The Burr XII modified Weibull distribution: model, properties and applications


A new distribution called Burr XII modified Weibull (BXIIMW or BMW) distribution is presented and its properties explored. This new distribution contains several new and well known sub-models, including Burr-Weibull, Burr-exponential, Burr-Rayleigh, Burr XII, Lomax modified Weibull, Lomax-Weibull, Lomax-exponential, Lomax-Rayleigh, Lomax, Weibull, Rayleigh, and exponential distributions. Some structural properties of the proposed distribution including the shapes of the density and hazard rate functions, moments, conditional moments, moment generating function, skewness and kurtosis are presented. Mean deviations, Lorenz and Bonferroni curves, R\'enyi entropy and the distribution of the order statistics are given. The maximum likelihood estimation technique is used to estimate model parameters and finally applications of the model to real data sets are presented to illustrate the usefulness of the proposed distribution.

DOI Code: 10.1285/i20705948v10n1p118

Keywords: Weibull distribution, Burr XII distribution, Burr XII modied Weibull distribution, Maximum likelihood estimation.


Aarset, M. (1987). How to identify bathtub hazard rate. IEEE Transactions on Reliability, 36(1):106-108.

Andrews, D. and Herzberg, A. (1985). A Collection of Problems from Many Fields for the Students and Research Workers. Springer Series in Statistics, New York: Springer.

Barakat, H. and Abdelkader, Y. H. (2004). Computing the moments of order statistics from nonidentical random variables. Statistical Methodology and Applications, 13.

Bidram, H., Behboodian, J., and Towhidi, M. (2013). The beta weibull geometric distribution. Journal of Statistical Computation and Simulation, 83(1):52-67.

Burr, I. and Cislak, P. (1968). On a general system of distributions: I. its curve shaped characteristics, ii . the sample median. The Annals of Mathematical Statistics, 63:627-635.

Burr, I. W. (1942). Cumulative frequency functions. The Annals of Mathematical Statistics, 13(2):215-232.

Chambers, J., Cleveland, W., Kleiner, B., and Tukey, P. (1983). Graphical Methods of Data Analysis. Chapman and Hall.

Chen, G. and Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27:154-161.

Gurvich, M., DiBenedetto, A., and Ranade, S. (1997). A new statistical distribution for characterizing the random strength of brittle materials. Journal of Materials Science, 32:2559-2564.

Hatke, M. (1949). A certain cumulative probability function. The Annals of Mathematical Statistics, 20:461-463.

Hoskings, J. R. M. (1990). L-moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, B 52:105-124.

Huang, S. and Oluyede, B. (2014). Exponentiated kumaraswamy-dagum distribution with applications to income and lifetime data. Journal of Statistical Distributions and Applications, 1(8).

Lai, C., Xie, M., and Murthy, D. (2003). A modied weibull distribution. IEEE Transactions on Reliability, 52(1):33-37.

Mudholkar, G. and Srivastava, D. (1993). Exponentiated weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42:299-302.

Nadarajah, S., Cordeiro, G. M., and Ortega, E. M. M. (2011). General results for the beta-modied weibull distribution. Journal of Statistical Computation and Simulations, 81(10):1211-1232.

Oluyede, B., Huang, S., and Pararai, M. (2014). A new class of generalized dagum distribution with applications to income and lifetime data. Journal of Statistical and Econometric Methods, 3(2):125-151.

Oluyede, B., Huang, S., and Yang, T. (2015). A new class of generalized modied weibull distribution with applications. Austrian Journal of Statistics, 44:45-68.

Percontini, A., Blas, B., and Cordeiro, G. (2013). The beta weibull poisson distribution. Chilean Journal of Statistics, 4(2):3-26.

Pham, H. and Lai, C. (2007). On recent generalizations of the weibull distribution. IEEE Transaction on Reliability, 56:454-458.

Renyi, A. (1960). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pages 547-561.

Rodrigues, R. (1977). A guide to the burr type xii distributions. Biometrika, 64:129-134.

Santos Silva, J. M. and Tenreyro, S. (2010). On the existence of maximum likelihood estimates in poisson regression. Econ. Lett., 107:310-312.

Seregin, A. (2010). Uniqueness of the maximum likelihood estimator for k-monotone densities. Proc. Amer. Math. Soc, 138:4511-4515.

Soliman, A. (2005). Estimation of parameters of life from progressively censored data using burr-xii model. IEEE Transactions on Reliability, 54:34-42.

Tadikamalla, P. (1980). A look at the burr and related distributions. Int. Stat. Rev., 48:337-344.

Team, R. D. C. (2011). A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.

Weibull, W. (1951). Statistical distribution function of wide applicability. Journal of Applied Mechanics, 18:293-296.

Xia, J., Mi, J., and Zhou, Y. Y. (2009). On the existence and uniqueness of the maximum likelihood estimators of normal and log-normal population parameters with grouped data. J. Probab. Statist, Article id 310575:16 pages.

Zhou, C. (2009). Existence and consistency of the maximum likelihood estimator for the extreme index. J. Multivariate Analysis, 100:794-815.

Full Text: pdf

Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.