Abstract
Modeling of extreme values like annual maxima’s is important in many applications. Pearson Type-3 (PE3) distribution is an important probability distribution, widely used for modeling of extreme values with a variety of estimation methods. The focus of this study is to assess the effects of three methods of estimation of parameters for PE3 distribution namely L-moments (LM), maximum likelihood estimation (MLE) and maximum product of spacing (MPS). Assessment is based on a two-step approach. The first step uses simulation experiments while the second is based on empirical analyses, by varying size and shape characteristics of the sample. The study concluded that the estimates using LM method have low bias in case of small sample and when data exhibits small to moderate skewness and kurtosis. MPS is a reasonable alternative and provides efficient estimates, especially when the data shows large skewness and kurtosis with small to moderate size of sample. MLE method is useful in case of very large sample size with low values of shape characteristics of data. The results of this study provide useful guidelines for fitting PE3 distribution, especially to extreme values.
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Data Availability
Data of annual maxima’s of river discharges of four sites of KPK Pakistan, used in this study and all the other relevant material will be available for the editorial office at any stage for non-commercial purpose.
References
Arns A, Wahl T, Haigh ID, Jensen J, Pattiaratchi C (2013) Estimating extreme water level probabilities: A comparison of the direct methods and recommendations for best practise. Coast Eng 81:51–66
Asquith WH (2020) L-moments, censored L-moments, trimmed L-loments, L-comoments, and many distributions. R package version 2.3.6, Texas Tech University, Lubbock, Texas. https://cran.r-project.org/web/packages/lmomco/lmomco.pdf. Accessed 18 May 2000
Asquith WH, Kiang JE, Cohn TA (2017) Application of at-site peak-streamflow frequency analyses for very low annual exceedance probabilities (no. 2017–5038). US geological survey
Beirlant J, Goegebeur Y, Teugels J, Segers J (2004) Statistics of extremes. Wiley series in probability and statistics
Bobee B, Ashkar F (1991) The Gamma family and derived distributions applied in hydrology (no. GB656. 2. M34. B63 1991)
Chang SK, Moore SM (1983) Flood frequency analysis for mall watersheds in Southern Illinois. Water Resour Res 19(2):277–282
Cheng RCH, Amin NAK (1983) Estimating parameters in continuous univariate distributions with a shifted origin. J R Stat Soc Ser B Methodol 45(3):394–403
Coles S, Bawa J, Trenner L, Dorazio P (2001) An introduction to statistical modeling of extreme values, vol 208. Springer, London, p 208
Cook NJ (1985) The designer’s guide to wind loading of building structures part 1: background. Damage survey, wind data and structural classification building research establishment, Garston and Butterworths London
Csörgő S, Faraway JJ (1996) The exact and asymptotic distributions of cramér-von mises statistics. J R Stat Soc Ser B Methodol 58(1):221–234
Drissia TK, Jothiprakash V, Anitha AB (2019) Flood frequency analysis using L moments: a comparison between at-site and regional approach. Water Resour Manag 33(3):1013–1037
El-Sherpieny ESA, Almetwally EM, Muhammed HZ (2020) Progressive type-II hybrid censored schemes based on maximum product spacing with application to power lomax distribution. Phys A Stat Mech Appl 553:124251
Ferreira A, De Haan L (2015) On the block Maxima method in extreme value theory: PWM estimators. Ann Stat 43(1):276–298
Hosking JR (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B Methodol 52(1):105–124
Hosking JRM, Wallis JR (1997) Regional frequency analysis—an approach based on L-moments. Cambridge University Press, New York
Hussain Z (2017) Estimation of flood quantiles at gauged and ungauged sites of the four major rivers of Punjab, Pakistan. Nat Hazards 86(1):107–123
Hussain Z, Shahzad MN, Abbas K (2017) Application of regional rainfall frequency analysis on seven sites of Sindh, Pakistan. KSCE J Civ Eng 21(5):1812–1819
Interagency Advisory Committee on Water Data [IACWD] (1982) Guidelines for determining flood flow frequency: Bulletin 17b of the hydrology subcommittee, office of water data coordination, U.S. geological survey, Reston, Va., 183 p
Jan NAM, Shabri A (2017) Estimating distribution parameters of annual maximum streamflows in Johor, Malaysia using TL-smoments approach. Theor Appl Climatol 127(1–2):213–227
Katz RW, Parlange MB, Naveau P (2002) Statistics of extremes in hydrology. Adv Water Resour 25(8–12):1287–1304
Khan MSR, Hussain Z, Ahmad I (2019) A comparison of quadratic regression and artificial neural networks for the estimation of quantiles at ungauged sites in regional frequency analysis. Appl Ecol Environ Res 17(3):6937–6959
Kite GW (1988) Frequency and risk analyses in hydrology. Water Resources Publications, Littleton
Koutrouvelis IA, Canavos GC (1999) Estimation in the Pearson type 3 distribution. Water Resour Res 35(9):2693–2704
Kumar Singh R, Kumar Singh S, Singh U (2016) Maximum product spacings method for the estimation of parameters of generalized inverted exponential distribution under progressive type II censoring. J Stat Manag Syst 19(2):219–245
Lee DH, Kim NW (2019) Regional flood frequency analysis for a poorly gauged basin using the simulated flood data and L-moment method. Water 11(8):1717
Lei GJ, Yin JX, Wang WC, Wang H (2018) The analysis and improvement of the fuzzy weighted optimum curve-fitting method of Pearson–type III distribution. Water Resour Manag 32(14):4511–4526
Li W, Zhou J, Sun H, Feng K, Zhang H, Tayyab M (2017) Impact of distribution type in bayes probability flood forecasting. Water Resour Manag 31(3):961–977
Matalas NC, Wallis JR (1973) Eureka! It fits a Pearson type: 3 distribution. Water Resour Res 9(2):281–289
Murage P, Mung’atu J, Odero E (2019) Optimal threshold determination for the maximum product of spacing methodology with ties for extreme events. Open J Model Simul 7(03):149
Naghettini M (ed) (2017) FFundamentals of statistical hydrology. Springer International Publishing, Cham
NIST/SEMATECH (2012) E-handbook of statistical methods. NIST/SEMATECH 2012. http://www.itl.nist.gov/div898/handbook/. Accessed in Dec 2019
Palutikof JP, Brabson BB, Lister DH, Adcock ST (1999) A review of methods to calculate extreme wind speeds. Meteorol Appl 6(2):119–132
Ranneby B (1984) The maximum spacing method. An estimation method related to the maximum likelihood method. Scand J Stat 1(2):93–112
Rutkowska A, Żelazny M, Kohnová S, Łyp M, Banasik K (2018) Regional L-moment-based flood frequency analysis in the upper Vistula River basin, Poland. In: Geoinformatics and atmospheric science. Birkhäuser, Cham, pp 243–263
Singh U, Singh SK, Singh RK (2014) A comparative study of traditional estimation methods and maximum product spacings method in generalized inverted exponential distribution. J Stat Appl Probab 3(2):153
Song D, Ding J (1988) The application of probability weighted moments in estimating the parameters of the Pearson type three distribution. J Hydrol 101(1-4):47–61
Soukissian TH, Tsalis C (2015) The effect of the generalized extreme value distribution parameter estimation methods in extreme wind speed prediction. Nat Hazards 78(3):1777–1809
Vivekanandan N (2015) Flood frequency analysis using method of moments and L-moments of probability distributions. Cogent Eng 2(1):1018704
Wang QJ (1990) Estimation of the GEV distribution from censored samples by method of partial probability weighted moments. J Hydrol 120(1–4):103–114
Wong TST, Li WK (2006) A note on the estimation of extreme value distributions using maximum product of spacings. In: Time series and related topics. Institute of mathematical statistics 52:272–283
Acknowledgements
We acknowledge the support of Irrigation department of KPK, Pakistan for providing data for the study. We are also thankful to the handling editor, associate editor and anonymous reviewers for their constructive comments to improve the quality of the paper.
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Conceptual idea was initiated by [Muhammad Shafeeq ul Rehman Khan] and [Ishfaq Ahmad], Supervision [Ishfaq Ahmad], Data collection and analyses were performed by [Muhammad Shafeeq ul Rehman Khan] and [Zamir Hussain], original draft was written by [Muhammad Shafeeq ul Rehman Khan] and [Zamir Hussain], final draft was reviewed and edited by [Ishfaq Ahmad] and [Zamir Hussain].
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Khan, M.S.u.R., Hussain, Z. & Ahmad, I. Effects of L-Moments, Maximum Likelihood and Maximum Product of Spacing Estimation Methods in Using Pearson Type-3 Distribution for Modeling Extreme Values. Water Resour Manage 35, 1415–1431 (2021). https://doi.org/10.1007/s11269-021-02767-w
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DOI: https://doi.org/10.1007/s11269-021-02767-w