Book contents
- Generalized Frequency Distributions for Environmental and Water Engineering
- Generalized Frequency Distributions for Environmental and Water Engineering
- Copyright page
- Dedication
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- 2 Burr–Singh–Maddala Distribution
- 3 Halphen Type A Distribution
- 4 Halphen Type B Distribution
- 5 Halphen Inverse B Distribution
- 6 Three-Parameter Generalized Gamma Distribution
- 7 Generalized Beta Lomax Distribution
- 8 Feller–Pareto Distribution
- 9 Kappa Distribution
- 10 Four-Parameter Exponential Gamma Distribution
- 11 Summary
- Appendix General Procedure for Goodness-of-Fit Study, Construction of Confidence Interval, and 95% Confidence Bound with Parametric Bootstrap Method
- Index
- References
2 - Burr–Singh–Maddala Distribution
Published online by Cambridge University Press: 13 April 2022
Book contents
- Generalized Frequency Distributions for Environmental and Water Engineering
- Generalized Frequency Distributions for Environmental and Water Engineering
- Copyright page
- Dedication
- Contents
- Preface
- Acknowledgments
- 1 Introduction
- 2 Burr–Singh–Maddala Distribution
- 3 Halphen Type A Distribution
- 4 Halphen Type B Distribution
- 5 Halphen Inverse B Distribution
- 6 Three-Parameter Generalized Gamma Distribution
- 7 Generalized Beta Lomax Distribution
- 8 Feller–Pareto Distribution
- 9 Kappa Distribution
- 10 Four-Parameter Exponential Gamma Distribution
- 11 Summary
- Appendix General Procedure for Goodness-of-Fit Study, Construction of Confidence Interval, and 95% Confidence Bound with Parametric Bootstrap Method
- Index
- References
Summary
The Burr–Singh–Maddala (BSM) probability distribution is a generalization of the Pareto distribution and the Weibull distribution that are used for frequency analyses of a variety of hydrologic and hydrometeorologic data. This distribution possesses a number of interesting characteristics that are discussed in this chapter. The BSM distribution is derived using the entropy theory, which then is applied to derive the BSM distribution parameters. Real-world data are used to illustrate the application of the distribution.
- Type
- Chapter
- Information
- Publisher: Cambridge University PressPrint publication year: 2022
References
Brouers, F. (2014a). Statistical foundations of empirical isotherms. Open Journal of Statistics, Vol. 4, pp. 687–701.Google Scholar
Brouers, F. (2014b). The fractal (BSF) kinetics equation and its applications. Journal of Modern Physics, Vol. 5, pp. 1954–1998.CrossRefGoogle Scholar
Brouers, F. (2015). The Burr XII distribution family and the maximum entropy principle: Power-law phenomena are not necessarily “nonextensive.” Open Journal of Statistics, Vol. 5, pp. 730–741.Google Scholar
Brouers, F. and Al-Musawi, T.J. (2015). On the optimum use of isotherms model for the characterization of biosorption of lead onto algae. Journal of Molecular Liquids, Vol. 212, pp. 46–51.CrossRefGoogle Scholar
Brouers, F. and Sotolongo-Costa, O. (2005). Relaxation in heterogeneous systems: A rare events dominated phenomenon. Physica A: Statistical Mechanics and Its Applications, Vol. 356, pp. 359–374.CrossRefGoogle Scholar
Burr, I.W. (1942). Cumulative frequency functions. The Annals of Mathematical Statistics, Vol. 13, No. 2, pp. 215–232.CrossRefGoogle Scholar
Dubey, M.J. and Gove, J.H. (2015). Size-based distribution in the generalized beta distribution family, with applications to forestry. Forestry, Vol. 88, pp. 141–151.Google Scholar
Fisk, P.R. (1961). The graduation of income distributions. Econometrica, Vol. 29, No. 2, pp. 171–185.Google Scholar
Greenwood, J.A., Landwehr, J.M., and Matalas, N.C. (1979). Probability weighted moments: definition and relation to parameters of several distributions expressable in inverse form. Water Resources Research, Vol. 15, No. 5, pp. 1049–1054.CrossRefGoogle Scholar
Hosking, J.R.M. (1990). L-moments: Analysis and estimation of distribution using linear combinations of order statistics. Journal of the Royal Statistical Society: Series B (Methodological), Vol. 52, No. 1, 105–124.Google Scholar
Papalexiou, S.M. and Koutsoyiannis, D. (2012). Entropy-based derivation of probability distributions: A case study to daily rainfall. Advances in Water Resources, Vol. 45, pp. 51–57.CrossRefGoogle Scholar
Singh, S.K. and Maddala, G.S. (1976). A function for the size distribution of incomes. Econometrica, Vol. 44, pp. 963–970.CrossRefGoogle Scholar
Singh, V.P. and Rajagopal, A.K. (1986). A new method of parameter estimation for hydrologic frequency analysis. Hydrological Science and Technology, Vol. 2, No. 3, pp. 33–40.Google Scholar
Sornette, D. (2003). Critical Phenomena in Natural Sciences. 2nd edition, chapter 14, Springer, Heidelberg.Google Scholar
Tsallis, C. (1988). Possible generalization of Boltzmann–Gibbs statistics. Journal of Statistical Physics Vol. 52, Nos. 1–2, pp. 479–487.Google Scholar
Weron, K. and Kotulski, M. (1997). On the equivalence of the parallel channel and the correlated cluster relaxation models. Journal of Statistical Physics, Vol. 88, Nos. 5/6, pp. 1241–1256.CrossRefGoogle Scholar