Abstract
Two entropy-based methods, called ordinary entropy (ENT) method and parameter space expansion method (PSEM), both based on the principle of maximum entropy, are applied for estimating parameters of the extended Burr XII distribution. With the parameters so estimated, the Burr XII distribution is applied to six peak flow datasets and quantiles (discharges) corresponding to different return periods are computed. These two entropy methods are compared with the methods of moments (MOM), probability weighted moments (PWM) and maximum likelihood estimation (MLE). It is shown that PSEM yields the same quantiles as does MLE for discrete cases, while ENT is found comparable to the MOM and PWM. For shorter return periods (<10–30 years), quantiles (discharges) estimated by these four methods are in close agreement, but the differences amongst them grow as the return period increases. The error in quantiles computed using the four methods becomes larger for return periods greater than 10–30 years.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burr IW (1942) Cumulative frequency functions. Ann Math Stat 13(2):215–232
Burr IW, Cislak PJ (1968) On a general system of distribution I. Its curve characteristics II. The sample median. J Am Stat Assoc 63:627–638
Fiorentino M, Arora K, Singh VP (1987) The two-component extreme value distribution for flood frequency analysis: Derivation of a new estimation method. Stoch Hydrol Hydraul (now SERRA) 1:199–208
Greenwood JA, Landwehr JM, Matalas NC, Wallis JR (1979) Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form. Water Resour Res 15:1049–1054
Hosking JRM (1986) The theory of probability weighted moments. Res Rep RC 12210:160
Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620–630
Jaynes ET (1982) On the rationale of maximum-entropy methods. Proc IEEE 70(9):939–952
Kappenman RF (1989) The use of power transformation for improved entropy estimation. Commun Stat Theory Methods 18(9):3355–3364
Kapur JN, Kesavan HK (1992) Entropy optimization principles with applications. Academic Press, Inc., New York
Kesavan HK, Kapur JN (1989) The generalized maximum entropy principle. IEEE Trans Syst Man Cybern 19(5):1042–1052
Kullback S, Leibler RA (1951) On infromation and sufficiency. Ann Math Stat 22:79–86
Landwehr JM, Matalas NC, Wallis JR (1979) Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resour Res 15(5):1055–1064
Li ZW, Zhang YK (2008) Multi-scale entropy analysis of Mississippi river flow. Stoch Environ Res Risk Assess 22(4):507–512
Lind NC, Hong HP, Solana V (1989) A cross entropy method for flood frequency analysis. Stoch Hydrol Hydraul 3(3):191–202
Lindsay SR, Wood GR, Woollons RC (1996) Modelling the diameter distribution of forest stands using the Burr distribution. J Appl Stat 23(6):609–620
Rodriguez RN (1977) A guide to the Burr type XII distributions. Biometrika 64(1):129–134
Shannon CE (1948) A mathematical theory of communications. Bell System Technical Journal 27:379–443
Shao QX, Wong H, Xia J, Wai-Cheung IP (2004) Models for extremes using the extended three-parameter Burr XII system with application to flood frequency analysis. J Hydrol Sci 49(4):685–701
Shore JE, Johnson RW (1980) Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans Inform Theory IT 26(1):26–37
Singh VP (1997) The use of entropy in hydrology and water resources. Hydrol Process 11(6):587–626
Singh VP (1998) Entropy-based parameter estimation in hydrology. Kluwer, Dordrecht
Singh VP, Deng ZQ (2003) Entropy-based parameter estimation for Kappa distribution. Journal of Hydrologic Engineering 8(2):81–92
Singh VP, Rajagopal AK (1986) A new method of parameter estimation for hydrologic frequency analysis. Hydrological Science and Technology 2(3):33–40
Singh VP, Guo H, Yu FX (1993) Paramter estimation for 3-parameetr log-logistic distribution (LLD3) by POME. Stoch Hydrol Hydraul (now SERRA) 7:163–177
Tadikamalla PR (1980) A look at the Burr and related distributions. Int Stat Rev 48(3):337–344
Wang FK, Keats JB, Zimmer WJ (1996) Maximum likelihood estimation of the Burr XII parameters with censored and uncensored data. Microelectron Reliab 36(3):359–362
Weidemann HL, Stear EB (1969) Entropy analysis of parameter estimation. Inf Control 14:493–506
Wingo DR (1993) Maximum likelihood estimation of Burr XII distribution parameters under type II censoring. Microelectron Reliab 33(9):1251–1257
Xingsi L (1992) An entropy-based aggregate method for minimax optimization. Eng Optim 18:277–285
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hao, Z., Singh, V.P. Entropy-based parameter estimation for extended Burr XII distribution. Stoch Environ Res Risk Assess 23, 1113–1122 (2009). https://doi.org/10.1007/s00477-008-0286-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-008-0286-7