# Beta Kumaraswamy Burr Type X Distribution and Its Properties

## DOI:

https://doi.org/10.53797/aseana.v2i1.2.2022## Keywords:

Bayesian, Beta Kumaraswamy-G, Burr Type X, Kumaraswamy-G, quantile function, maximum likelihood estimation, inverse CDF method of simulation## Abstract

We proposed a so-called Beta Kumaraswamy Burr Type X distribution which gives the extension of the Kumaraswamy-G class of family distribution. Some properties of this proposed model were provided, like the expansion of densities and quantile function. We considered the Bayes and maximum likelihood methods to estimate the parameters and also simulate the model parameters to validate the methods based on different set of true values. Some real data sets were employed to show the usefulness and flexibility of the model which serves as generalization to many sub-models in the fields of engineering, medical, survival and reliability analysis.

### Downloads

## References

Gupta, A.K. and Nadarajah, S. eds., 2004. Handbook of beta distribution and its applications. CRC press.

Kumaraswamy, P., 1980. A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1), pp.79-88.

Eugene N., Lee C. and Famoye F., 2002. Beta-normal distribution and its applications. Commun Statist Theor Meth 31: 497-512.

Jones, M.C., 2004. Families of distributions arising from distributions of order statistics. Test, 13(1), pp.1-43.

Jones, M.C., 2009. Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), pp.70-81.

Cordeiro G.M. and De Castro M., 2011. A new family of generalized distributions. J Stat Comput Simul 81: 883-893.

Alzaatreh, A., Lee, C. and Famoye, F., 2013. A new method for generating families of continuous distributions. Metron, 71(1), pp.63-79.

Handique, L. and Chakraborty, S., 2016. Beta generated Kumaraswamy-G and other new families of distributions. arXiv preprint arXiv:1603.00634.

Burr, I.W., 1942. Cumulative frequency functions. The Annals of mathematical statistics, 13(2), pp.215-232.

Raqab, M.Z., 1998. Order statistics from the Burr type X model. Computers and Mathematics with Applications, 36(4), pp.111-120.

Raqab, M.Z. and Kundu, D., 2006. Burr type X distribution: revisited. Journal of Probability and Statistical Sciences, 4(2), pp.179-193.

Surles, J.G. and Padgett, W.J., 2001. Inference for reliability and stressstrength for a scaled Burr Type X distribution. Lifetime Data Analysis, 7(2), pp.187-200.

Surles, J.G. and Padgett, W.J., 2005. Some properties of a scaled Burr type X distribution. Journal of Statistical Planning and Inference, 128(1), pp.271-280.

Merovci, F., Khaleel, M.A., Ibrahim, N.A. and Shitan, M., 2016. The beta Burr type X distribution properties with application. SpringerPlus, pp.118.

Khaleel, M.A., Ibrahim, N.A., Shitan, M. and Merovci, F., 2016. Some properties of Gamma Burr type X distribution with Application. AIP Conference proceedings.

Cordeiro, G.M., Ortega, E.M. and Nadarajah, S., 2010. The Kumaraswamy Weibull distribution with application to failure data. Journal of the Franklin Institute, 347(8), pp.1399-1429.

Cordeiro, G. M., Nadarajah, S. and Ortega, E. M. 2013c. General Results for the Beta Weibull Distribution. Journal of Statistical Computation and Simulation 83 (6): 10821114.

David Osborn and Richard Madey, 1968. The incomplete beta function and its ratio to the complete beta function. Mathematics of Computation Vol. 22, No. 101 pp. 159-162+s16-s32.

Bornemann, Folkmar and Weisstein, Eric W., 2016 Power series From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com.

Greenwood, J.A., Landwehr, J.M., Matalas, N.C. and Wallis, J.R., 1979. Probability weighted moments: definition and relation to parameters of several distributions expressable in inverse form. Water Resources Research, 15(5), pp.1049-1054.

Nadarajah, S. and Zografos, K. 2003. Formulas for Renyi Information and Related Measures for Univariate Distributions. Information Sciences 155

(1): 119138.

Nadarajah, S., Cordeiro, G.M. and Ortega, E.M., 2012. General results for the Kumaraswamy-G distribution. Journal of Statistical Computation and Simulation, 82(7), pp.951-979.

Dagum C., 1980. The generation and distribution of income, the Lorenz curve and the Gini ratio. Econ Appl XXXIII: 327367.

Song, K.S., 2001. Rnyi information, loglikelihood and an intrinsic distribution measure. Journal of Statistical Planning and Inference, 93(1), pp.51-69.

Kenney J.F., Keeping and E.S., 1962. Mathematics of Statistics, Part 1. Van Nostrand, New Jersey, 3rd edition.

Moors, J.J.A., 1988. A quantile alternative for kurtosis. The statistician, pp.25-32.

Andrews, D.F. and Herzberg, A.M., 1985. Stress-rupture life of kevlar 49/epoxy spherical pressure vessels. In Data (pp. 181-186). Springer New York.

Hosmer, D.W. and Lemeshow, S. and May, S. 2008. Applied Survival Analysis: Regression Modeling of Time to Event Data: Second Edition, John Wiley and Sons Inc., New York, NY.

Shiffman, S., Waters, A.J. and Hickcox, M., 2004. The nicotine dependence syndrome scale: a multidimensional measure of nicotine dependence. Nicotine and Tobacco Research, 6(2), pp.327-348.

Watson, G.S., 1961. Goodness-of-fit tests on a circle. Biometrika, 48(1/2), pp.109-114.

Yusuf, M.U. and Bakar, M.R.B.A., 1739. A Bayesian estimation on right censored survival data with mixture and non-mixture cured fraction model based on Beta-Weibull distribution. In AIP Conference Proceedings (Vol. 20079, No. 2016).

Pescim, R. R., Cordeiro, G. M., Demetrio, C. G., Ortega, E. M. and Nadarajah, S. 2012. The New Class of Kummer Beta Generalized Distributions. Statistics and Operations Research Transactions (SORT) 36 (2): 153(180)

Nadarajah, S. and Zografos, K. 2003. Formulas for Renyi Information and Related Measures for Univariate Distributions. Information Sciences 155 (1): 119(138)

Zografos, K. and Nadarajah, S. 2005. Expressions for Renyi and Shannon Entropies for Multivariate Distributions. Statistics and Probability Letters 71 (1):

## Downloads

## Published

## How to Cite

*ASEANA Science and Education Journal*,

*2*(1), 9-36. https://doi.org/10.53797/aseana.v2i1.2.2022

## Issue

## Section

## License

Copyright (c) 2022 Umar Yusuf Madaki, Mohd Rizam Abu Bakar, Laba Handique

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.