# Generating Distribution Functions Based on Burr Differential Equation

## Abstract

One of the most prominent families of statistical distributions is the Burr’s system. Recent renewed interest in developing more flexible statistical distributions led to the re-examination of Burr’s system. Solutions of Burr differential equation are expressed in terms of distribution functions. Burr  considered only 12 distribution functions known in literature as the Burr system of distributions, yet there are more than that in number. Studying the Burr system, it was realized that 9 of the Burr distributions are powers of cdf ′s, popularly now known as exponentiated distributions. The remaining 3 are direct solutions in terms of cdf′s. Detailed studies using generator approach techniques to generate Burr distributions has not been undertaken in literature. This motivated us to generalize solutions of Burr differential equation by generator approach. With this aim in mind, beta generator method, exponentiated generator method and beta-exponentiated generator method (a combination of beta and exponentiated generator methods) was proposed. However in this paper, we will focus on exponentiated generator technique as it generates cdf ′s. The other two generator approach techniques generate pdf′s and distributions of order statistics.

Keywords:
Cumulative distribution function (cdf), F (x), probability density function (pdf), f(x), beta generator, exponentiated generator, beta-exponentiated generator, income elasticity, reverse hazard function, hazard function; order statistics.

## Article Details

How to Cite
Momanyi, R. O., & Ottieno, J. A. M. (2020). Generating Distribution Functions Based on Burr Differential Equation. Journal of Advances in Mathematics and Computer Science, 35(8), 55-64. https://doi.org/10.9734/jamcs/2020/v35i830313
Section
Minireview Article

## References

Burr IW. Cumulative frequency functions. Annals of Mathematical Statistics. 1942;13:215-232.

Kotz S, Vicari D. Survey of developments in the theory of continuous skewed distributions.Metron. 2005;LXIII:225-261.

Pearson K. Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material. Transactions of the Royal Society London. 1895;A186:343-415.

Johnson NL, Kotz S. Continuous univariate distributions. Houghton Mifflin, Boston. 1970;1.

Stoppa G. A new generating system of income distribution models. Quaderni di statistica ematematica applicata alle scienze economico-sociali. 1990a;12:47-55.

Kleiber C, Kotz S. Statistical size distributions in economics and actuarial sciences. John Wiley & Sons, Hoboken, New Jersey; 2003.

Olapade AK. Some properties of the type I generalized logistic distribution. Intro Stats, 2;2000.

Chiang Ch. L. Introduction to stochastic prcoesses in biostatistics. Wiley & Sons, Inc., New York, London, Sydney; 1968. 301 S, 1 Abb, 25 Tab, Preis 131 s.

Mudholkar GS, Srivastava DK. Exponentiated Weibull family for analyzing bathtub failurerate data. IEEE Transactions on Reliability. 1993;42:299-302.