# zbMATH — the first resource for mathematics

On spatial extremes: with application to a rainfall problem. (English) Zbl 1273.62258
Summary: We consider daily rainfall observations at 32 stations in the province of North Holland (the Netherlands) during 30 years. Let $$T$$ be the total rainfall in this area on one day. An important question is: what is the amount of rainfall $$T$$ that is exceeded once in 100 years? This is clearly a problem belonging to extreme value theory. Also, it is a genuinely spatial problem. Recently, a theory of extremes of continuous stochastic processes has been developed. Using the ideas of that theory and much computer power (simulations), we have been able to come up with a reasonable answer to the question above.

##### MSC:
 62P12 Applications of statistics to environmental and related topics 62G32 Statistics of extreme values; tail inference
Full Text:
##### References:
 [1] Alila, Y. (1999). A hierarchical approach for the regionalization of precipitation annual maxima in Canada., J. Geophys. Res. 104 31,645-31,655. [2] Allen, R. J. and DeGaetano, A. T. (2005). Considerations for the use of radar-derived precipitation estimates in determining return intervals for extreme areal precipitation amounts., J. Hydrol. 315 203-219. [3] Bacchi, B. and Ranzi, R. (1996). On the derivation of the areal reduction factor of storms., Atmos. Res. 43 123-135. [4] Balkema, A. and de Haan, L. (1974). Residual life time at great age., Ann. Probab. 2 792-804. · Zbl 0295.60014 [5] Bell, F. C. (1976). The areal reduction factor in rainfall frequency estimation. Technical Report 35, Cent. for Ecol. and Hydrol., Wallingford, UK. [6] Buishand, T. A. (1983). Extremely high rainfall amounts and the theory of extreme values., Cultuurtechnisch Tijdschrift 23 9-20. (In Dutch.) [7] Buishand, T. A. (1991). Extreme rainfall estimation by combining data from several sites., Hydrological Sciences J. 36 345-365. [8] Coles, S. G. (1993). Regional modelling of extreme storms via max-stable processes., J. Roy. Statist. Soc. Ser. B 55 797-816. JSTOR: · Zbl 0781.60041 [9] Coles, S. G. and Tawn, J. (1994). Statistical methods for multivariate extremes: An application to structural design (with discussion)., Appl. Statist. 43 1-48. · Zbl 0825.62717 [10] Coles, S. and Tawn, J. (1996). Modelling extremes of the areal rainfall process., J. Roy. Statist. Soc. Ser. B 58 329-347. JSTOR: · Zbl 0863.60041 [11] de Haan, L. (1984). A spectral representation for max-stable processes., Ann. Probab. 12 1194-1204. · Zbl 0597.60050 [12] de Haan, L. and Ferreira, A. (2006)., Extreme Value Theory : An Introduction . Springer, New York. · Zbl 1101.62002 [13] de Haan, L. and Lin, T. (2001). On convergence towards an extreme value distribution in, C [0, 1]. Ann. Probab. 29 467-483. · Zbl 1010.62016 [14] de Haan, L. and Pereira, T. T. (2006). Spatial extremes: Models for the stationary case., Ann. Statist. 34 146-168. · Zbl 1104.60021 [15] de Haan, L. and Zhou, C. (2008). On extreme value analysis of a spatial process., Revstat 6 71-81. · Zbl 1153.62074 [16] Drees, H. and Huang, X. (1998). Best attainable rates of convergence for estimators of the stable tail dependence function., J. Multivariate Anal. 64 25-47. · Zbl 0953.62046 [17] Fowler, H. J. and Kilsby, C. G. (2003). A regional frequency analysis of United Kingdom extreme rainfall from 1961 to 2000., Int. J. Climatol. 23 1313-1334. [18] Gellens, D. (2002). Combining regional approach and data extension procedure for assessing GEV distribution of extreme precipitation in Belgium., J. Hydrol. 268 113-126. [19] Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire., Ann. Math. 44 423-453. JSTOR: · Zbl 0063.01643 [20] Huang, X. (1992). Statistics of bivariate extreme values. Ph.D. thesis, Tinbergen, Institute. [21] Leander, R. and Buishand, T. A. (2007). Resampling of regional climate model output for the simulation of extreme river flows., J. Hydrol. 332 487-496. [22] Natural Environment Research Council (NERC) (1975). Flood studies report. II. Meteorological studies, Cent. for Ecol. and Hydrol., Wallingford, UK. [23] Øksendal, B. (1992)., Stochastic Differential Equations , 3rd ed. Springer, New York. · Zbl 0747.60052 [24] Pickands III, J. (1975). Statistical inference using extreme order statistics., Ann. Statist. 3 119-131. · Zbl 0312.62038 [25] Resnick, S. I. (1987)., Extreme Values , Regular Variation , and Point Processes . Springer, New York. · Zbl 0633.60001 [26] Schlather, M. (2002). Models for stationary max-stable random fields., Extremes 5 33-44. · Zbl 1035.60054 [27] Sivapalan, M. and Blöschl, G. (1998). Transformation of point to areal rainfall: Intensity-duration-frequency curves., J. Hydrol. 204 150-167. [28] Smith, R. (1990). Max-stable processes and sptial extremes. Unpublished, manuscript. [29] Stewart, E. J. (1989). Areal reduction factors for design storm construction: Joint use of raingauge and radar data. In, New Directions for Surface Water Modeling ( Proceedings of the Baltimore Symposium , May 1989 ) 31-40. IAHS Publ. no. 181. International Association of Hydrological Sciences (IAHS). [30] Veneziano, D. and Langousis, A. (2005). The areal reduction factor: A multifractal analysis., Water Resour. Res. 41 W07008. DOI: 10.1029/2004WR003765. · Zbl 1302.28021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.