Abstract
Analysis of extreme rainfall events can be performed using two main approaches; fitting Generalized Extreme Value distribution to the yearly peaks of events in the observation period or the annual maximum series, and fitting Generalized Pareto distribution to the peaks of events that exceed a given threshold or the partial duration series. Even though partial duration series are able to reduce sampling uncertainty and are useful for analyzing extreme values and tail asymmetries, the series require an optimal threshold. The objective of this study is to compare and determine the best method for selecting the optimal threshold of partial duration series using hourly, 12-hour and 24-hour data of rainfall time series in Peninsular Malaysia. Nine semi-parametric second order reduced-bias estimators are applied to estimate extreme value index and six estimators are used for the external estimation of the second order parameter. A semi-parametric bootstrap is used to estimate mean square error of the estimator at each threshold and the optimal threshold is then selected based on the smallest mean square error. Based on the plots of extreme value index and mean square error, several second order reduced-bias estimators behave reasonably well compared to Hill estimator, as indicated by their stable sample paths and flatter mean square errors.
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References
AghaKouchak A, Nasrollahi N (2010) Semi-parametric and parametric inference of extreme value models for rainfall data. Water Resour Manag 24: 1229–1249
Anagnostopoulou C, Tolika K (2012) Extreme precipitation in europe: statistical threshold selection based on climatological criteria. Theor Appl Climatol 107: 479–489
Balkema AA, de Haan L (1974) Residual life time at great age. Ann Probab 2: 792–804
Begueria S (2005) Uncertainties in partial duration series modelling of extremes related to the choice of the threshold value. J Hydrol 303: 215–230
Beirlant J, Dierckx G, Goegebeur Y, Matthys G (1999) Tail index estimation and an exponential regression model. Extremes 2: 177–200
Beirlant J, Dierckx G, Guillou A, Starica C (2002) On exponential representations of log-spacings of extreme order statistics. Extremes 5: 157–180
Beirlant J, Figueiredo F, Gomes M I, Vandewalle B (2008) Improved reduced-bias tail index and quantile estimators. J Stat Planning Infer 138: 1851–1870
Brodie I (2013) Using volume delivery time to identify independent partial series events. Water Resour Manag 27 (10): 3727–3738
Caeiro F, Gomes M I (2006) A new class of estimators of a “scale” second order parameter. Extremes 9: 193–211
Caeiro F, Gomes M I, Pestana D (2005) Direct reduction of bias of the classical hill estimator. Revstat 3 (2): 113–136
Caers J, Maes M A (1998) Identifying tails, bounds and end-points of random variables. Struct Saf 20 (1): 1–23
Coles S G (2001) An introduction of statistical modeling of extreme values. Springer, New York
Deidda R (2010) A multiple threshold method for fitting the generalized pareto distribution to rainfall time series. Hydrol Earth Syst Sci 14: 2559–2575
Deidda R, Puliga M (2006) Sensitivity of goodness-of-fit statistics to rainfall data rounding off. Phys Chem Earth 31: 1240–1251
de Haan L, Peng L (1998) Comparison of tail index estimators. Stat Neerl 52: 60–70
de Haan L, Stadtmuller U (1996) Generalized regular variation of second order. J Aust Math Soc Ser A 61 (3): 381–395
Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Applications of mathematics. Springer
Fisher R A, Tippett L H C (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math Proc Camb Philos Soc 24 (2): 180–190
Floris M, D’Alpaos A, Squarzoni C, Genevois R, Marani M (2010) Recent changes in rainfall characteristics and their influence on thresholds for debris flow triggering in the dolomitic area of cortina d’ampezzo, north-eastern italian alps. Nat Hazards Earth Syst 10: 571–580
Fraga Alves M I, Gomes M I, de Haan L (2003) A new class of semi-parametric estimators of the second order parameter. Portugaliae Math 60 (2): 193–213
Gomes M I, Martins M J (2002) Asymptotically unbiased estimators of the tail index based on external estimation of the second order parameter. Extremes 5 (1): 5–31
Gomes M I, Oliveira O (2001) The bootstrap methodology in statistics of extremes: Choice of the optimal sample fraction. Extremes 4: 331–358
Gomes M I, Martins M J, Neves M (2000) Alternatives to a semi-parametric estimator of parameters of rare events: the jackknife methodology. Extremes 3 (3): 207–229
Gomes M I, Fraga Alves M I, Santos P A (2008a) Port hill and moment estimators for heavy-tailed models. Commun Stat Simul Comput 37 (7): 1281–1306
Gomes M I, de Haan L, Rodrigues LH (2008b) Tail index estimation for heavy-tailed models: accomodation of bias in weighted log-excesses. J Royal Stat Soc Series B (Methodol) 70: 31–52
Gomes MI, Rodrigues LH, Pereira H, Pestana D (2008c) A semi-parameter estimator of a scale and second order parameter based upon the log-excesses. In: Information technology interfaces, 2008. 30th international conference on information technology interfaces. pp 329–334
Goyal M (2014) Statistical analysis of long term trends of rainfall during 1901-2002 at Assam, India. Water Resour Manag 28 (6): 1501–1515
Gray HL, Schucany WR (1972) The generalized Jackknife statistic. Marcel Dekker
Hall P, Welsh A H (1985) Adaptive estimates of parameters of regular variation. Ann Stat 13: 331–341
Harasawa H, Nishioka S E (2003) Climate change on Japan. KokonShoin Publications, Tokyo
Hill B M (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3 (5): 1163–1174
IPCC (2007) Climate change - the physical science basis. Cambridge University Press, United Kingdom
Lang M, Ouarda T B M J, Bobee B (1999) Towards operational guidelines for over-threshold modeling. J Hydrol 225: 103–117
MacDonald A, Scarrott C J, Lee D, Darlow B, Reale M, Russell G (2011) A flexible extreme value mixture model. Comp Stat Data Anal 55: 2137–2157
Peng L (1998) Asymptotically unbiased estimator for the extreme-value index. Stat Probab Lett 38 (2): 107–115
Pickands J (1975) Statistical inference using extreme order statistics. Annals Stat 3 (1): 119–131
Scarrott C, MacDonald A (2012) A review of extreme value threshold estimation and uncertainty quantification. Revstat 10 (1): 33–60
Shinyie W L, Ismail N (2012) Analysis of t-year return level for partial duration rainfall series. Sains Malaysiana 41 (11): 1389–1401
Shinyie W L, Ismail N, Jemain AA (2013) Semi-parametric estimation for selecting optimal threshold of extreme rainfall events. Water Resour Manag 27(7):2325–2352
Tancredi A, Anderson C W, O’Hagan A (2006) Accouting for threshold uncertainty in extreme value estimation. Extremes 9: 87–106
Um M J, Cho W, Heo J H (2010) A comparative study of the adaptive choice of thresholds in extreme hydrologic events. Stoch Environ Res Risk Assess 24: 611–623
von Mises R (1936) La distribution de la plus grande de n valeurs. Revue Math Union Interbalcanique 1:141–160, reprinted in selected papers of Richard von Mises II (1964). Am Math Soc 2: 271–294
Acknowledgements
We acknowledge the financial support of the Universiti Kebangsaan Malaysia (DPP-2013-081 and 06-01-02-SF0953). We thank the Department of Irrigation and Drainage for providing the hourly rainfall data. We are also immensely grateful to the anonymous reviewer for the helpful comments. We would like to thank Dr Wan Zawiah Wan Zin for her support in postdoctoral supervision.
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Shinyie, W.L., Ismail, N. & Jemain, A.A. Semi-parametric Estimation Based on Second Order Parameter for Selecting Optimal Threshold of Extreme Rainfall Events. Water Resour Manage 28, 3489–3514 (2014). https://doi.org/10.1007/s11269-014-0684-1
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DOI: https://doi.org/10.1007/s11269-014-0684-1