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K = ill - e - 2 il2 or the mean and variance formula in Table 18.2.1. In general for K < estimators are preferable. Hosking and Wallis 70 review alternative esti dures and their precision. Section 18.6.3 develops a relationship bet and GEV distributions. If e must be estimated, the smaller obsery estimator. ex = (ill - ex I + K) FREQUENCY ANALYSIS OF EXTREME EVENTS 18.23 18.22 CHAPTER EIGHTEEN and 'on 2 is important in hydrologic applications and has been the subject of 'es (Ref. 29; examples include Refs. 32, 87, 90, 112, 124). At one time the n that best fitted each data set was used for frequency analysis at that site, roach has now been largely abandoned. Such a procedure is too sensitive g variations in the data. Operational procedures adopted by different ,od-frequency studies for use in their respective countries should be based . ation of regionalization of some parameters and split-sample/Monte . ons of different estimation procedures to find distribution-estimation mbinations which yield reliable flood quantile and risk estimates. Such ca~ed robust because they perform reasonably well for a wide range of nlted States, the log-Pearson type 3 distribution with weighted skew . adopted; an index-flood GEV procedure was selected for the British 18.7.2 and 18.7.3). This principle also applies to frequency analyses of na. 18.2.4 Generalized Pareto Distribution The generalized Pareto distribution (GPO) is a simple distribution useful fo, ing events which exceed a specified lower bound, such as a~ flo~ds above a or daily flows above zero. Moments of the GPO m:e de~~bed.m Tables 1 18.2.1. A special case is the 2-parameter exponentIal dlstnbutlOn (for /( For a given lower bound e, the shape K and scale ex parameters can b easily with L-moments from 18.3.1 Principles and Issues in Selecting a Probability plots are extremely useful for visually revea~ing thep Plots are an effective way to see what the data look hke and; distributions appear consistent with the data. Analytical goo. useful for gaining an appreciation for whether the lack of n sample-to-sample variability, or whether a particular dep~ model is statistically significant. In most cases several dls statistically acceptable fits to the available data so that good . to identify the "true" or "best" ~stri~ution to use: Such ~es can demonstrate that some distnbutlons appear mconslSte Several fundamental issues arise when selecting a distri . tinguish between the following questions: 1. What is the true distribution from which the observati. 2. What distribution should be used to obtain reasona mates of design quantiles and hydrologic risk? 3. Is a proposed distribution consistent with the availa Question I is often asked. Unfortunately, the true complex to be of practical use. Still, L-moment skew diagrams discussed in Sees. 18.1.~ and 1~.3.3 a~egQo families of distributions are conslstent With avatlab goodness-of-fit statistics, such as probability plot co, also been used to see how well a member of each sample. Unfortunately, such goodness-of-fit stati actual family from which the samples are drawn generally fit the data best. Regional L-moment d sample statistics which des,,~ribe the "parent" rather than goodness-of-fit. Coodness-of-fit tests. g Positions and Probability Plots . ation of the adequacy of a fitted distribution is generally per- the observations so that they would fall approximately on a tulated distribution were the true distribution from which the rawn. This can be done with the use of special commercially papers for some distributions, or with the more general tech- on which such special papers are based.30 Section 17.2.2 also display of data. observed values and X(il' the ith largest value in a sample, so . :5X(I)' The random variable Vi defined as ( .'. c: ~..,,: W /.,;' 1<...... 18.3 PROBABILITY PLOTS AND GOODNESS-OF-FIT'TESTS Vi = 1 - F x[X(i)] (18.3.1) ?flnce probability associated with the ith largest observation. ,s were independent, in repeated sampling the Vi have a an i E[ VJ = n + I ( 18.3.2) ( 18.3.3) e exceedance probabilities Vi' one can develop esti- n be used to plot each X(i) against a probability scale. r the events. A visual comparison of the data and a a plot of the ith largest observed event Xli) versus an ould be. If G(x) is the distribution of X, the value of "ly G-I( 1 - qi)' where the probability-plotting posi- e points [G-I(l - qj), X(i)] when plotted would, ooa straight line through the origin. Such a plot uallydisplays[-'(1 - qi),logX(i)]' The exceed-