Iqbal, Z. (2018). The Burr X Frechet distribution with its properties and applications. Journal of Applied Probability, 13, 23-51.

Afy, A. Z. and Mohamed, O. A. (2020). A new three-parameter exponential distribution with variable shapes for the hazard rate: estimation and applications. Mathematics, 8, 1-17.

Afy, A. Z., Nassar, M., Cordeiro, G. M. and Kumar, D. (2020). The Weibull Marshall Olkin Lindley distribution: properties and estimation. Journal of Taibah University for Science, 14, 192-204.

Afy, A. Z., Yousof, H. M., Cordeiro, G. M. Ortega, E. M. M. and Nofal, Z. M. (2016). The Weibull Frechet distribution and its applications. Journal of Applied Statistics, 43,


Anderson, T. W. and Darling, D. A. (1952). Asymptotic theory of certain goodness of t

criteria based on stochastic processes. The annals of mathematical statistics, 23, 193-212.

Aryal, G. and Elbatal, I. (2015). Kumaraswamy modied inverse Weibull distribution.

Appl. Math. Inf. Sci., 9, 651-660.

Basheer, A. M. (2019) Alpha power inverseWeibull distribution with reliability application.

Journal of Taibah University for Science, 13, 423-432.

Bi, Q. and Gui, W. (2017). Bayesian and classical estimation of stress-strength reliability

for inverse Weibull lifetime models. Algorithms, 10, 1-16.

Casella, G. and Berger, R. L. (1990). Statistical Inference. Brooks. Cole Publishing, CADe-

nux T, Masson MH, Hbert PA (2005) Nonparametric rank-based statistics and signicance

tests for fuzzy data. Fuzzy Sets Syst, 153, p.128 Dubois.

Cheng, R. C. H., and Amin, N. A. K. (1979). Maximum product of spacings estimation with

applications to the lognormal distribution. Technical report, Department of Mathematics,

University of Wales.

Cheng, R. C. H., and Amin, N. A. K. (1983). Estimating parameters in continuous uni-

variate distributions with a shifted origin. J. R. Statist. Soc. B, 45, 394-403.

De Gusmao, F. R., Ortega, E. M., Cordeiro, G. M. (2011). The generalized inverse Weibull

distribution. Statistical Papers, 52, 591-619.

Dey, S., Dey, T., Ali, S. and Mulekar, M. S. (2016). Two-parameter Maxwell distribution:

properties and different methods of estimation. J. Stat. Theory. Prac., 10, 291-310.

Dey, S., Raheem, E., Mukherjee, S. and Ng, , H. K. T. (2017). Two parameter

exponentiated-Gumbel distribution: properties and estimation with

ood data example. J.

Statist. Manag. Sys., 20, 197-233.

do Espirito Santo, A. P. J. and Mazucheli, J. (2015). Comparison of estimation methods for

the MarshallOlkin extended Lindley distribution. J. Stat. Comput. Simul., 85, 3437-3450.

Drapella, A. (1993). Complementary Weibull distribution: unknown or just forgotten.

Qual. Reliab. Eng. Int., 9, 383-385.

Feigl, P. and Zelen, M. (1965). Estimation of exponential probabilities with concomitant

information. Biometrics, 21, 826-838.

Frechet, M. (1924). Sur la Loi des Erreurs d'Observation. Bulletin de la Soci et e Math

ematique de Moscou, 33, 5-8.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Distributions. In Continuous Uni-

variate Distributions, 2nd ed., Vol. 2, Wiley, New York, NY, USA.

Kao, J. H. K. (1958). Computer methods for estimating Weibull parameters in reliability

studies. Trans. IRE Reliability Quality Control, 13,15-22.

Kao, J. H. K. (1959). A graphical estimation of mixed Weibull parameters in life testing

electron tube. Technometrics, 1, 389-407.

Keller, A. Z., Kamath, A. R. R. and Perera, U. D. (1982). Reliability analysis of CNC

machine tools. Reliab. Eng., 3, 449-473.

Khan, M. S. (2010). The beta inverse Weibull distribution. International Transactions in

Mathematical Sciences and Computer, 3, 113-119.

Khan, M. S. and King, R. (2012). Modied inverseWeibull distribution. Journal of statistics

applications & Probability, 1, 115-132.

Khan, M. S., Pasha, G. R. and Pasha, A. H. (2008). Theoretical analysis of inverse Weibull

distribution. Weas Trans.Math., 7, 30{38.

Krishna, E., Jose, K. K., Alice, T. and Ristic, M. M. (2013). The Marshall-Olkin Frechet

distribution. Commun. Stat. Theory Methods, 42, 4091-4107.

Kundu, D. and Raqab, M. Z. (2005). Generalized Rayleigh distribution: dierent methods

of estimations. Comput. Stat. Data Anal., 49, 187-200.

Li, C. and Hao, H. (2017). Reliability of a stress{strength model with inverse Weibull

distribution. IAENG International Journal of Applied Mathematics, 47, 302-306.

Mansour, M. M., Elrazik, E. M. A., Altun, E., Afy, A. Z. and Iqbal, Z. (2018). A new

three-parameter Frechet distribution: properties and applications. Pak. J. Stat., 34, 441-

Mazucheli, J., Louzada, F. and Ghitany, M. E. (2013). Comparison of estimation methods

for the parameters of the weighted Lindley distribution. Appl. Math. Comput., 220 (2013),


Mead, M. E., Afy, A. Z., Hamedani, G. G. and Ghosh, I. (2017). The beta exponential

Frechet distribution with applications. Austrian Journal of Statistics, 46, 41-63.

Nadarajah, S. and Kotz, S. (2003). The exponentiated Frechet distribution. Interstat Elec-

tronic Journal, 1-7.

Nassar, M., Afy, A. Z., Dey, S. and Kumar, D. (2018a). A new extension of Weibull

distribution: properties and dierent methods of estimation. J. Comput. Appl. Math.,

, 439-457.

Nassar, M., Afy, A. Z. and Shakhatreh, M. (2020). Estimation methods of alpha power ex-

ponential distribution with applications to engineering and medical data. Pakistan Journal

of Statistics and Operation Research, 16, 149-166.

Nassar, M., Dey, S. and Kumar, D. (2018b). A new generalization of the exponentiated

Pareto distribution with an application. Amer. J. Math. Manag. Sci., 37, 217-242.

Nassar, M., Dey, S. and Kumar, D. (2018c). Logarithm transformed Lomax distribution

with applications. Calcutta Statistical Association Bulletin, 70, 122-135.

Okasha, H. M., El-Baz, A. H., Tarabia, A. M. K. and Basheer, A. M. (2017). Extended

inverse Weibull distribution with reliability application. Journal of the Egyptian Mathe-

matical Society, 25, 343-349.

Oluyede, B. O. and Yang, T. (2014). Generalizations of the inverse Weibull and related

distributions with applications. Electronic Journal of Applied Statistical Analysis, 7, 94-

Pararai, M., Warahena-Liyanage, G. and Oluyede, B. O. (2014). A new class of general-

ized inverse Weibull distribution with applications. Journal of Applied Mathematics and

Bioinformatics, 4, 17-35.

Sen, S., Afy, A. Z., Al-Mo

eh, H. and Ahsanullah, M. (2019). The quasi xgamma-

geometric distribution with application in medicine. Filomat, 33, 5291-5330.

Swain, J., Venkatraman, S., and Wilson, J. (1988). Least squares estimation of distri-

bution function in Johnsons translation system. Journal of Statistical Computation and

Simulation, 29, 271-297.

Tablada, C. J. and Cordeiro, G. M. (2017). The modied Frechet distribution and its

properties. Commun. Stat. Theory Methods, 46, 10617-10639.