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Entropy-based parameter estimation for extended Burr XII distribution. (English) Zbl 1416.62124
Summary: 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.

62E15 Exact distribution theory in statistics
62P12 Applications of statistics to environmental and related topics
60E05 Probability distributions: general theory
Full Text: DOI
[1] Burr, IW, Cumulative frequency functions, Ann Math Stat, 13, 215-232, (1942) · Zbl 0060.29602
[2] Burr, IW; Cislak, PJ, On a general system of distribution I. Its curve characteristics II. The sample median, J Am Stat Assoc, 63, 627-638, (1968)
[3] Fiorentino, M.; Arora, K.; Singh, VP, The two-component extreme value distribution for flood frequency analysis: Derivation of a new estimation method, Stoch Hydrol Hydraul (now SERRA), 1, 199-208, (1987) · Zbl 0662.62027
[4] Greenwood, JA; Landwehr, JM; Matalas, NC; Wallis, JR, Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form, Water Resour Res, 15, 1049-1054, (1979)
[5] Hosking, JRM, The theory of probability weighted moments, Res Rep RC, 12210, 160, (1986)
[6] Jaynes, ET, Information theory and statistical mechanics, Phys Rev, 106, 620-630, (1957) · Zbl 0084.43701
[7] Jaynes, ET, On the rationale of maximum-entropy methods, Proc IEEE, 70, 939-952, (1982)
[8] Kappenman, RF, The use of power transformation for improved entropy estimation, Commun Stat Theory Methods, 18, 3355-3364, (1989) · Zbl 0696.62179
[9] Kapur JN, Kesavan HK (1992) Entropy optimization principles with applications. Academic Press, Inc., New York
[10] Kesavan, HK; Kapur, JN, The generalized maximum entropy principle, IEEE Trans Syst Man Cybern, 19, 1042-1052, (1989)
[11] Kullback, S.; Leibler, RA, On infromation and sufficiency, Ann Math Stat, 22, 79-86, (1951) · Zbl 0042.38403
[12] Landwehr, JM; Matalas, NC; Wallis, JR, Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles, Water Resour Res, 15, 1055-1064, (1979)
[13] Li, ZW; Zhang, YK, Multi-scale entropy analysis of Mississippi river flow, Stoch Environ Res Risk Assess, 22, 507-512, (2008) · Zbl 1159.92316
[14] Lind, NC; Hong, HP; Solana, V., A cross entropy method for flood frequency analysis, Stoch Hydrol Hydraul, 3, 191-202, (1989)
[15] Lindsay, SR; Wood, GR; Woollons, RC, Modelling the diameter distribution of forest stands using the Burr distribution, J Appl Stat, 23, 609-620, (1996)
[16] Rodriguez, RN, A guide to the Burr type XII distributions, Biometrika, 64, 129-134, (1977) · Zbl 0354.62017
[17] Shannon, CE, A mathematical theory of communications, Bell System Technical Journal, 27, 379-443, (1948)
[18] Shao, QX; Wong, H.; Xia, J.; Wai-Cheung, IP, Models for extremes using the extended three-parameter Burr XII system with application to flood frequency analysis, J Hydrol Sci, 49, 685-701, (2004)
[19] Shore, JE; Johnson, RW, Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy, IEEE Trans Inform Theory IT, 26, 26-37, (1980) · Zbl 0429.94011
[20] Singh, VP, The use of entropy in hydrology and water resources, Hydrol Process, 11, 587-626, (1997)
[21] Singh VP (1998) Entropy-based parameter estimation in hydrology. Kluwer, Dordrecht
[22] Singh, VP; Deng, ZQ, Entropy-based parameter estimation for Kappa distribution, Journal of Hydrologic Engineering, 8, 81-92, (2003)
[23] Singh, VP; Rajagopal, AK, A new method of parameter estimation for hydrologic frequency analysis, Hydrological Science and Technology, 2, 33-40, (1986)
[24] Singh, VP; Guo, H.; Yu, FX, Paramter estimation for 3-parameetr log-logistic distribution (LLD3) by POME, Stoch Hydrol Hydraul (now SERRA), 7, 163-177, (1993) · Zbl 0784.62020
[25] Tadikamalla, PR, A look at the Burr and related distributions, Int Stat Rev, 48, 337-344, (1980) · Zbl 0468.62013
[26] Wang, FK; Keats, JB; Zimmer, WJ, Maximum likelihood estimation of the Burr XII parameters with censored and uncensored data, Microelectron Reliab, 36, 359-362, (1996)
[27] Weidemann, HL; Stear, EB, Entropy analysis of parameter estimation, Inf Control, 14, 493-506, (1969) · Zbl 0212.23301
[28] Wingo, DR, Maximum likelihood estimation of Burr XII distribution parameters under type II censoring, Microelectron Reliab, 33, 1251-1257, (1993)
[29] Xingsi, L., An entropy-based aggregate method for minimax optimization, Eng Optim, 18, 277-285, (1992)
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