The Burr-Weibull Power Series Class of Distributions

  • Broderick Oluyede Georgia Southern University
  • Precious Mdlongwa
  • Boikanyo Makubate Botswana International University of Science and Technology
  • Shujiao Huang University of Houston

Abstract

A new generalized class of distributions called the Burr-Weibull Power Series (BWPS) class of distributions is developed and explored. This class of distributions generalizes the Burr power series and Weibull power series classes of distributions, respectively. A special model of the BWPS class of distributions, the new Burr-Weibull Poisson (BWP) distribution is considered and some of its mathematical properties are obtained. The BWP distribution contains several new and well known sub-models, including Burr-Weibull, Burr-exponential Poisson, Burr-exponential, Burr-Rayleigh Poisson, Burr-Rayleigh, Burr-Poisson, Burr, Lomax-exponential Poisson, Lomax-Weibull, Lomax-exponential, Lomax-Rayleigh, Lomax-Poisson, Lomax, Weibull, Rayleigh and exponential distributions. Maximum likelihood estimation technique is used to estimate the model parameters followed by a Monte Carlo simulation study. Finally an application of the BWP model to a real data set is presented to illustrate the usefulness of the proposed class of distributions.

Author Biographies

Broderick Oluyede, Georgia Southern University
Broderick O. Oluyede, Ph.D.Professor and Consulting StatisticianDirector, Statistical Consulting Unit (SCU)Department of Mathematical SciencesGeorgia Southern UniversityStatesboro, GA 30460Telephone: (912) 478 5427Email: Boluyede@GeorgiaSouthern.edu
Boikanyo Makubate, Botswana International University of Science and Technology

Senior Lecturer

Department of Mathematics and Computational Sciences

Botswana International University of Science and Technology
Palapye, BW

Shujiao Huang, University of Houston

Research Associate

University of Houston

References

Abd-Elfattah AM, Alaboud FM, Alharby AH (2007). On sample size estimation for Lomax distribution. Australian Journal of Basic Applied Sciences, 1(1), 373-378.

Abd-Ellah AH (2003). Bayesian one sample prediction bounds for the Lomax distribution. Indian Journal of Pure and Applied Mathematics, 30(1), 101-109.

Al-Awadhi SA, Ghitany ME (2001). Statistical Properties of Poisson-Lomax Distribution and Its Application to Repeated Accidents Data. Jornal of Applied Statistcal Sciences, 10(1), 365-372.

Arnold BC (1983). Pareto Distributions. International Cooperative Publishing House, Fairland, Maryland.

Balakrishnan N, M Ahsanullah M (1994). Relations for single and product moments of record values from Lomax distribution. Sankhya B, 56(1), 140-146.

Balkema AA, L de Haan L (1974). Residual life time at great age. Annals of Probability, 2(1), 792-804.

Bidram H, Behboodian J, Towhidi M (2013). The Beta Weibull Geometric Distribution. Journal of Statistical Computation and Simulation, 83(1), 52-67.

Burr IW (1942). Cumulative Frequency Functions. The Annals of Mathematical Statistics, 13(2), 215-232.

Burr IW (1973). Pameters for a General System of Distributions to Match a Grid of alpha3 and alpha4. Communications in Statistics-Theory and Methods, 2(1), 1-21.

Chen G, Balakrishnan N (1995). A General Purpose Approximate Goodness-of-t Test." Journal of Quality Technology, 27(2), 154-161.

Childs A BN, Moshref M (2001). Order statistics from non-identical right truncated Lomax random variables with applications. Statistical Papers, 42(1), 187-206.

Ghitany ME, Al-Awadhi FA, Alkhalfan LA (2007). Marshall-Olkin extended Lomax distribution and its application to censored data. Communications in Statistics-Theory and Methods, 36, 1855-1866.

Howlader HA, Hossain AM (2002). Bayesian survival estimation of Pareto distribution of the second kind based on failure-censored data. Computational Statistics & Data Analysis, 38, 301-314.

Huang S, Oluyede BO (2014). Exponentiated Kumaraswamy-Dagum Distribution with Applications to Income and Lifetime Data. Journal of Statistical Distributions and Applications, 1(1), 1-8.

Johnson NL, Kotz S, Balakrishnan N (1994). Continuous Univariate Distributions, volume 1. John Wiley & Sons, New York.

Jones MC (2004). Families of Distributions Arising from Distributions of Order Statistics. Test, 13(1), 1-43.

Lai CD, Xie M, Murthy DNP (2003). A Modied Weibull Distribution. IEEE Transactions on Reliability, 52(1), 33-37.

Lomax K (1954). Business failures. Another example of the analysis of failure data. Journal of the American Statistical Association, 49, 847-852.

Marshall AW, Olkin I (1997). A New Method for Adding a Parameter to a Family of Distributions with Applications to the Exponential and Weibull Families. Biometrika, 84, 641-652.

Murthy DNP, Xie M, Jiang R (2004). Weibull Models, volume 505. John Wiley & Sons.

Nadarajah S (2005). Sums, products, and ratios for the bivariate Lomax distribution. Computational Statistics & Data Analysis, 49, 109-129.

Oluyede BO, Huang S, 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 BO, Makubate B, Warahena-Liyanage G, Huang S (2016). A New Model for Lifetime Data: The Burr XII-Weibull Distribution, Properties and Applications. Submitted, pp. 0000-0000.

Percontini A, Blas B, Cordeiro GM (2013). The beta Weibull Poisson distribution. Chilean Journal of Statistics, 4(2), 3-26.

Petropoulos C, Kourouklis S (2004). Improved estimation of extreme quantiles in the multivariate Lomax (Pareto II)distribution. Metrika, 60, 15-24.

R Development Core Team (2011). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2011.

Renyi A (1960). On Measures of Entropy and Information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pp. 547-561.

Rodriguez RN (1977). A guide to the Burr Type XII distributions. Biometrika, 64, 129-134.

Roy D, Gupta RP (1996). Bivariate extension of Lomax and nite range distributions through characterization approach. Journal of Multivariate Analysis, 59, 22-33.

Seregin A (2010). Uniqueness of the Maximum Likelihood Estimator for k-Monotone Densities. Proceedings of the American Mathematical Society, 138(12), 4511-4515.

Silva JMCS, Tenreyro S (2010). On the Existence of the Maximum Likelihood Estimates in Poisson Regression. Economics Letters, 107(2), 310-312.

Tadikamalla PR (1980). A look at the Burr and related distributions. International Statistical Review, 48(3), 337-344.

Vidondo B, Prairie YT, Blanco JM, Duarte CM (1997). Some aspects of the analysis of size spectra in aquatic ecology." Limnol. Oceanography, 42, 184-194.

Xia J, Mi J, Zhou Y (2009). On the Existence and Uniqueness of the Maximum Likelihood Estimators of Normal and Log-normal Population Parameters with Grouped Data. Journal of Probability and Statistics, 2009.

Xu K, Xie M, Tang L, Ho S (2003). Application of Neural Networks in Forecasting Engine Systems Reliability. Applied Soft Computing, 2(4), 255-268.

Zhou C (2009). Existence and Consistency of the Maximum Likelihood Estimator for the Extreme Value Index. Journal of Multivariate Analysis, 100(4), 794-815.

Published
2018-12-17
How to Cite
Oluyede, B., Mdlongwa, P., Makubate, B., & Huang, S. (2018). The Burr-Weibull Power Series Class of Distributions. Austrian Journal of Statistics, 48(1), 1-13. https://doi.org/10.17713/ajs.v48i1.633
Section
Special Issue on Lifetime Data Modelling (closed)