Department of Applied Mathematics

Due to synoptic-climatological reasons as well as a specific configuration of mountain ranges, the northeast part of the Czech Republic is an area with an enhanced influence of low-pressure systems of the Mediterranean origin. They are associated with an upper-level advection of warm and moist air and often lead to heavy precipitation events. Particularities of this area are evaluated using a regional frequency analysis. The northeast region is identified as a homogeneous one according to tests on statistical characteristics of precipitation extremes (annual maxima of 1-to 7-day amounts), and observed distributions follow a different model compared to the surrounding area. Noteworthy is the heavy tail of distributions of multi-day events, reflected also in inapplicability of the L-moment estimators for the general 4-parameter kappa distribution utilized in Monte Carlo simulations in regional homogeneity and goodness-of-fit tests. We overcome this issue by using the maximum likelihood estimation. The Generalized Logistic distribution is identified as the most suitable one for modelling annual maxima; advantages of the regional over local approach to the frequency analysis consist mainly in reduced uncertainty of the growth curves and design value estimates. The regional growth curves are used to derive probabilities of recurrence of recent heavy precipitation events associated with major floods in the Odra river basin.

Regional analysis is utilized to improve estimates of probabilities of extreme precipitation events in the Czech Republic. The Generalized Extreme Value (GEV) distribution was identified as the most suitable one for modelling maximum annual 1-to 7-day precipitation amounts according to the Lmoment ratio diagram and goodness-of-fit tests; only in the northeast region, the Generalized Logistic (GLO) distribution should be preferred. The regional approach considerably lessens the between-site variation of estimates of the shape parameter of the GEV/GLO distribution compared to at-site procedures, and estimates of high quantiles (e.g. 50-yr return values) are more reliable and climatologically consistent in individual regions. Different shapes of the growth curves also indicate that the homogeneous regions are useful and reasonable for modelling probabilities of heavy precipitation. Particularly noteworthy is the heavy tail of distributions of multi-day extremes in the northeast region. KEY WORDS: extreme precipitation event -regional frequency analysis -L-moments -design values -central Europe -Czech Republic Hosking JRM (1994) The four-parameter kappa distribution. IBM J Res Develop 38: 251-258 Hosking JRM, Wallis JR (1993) Some statistics useful in regional frequency analysis. Water Resour the index-flood method. J Hydrol 255: 194-211 Lu L-H, Stedinger JR (1992) Sampling variance of normalized GEV/PWM quantile estimators and a regional homogeneity test. J Hydrol 138: 223-245 Pal JS, Giorgi F, Bi XQ (2004) Consistency of recent European summer precipitation trends and extremes with future regional climate projections. Geoph Res Lett 31: L13202 Smithers JC, Schulze RE (2001) A methodology for the estimation of short duration design storms in South Africa using a regional approach based on L-moments. J Hydrol 241: 42-52 Šamaj F, Valovič Š, Brázdil R, Gulčíková V (1982) Maximum daily atmospheric precipitation totals in Czechoslovakia. Meteorol Zpr 35: 129-135 [in Slovak, with English summary] Šamaj F, Valovič Š, Brázdil R (1983) Extreme daily precipitation totals in Czechoslovakia. Meteorol Zpr 36: 14-21 [in Slovak, with English summary] Štekl J, Brázdil R, Kakos V, Jež J, Tolasz R, Sokol Z (2001) Extreme daily precipitation on the territory of the

The problem of approximating by a circle of measured data points, registered by RICH detector is considered. The situation is formulated in terms of the Bayesian estimation problem, the solution is based on the Metropolis-Hastings algorithm.

Keywords: climate change extreme value analysis global climate models peaks-over-threshold method quantile regression Poisson process extreme temperatures The paper presents a methodology for estimating high quantiles of distributions of daily temperature in a non-stationary context, based on peaks-over-threshold analysis with a time-dependent threshold expressed in terms of regression quantiles. The extreme value models are applied to estimate 20-yr return values of maximum daily temperature over Europe in transient global climate model (GCM) simulations for the 21st century. A comparison of scenarios of changes in the 20-yr return temperatures based on the non-stationary peaks-over-threshold models with conventional stationary models is performed. It is demonstrated that the application of the stationary extreme value models in temperature data from GCM scenarios yields results that may be to a large extent biased, while the non-stationary models lead to spatial patterns that are robust and enable one to detect areas where the projected warming in the tail of the distribution of daily temperatures is largest. The method also allows splitting the projected warming of extremely high quantiles into two parts that reflect change in the location and scale of the distribution of extremes, respectively. Spatial patterns of the two components differ significantly in the examined climate change projections over Europe.

Several approaches to estimating distributions of precipitation extremes are compared by means of simulation experiments, and their applications into observed data in the Czech Republic are evaluated. Regional frequency models, which take into account data in fixed or flexible 'regions' when fitting a distribution at any site, lead to estimates with much smaller errors compared to a single-site analysis, and efficiently reduce random and climatologically irrelevant variations in the estimates of the model parameters and high quantiles. The region-of-influence (ROI) methodology with a built-in regional homogeneity test is recognized as a useful approach, with the model based on proximity of sites outperforming the Hosking-Wallis regional frequency analysis. Comparison of estimates of the return period of a heavy precipitation event on 24 June 2009, which triggered a disastrous local flash flood, illustrates that the at-site analysis leads to unrealistic and extremely uncertain estimates that strongly depend on whether or not a single outlying observation is involved in the sample, while all regional methods yield return periods in the same order of several hundreds of years, notwithstanding whether the 2009 data are included in the sample. The ROI method may be found useful for modelling probabilities of other meteorological variables, extremes of which are strongly influenced by sampling variability, and may also represent an efficient tool for 'smoothing' random variations in the estimates of model parameters and high quantiles of precipitation in high-resolution regional climate model simulations.

into a two-samples model. The proposed tests are based on two extremes of a splitted sample, and on the empirical distribution functions of their mean. The main idea behind the tests is that the tails of the test criterion distinguish sharply between the heavy and light tails of the basicdistribution. The asymptoticnull distribution of the test c riterion is normal, the tests are consistent against the Pareto type tails with a larger index as well as against the exponential tails. The simulation study shows that the tests distinguish the tails already for moderate samples. Abstrakt: Jurečková (2000) navrhla třídu testu indexu Paretových chvostu pro rozdělení chyb v lineárním regresním modelu. Testy je založeny na extrém- ních regresních kvantilech. V tomto příspěvku uvažujeme analogické testy pro model polohy. Simulační studie ukazuje, že testy rozlišují chvosty poměrně dobře i pro nepříliš rozsáhlé výběry.

. . . ; n, with unknown ðb; sÞ, b A R p , s > 0, and with i.i.d. errors e 1 ; . . . ; e n having a continuous distribution F, we test for the goodness-of-fit hypothesis H 0 : F ðeÞ 1 F 0 ðe=sÞ, for a specified symmetric distribution F 0 , not necessarily normal. Even the finite sample null distribution of the proposed test criterion is independent of unknown ðb; sÞ, and the asymptotic null distribution is normal, as well as the distribution under local (contiguous) alternatives. The proposed tests are consistent against a general class of (nonparametric) alternatives, including the case of F having heavier (or lighter) tails than F 0 . A simulation study illustrates a good performance of the tests.

We propose an estimator of the Pareto tail index m of a distribution, that competes well with the Hill, Pickands and moment estimators. Unlike the above estimators, that are based only on the extreme observations, the proposed estimator uses all observations; its idea rests in the tail behavior of the sample mean ¯ Xn, having a simple structure under heavy-tailed F. The observations, partitioned into N independent samples of sizes n, lead to N sample means whose empirical distribution function is the main estimation tool. The estimator is strongly consistent and asymptotically normal as N →∞ , while n remains fixed. Its behavior is illustrated in a simulation study.

The minimum risk equivariant estimator (MRE) of the regression parameter vector β 6 in the linear regression model enjoys the finite-sample optimality property, but its calculation is 7 difficult, with an exception of few special cases. We study some possible approximations of MRE, 8 with distribution of the errors being known or unknown: A finite-sample approximation uses the 9

Extreme high precipitation amounts are among environmental events with the most disastrous consequences for human society. This paper deals with the identification of 'homogeneous regions' according to statistical characteristics of precipitation extremes in the Czech Republic, i.e. the basic and most important step toward the regional frequency analysis. Precipitation totals measured at 78 stations over 1961−2000 are used as an input dataset. Preliminary candidate regions are formed by the cluster analysis of site characteristics, using the average-linkage clustering and Ward's method. Several statistical tests for regional homogeneity are utilized, based on the 10-yr event and the variation of L-moment statistics. In compliance with results of the tests, the area of the Czech Republic has been divided into four homogeneous regions. The findings are supported by simulation experiments proposed to evaluate stability of the test results. Since the regions formed reflect also climatological differences in precipitation regimes and synoptic patterns causing high precipitation amounts, their future application may not be limited to the frequency analysis of extremes.

Extreme high precipitation amounts are among environmental events with the most disastrous consequences for human society. Estimates of their return periods and design values are of great importance in hydrologic modelling, engineering practice for water resources and reservoirs design and management, planning for weather-related emergencies, etc. The L-moment based method of the regional frequency analysis of maximum annual 1-to 7-day precipitation totals is currently being utilized for the area of the Czech Republic; daily precipitation amounts over 1961-2000 measured at 78 stations are used as an input dataset. The first step of the regional analysis consists in an identification of homogeneous regions. This contribution deals with a comparison of methods used in statistical testing for homogeneity of regions; they include the Lmoment X10 test, tests based on L-moment ratios and on the variation of L-moment statistics. Candidate regions were formed by the cluster analysis of site characteristics (including longitude, latitude, elevation, mean annual precipitation, and mean ratio of summer half-year [May to October] to winter half-year [November to April] precipitation), using the average-linkage clustering and Ward's method. The final homogeneous regions enter the next steps of the regional frequency analysis which concern selection of the most appropriate distribution, and estimation of parameters and quantiles of the fitted distribution together with their uncertainty.

proposed a highly intuitive goodness-offit test of normality with nuisance location and scale parameters. The test has received a considerable attention in the literature; its asymptotic null distribution is covered by the results of de , and was recently studied by Sen .

Some results on M -tests in detection of changes in linear models are presented. The paper focuses on the application of modified permutational arguments in order to obtain approximations for critical values. Theoretical results are accompanied by a simulation study.

If we are interested in such events as the extreme intensity of the wind, high ood levels of the rivers or extreme values of environmental indicators, or maximal or minimal performance of foreign exchange rates or share prices, we should take an interest in the tails of the underlying probability distribution rather than in its central part. Many authors were