Some hydrological applications of small sample estimators of Generalized Pareto and Extreme Value distributions

Journal of Hydrology 301 (2005) 37–53 www.elsevier.com/locate/jhydrol

Some hydrological applications of small sample estimators of Generalized Pareto and Extreme Value distributions C. De Michelea,*, G. Salvadoria,b a

b

DIIAR (Sezione Idraulica), Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy Dipartimento di Matematica ‘Ennio De Giorgi’, Universita` di Lecce, Provinciale Lecce-Arnesano, P.O. Box 193, I-73100 Lecce, Italy Received 3 December 2002; revised 10 June 2004; accepted 15 June 2004

Abstract The Generalized Pareto (GP) and Generalized Extreme Value (GEV) distributions have been widely applied in the frequency analysis of numerous meteorological and hydrological events. There are several techniques for the estimation of the parameters, which use the total sample as a source of information. In this paper, we show how valuable estimates are also possible considering only a proper subset of the sample, and we identify the portion of the sample containing the most relevant information for estimating a given parameter. In turn, this may prevent the use of anomalous values, which may adversely affect standard techniques. Here, we illustrate original techniques (based on linear combinations of ‘selected’ order statistics) to estimate the position parameter, the scale parameter, the quantiles, and the possible scaling behavior of the GP and GEV distributions with negative shape parameters. These estimators are generally unbiased and Mean-Square-Error-consistent. In addition, weakly consistent estimators of quantiles are introduced, the calculation of which does not require the knowledge of any parameter. Some case studies illustrate the applicability of the new techniques in hydrologic practice, and comparisons with standard methods are presented. The new estimators proposed may provide a reasonable alternative to standard methods, and may serve, at least, as a methodology to cross-check the estimates resulting from the application of other techniques. q 2004 Elsevier B.V. All rights reserved. Keywords: Linear estimator; Generalized Extreme Value distribution; Generalized Pareto distribution; Order statistics; Position/scale parameter; Scaling

1. Introduction The Generalized Extreme Value (GEV) distribution, introduced by Jenkinson (1955), provides * Corresponding author. Tel.: C39-02-2399-6233; fax: C39-022399-6207. E-mail addresses: [email protected] (C. De Michele), [email protected] (G. Salvadori). 0022-1694/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2004.06.015

a general framework for the frequency analysis of extreme meteorological and hydrological events. Such a statistical model has been widely used in the analysis of hydrological extremes because of its flexibility in representing the three asymptotic types of extreme value probability distributions, first introduced by Gnedenko (1943) and further refined by Gumbel (1954). For example, the GEV distribution was selected to model extreme flows in the pioneering

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C. De Michele, G. Salvadori / Journal of Hydrology 301 (2005) 37–53

regional flood analysis of Great Britain by NERC (1975), and to model the rainfall frequency in the United States by Willeke et al. (1995)—see also ¨ no¨z and Gunasekara and Cunanne (1991) and O Bayazit (1995) for other hydrological applications. Pickands (1975) introduced the Generalized Pareto (GP) distribution, which provides a framework for the statistical modeling of excesses over thresholds. Such a distribution is closely related to the GEV: Smith (1984) showed that if a homogeneous Poisson process, with GP independent identically distributed excesses, is observed for a fixed time period, then the largest excess is GEV distributed, with the same shape parameter (see also Salvadori and De Michele (2001)). Furthermore, the GP distribution has been widely applied for modeling hydrological processes: see, for example, Hosking and Wallis (1987), Kottegoda and Rosso (1997), and Anderson and Meerschaert (1998). Quite a few techniques exist to estimate the three parameters of both the GP and the GEV distributions (namely, a position, a scale, and a shape parameter): the traditional Method of Moments (MM) and Maximum Likelihood, the often used Probability Weighted Moments/L-moments (LM) (Hosking et al., 1985; Hosking, 1990), and the recent LH-moments (Wang, 1997) and Generalized Maximum Likelihood (Martins and Stedinger, 2000)—for a comparison of the methods, see Hosking et al. (1985) and Madsen et al. (1997a,b). These techniques use all the available sample as a source of information to estimate the parameters of interest: this was historically motivated by the general small size of hydrological samples. However, valuable estimates are also possible considering only a proper subset of the sample, which ‘contains’ the most relevant information for estimating a given parameter; in turn, this may prevent the use of anomalous values, which may affect standard techniques. For example, the shape parameter (when negative) affects the decay rate of the right tail of GP–GEV distributions: Hill (1975) and Pickands (1975) identified in the highest sample values the most relevant portion of the sample to estimate such a parameter (for a review see, for example, Falk et al. (1994), Beirlant et al. (1996), and Reiss and Thomas (2001)). Indeed, in the statistical literature, several authors have made efforts to develop techniques, based on selected order statistics,

to estimate the shape parameter of Pareto-like distributions, but not the position and scale ones. Evidently, this provides incomplete information for evaluating the quantiles, which are important in hydrologic practice. 1.1. Goals of the paper (1) Provide original techniques, based on selected order statistics, to estimate the position and scale parameters of GP–GEV distributions: on the one hand, this fills a gap present in the literature; on the other hand, when combined with algorithms devised for estimating the shape parameter only, this yields a self-contained method to estimate the parameters (and consequently the quantiles) of GP–GEV distributions using only specific subsets of a sample. (2) Introduce, as an alternative, weak estimators of the quantiles: again, these are based on selected order statistics, and may provide valuable estimates of the quantiles without requiring the knowledge or the calculation of any parameter; in particular, as shown below in the application, Section 6, they may provide good estimates in a quasi-Gumbel context, i.e. when the shape parameter is close to zero. In the next sections, we first review the fundamental properties of GP–GEV distributions. Then, we illustrate original direct techniques to estimate the position parameter, the scale parameter, the quantiles, and the possible scaling behavior of both the GP and the GEV distributions: these methods only use proper subsets of the order statistics (hereafter o.s.), and seem to be competitive when compared to standard methods. Consequently, if robust techniques are used to estimate the shape parameter, the overall performance of such ‘mixed’ algorithms is expected to be better than that of other methods often used in the applications. Indeed, it is important to stress that, in some conventional techniques, the estimate of one parameter may depend upon the previous estimates of other parameters; below we introduce methods, which estimate a given parameter without using any information about the others. Incidentally, we observe that a technique, similar to the ones presented here, was already successfully used in a recent paper (De Michele and Salvadori, 2002) to estimate the scale parameter of a GP distribution, solving

C. De Michele, G. Salvadori / Journal of Hydrology 301 (2005) 37–53

a problem which otherwise would have been undetermined. In addition, under proper assumptions on the parameters, the GP–GEV distributions may feature temporal scaling properties, which can be detected simply through the estimates of only the position and the scale parameters; therefore, the new methods outlined later may provide a useful tool for investigating the possible scaling behavior of a process. Then, in Section 5, we derive original weak estimators of the quantiles, for which calculations do not require the knowledge of any parameter. Finally, in Section 6, we consider some case studies, in order to illustrate the applicability of the new technique’s in hydrologic practice, and comparisons with standard techniques—i.e. either the MM or the LM technique—are presented and discussed. Note that these latter algorithms are most frequently used in applications, although they suffer from several (sometimes even critical) drawbacks, as discussed below. The new estimators proposed may provide a reasonable alternative to standard methods, and may serve, at least, as a methodology to cross-check the estimates resulting by the application of other techniques. For all the mathematical details and proofs omitted we refer to Salvadori (2002a,b, 2003), where the new procedures are tested on simulated GP–GEV samples generated via random simulations. We anticipate that the practical tests and the results presented in this work support the theoretical findings of these papers.

2. Properties of GP and GEV distributions In this section, we provide a summary of the fundamental mathematical properties of GP and GEV distributions: here we only consider negative shape parameters (respectively, k and k) in order to deal with upper-unbounded r.v.’s, this case being of greatest interest in applications (as discussed, e.g. in Kottegoda and Rosso (1997) and later in Section 6—the cases k, kR0, although of some importance in some areas, are left for future studies). Indeed, this constraint of negative k, k yields, for both the GP and the GEV distributions (asymptotic) excess probability functions with a power-law asymptote,

39

i.e. for t[1, 1 K FGP ðtÞ zt1=k

and

1 K FGEV ðtÞ zt1=k ;

(1)

and hence only the moments of order less than K1/k or K1/k exist. It is just the presence of such heavy tails, which makes GP and GEV distributions useful for describing extreme phenomena. Indeed, such a tail behavior is typical of Le´vy-stable r.v.’s (see, for example, Feller (1971) and Samorodnitsky and Taqqu (1994)), an important class of random variables playing a fundamental role in modeling extreme events and (multiscaling) multifractal processes (Schertzer and Lovejoy, 1987). Furthermore, under proper conditions illustrated in the study by Salvadori and De Michele (2001), both the GP and the GEV distributions may feature temporal scaling properties, defined as follows. Let Dt and DtrZrDt denote two generic time scales, where rO0 is the scale ratio, and let WDt, WDtr be the observations of the process at such time scales; then W is strict sense simple scaling (Gupta and Waymire, 1990) if their distribution functions are related as FWDtr ðtÞ Z FWDt ðrKd tÞ;

(2)

where d2R is the scaling exponent. In turn, also the quantiles of W do scale, and W features wide sense simple scaling, i.e. those moments of W which exist also scale. It must be stressed that, in practical applications, the scaling regime (when present) usually holds only between an inner cutoff rK and an outer cutoff rC, with rK!rC. As shown below, under the assumption of a constant shape parameter, the scaling features can be detected simply through the estimates of only the position and the scale parameters, therefore, the new methods outlined later may provide a useful tool for investigating the possible scaling behavior of a process. Let X be a GP r.v.; its distribution function FX is given by
 x K b 1=k FX ðxÞ Z 1 K 1 K k ; xO b; (3) c where b2R is a position parameter, cO0 is a scale parameter, and k!0 is a shape parameter. Let q2(0,1); the quantile xq Z FK1 X ðqÞ of order q of X is

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given by xq Z b C c

1 K ð1 K qÞk Z b C cjX ðq; kÞ: k

(4)

Note that the expression for xq is non-linear in q (because of the non-linearity of jX), and diverges when q/1K, as expected, since k!0 and therefore X is not bounded above. Also, considering the time scales Dt and Dtr, if the following power-law relations hold, 8 b Z bDt r d ; > > < Dtr (5) cDtr Z cDt r d ; > > : kDtr Z kDt ; then the GP process is strict sense simple scaling, that is XDtr wr d XDt (where ‘w’ means equality in distribution). Let Z be a GEV r.v.; its distribution function FZ is given by 1=k

zKx FZ ðzÞ Z eKð1Kk a Þ ;

a zO x C ; k

(6)

where x2R is a position parameter, aO0 is a scale parameter, and k!0 is a shape parameter. Let q2(0,1); the quantile zq Z FK1 Z ðqÞ of order q of Z is given by zq Z x C a

1 K ðKln qÞk Z x C ajZ ðq; kÞ: k

(7)

Note that the expression for zq is non-linear in q (because so is the function jZ), and diverges with q/1K, as expected, since k!0 and therefore Z is not bounded above. Also, considering the time scales Dt and Dtr, if the following power-law relations hold, 8 d > < xDtr Z xDt r ; (8) aDt Z aDt r d ; > : r kDtr Z kDt ; then the GEV process is strict sense simple scaling, that is ZDtr wr d ZDt : In the following three sections, we concentrate on the estimation of the parameters and the quantiles of GP and GEV distributions with negative shape parameters. As a fundamental property of all the new estimators proposed, we consider their unbiasedness and Mean-Square-Error (MSE) consistency,

stating whether or not they are distorted and their variance converges to zero when the sample size increases. Since the estimators are essentially linear combinations of order statistics, we use the notation LOS. For all the mathematical details and proofs omitted, see Salvadori (2002a,b, 2003). Hereafter, the Weibull plotting positions Fi Z i=nC 1; iZ1,.,n, will denote the ith non-exceedance probability estimated for a sample of size n: the use of such plotting positions is justified by their mathematical properties. In addition, b$c will denote the integer-part function. For the sake of clarity, Table 1 shows a list of the relevant mathematical notation used.

3. LOS estimators of GP parameters and quantiles In this section, we briefly outline new techniques to estimate the position parameter, the scale parameter, and the quantiles of the GP distribution, in case the shape parameter k is not known (the case ‘k known’, mainly of theoretical interest, is thoroughly discussed in Salvadori (2002a, 2003)) Let X(1)!/!X(n) be the o.s.’s associated with the available i.i.d. GP data {Xi}, iZ1,.,n, where the sample size n is fixed. Let us define the coefficients, for iZ1,.,nK1,
 1 K Fi lnð1 K Fi Þ : 4c;i Z ln and 4b;i Z lnð1 K FiC1 Þ 1 K FiC1 (9) Considering the parameters b and c, and the quantile xq (related in Eq. (4)), we may state the following result (Salvadori, 2002a, 2003). Result 1. The r.v. b^ given by 4b;1 Xð2Þ K Xð1Þ b^ Z 4b;1 K 1

(10)

is an (asymptotically) unbiased MSE-consistent estimator of b, provided that kOKðnK 1Þ=2: Let 1%m!n be fixed. The r.v. c^ given by c^ Z

m 1 X 1 ðX K XðiÞ Þ m iZ1 4c;i ðiC1Þ

(11)

C. De Michele, G. Salvadori / Journal of Hydrology 301 (2005) 37–53 Table 1 List of the relevant mathematical notation used Symbol

Equation

Xi X(i) FX

(3)

b

(3)

c k xq

(3) (3) (4)

jX

(4)

fb,i fc,i b^ c^ k^

(9) (9) (10) (11) (12)

x^q Zi

(12)

Z(i) FZ

(6)

x

(6)

a

(6)

k

(6)

zq

(7)

jZ

(7)

fx,i fa,i x^ a^ k^

(16) (16) (17) (18) (19)

z^q ix

(19) (14)

x^ix

(15)

iq y^iq

(22) (23)

Explanation ith GP observation ith GP order statistics GP distribution function GP position parameter GP scale parameter GP shape parameter GP quantile of order q GP quantile function Coefficient for b^ Coefficient for c^ LOS estimator of b LOS estimator of c Generic estimator of k LOS estimator of xq ith GEV observation ith GEV order statistics GEV distribution function GEV position parameter GEV scale parameter GEV shape parameter GEV quantile of order q GEV quantile function Coefficient for x^ Coefficient for a^ LOS estimator of x LOS estimator of a Generic estimator of k LOS estimator of zq LOS index for ^ a^ x^ix ; x; LOS weak estimator of x LOS index for y^iq LOS non-parametric weak estimator of xq or zq

41

Table 1 (continued) Symbol

Equation

Explanation

y^q

(24)

d

(2)

LOS semi-parametric weak estimator of xq or zq Scaling exponent

is an (asymptotically) unbiased MSE-consistent estimator of c, provided that kOKðnK mÞ=2: Let q2(0,1), and let k^ be an (asymptotically) unbiased MSE-consistent estimator of k. The r.v. x^q given by ^ ^ X ðq; kÞ x^q Z b^ C cj

(12)

is an (asymptotically) unbiased MSE-consistent estimator of xq. Empirically, via numerical simulations and tests, it is found that m zbn=3c may represent a satisfactory practical choice (although a precise mathematical justification is not yet available at present). Therefore, the left tail of the GP distribution turns out to be that part of the sample containing the most relevant information for the estimate of the quantities of interest. Also, the constraint kOKðnK mÞ=2; is a mathematical requirement which guarantees the existence of the variance of the corresponding LOS estimators. It is important to stress that neither c^ nor b^ require the knowledge of any of the remaining GP parameters (respectively, {k, b} or {k, c}), which may represent a considerable advantage in applications. Indeed, no pre-estimates of k (the most difficult parameter to estimate) are needed; thus, compared to, e.g. the MM or the LM techniques (where the calculations form a chain estimating, sequentially, k/c/b), the estimates of all the relevant parameters are independent of (uncorrelated with) one another, and possible errors in the estimate of one parameter do not affect the calculation of the others. Incidentally, we observe that a further remarkable advantage is represented by the fact that the constraints on k for the validity (applicability) of the LOS techniques are less strict than for standard methods. Note that, in order to reduce the bias on the estimate of c, the following alternative corrected

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expression for fc,i could be used:
 1 0 4c;i Z ln 1 C ; n Ki K3

(13)

where 32(0,1). Such a correction works well for small sample sizes (especially using 3z1), and is practically ineffective for large sample sizes. In addition it must be noted that, on the one hand, the ‘external’ estimator of the shape parameter k^ can be freely chosen among many competitive estimators; on the other hand, the ^ goodness of x^q will strongly depend upon that of k; which determines the estimated value of the function jX. For instance, the estimate of k could be carried out using a robust QQ-plot technique (see, for example, Falk et al. (1991), Bierlant et al. (1996), and Reiss and Thomas (2001)), which uses only a limited portion of the sample, the same philosophy adopted by the GP–LOS method illustrated here.

4. LOS estimators of GEV parameters and quantiles In this section, we briefly outline new techniques to estimate the position parameter, the scale parameter, and the quantiles of the GEV distribution, assuming that the shape parameter k is not known (the case ‘k known’, mainly of theoretical interest, is thoroughly discussed in Salvadori (2002b, 2003)) Henceforth, let Z(1)!/!Z(n) be the o.s.’ associated with the available i.i.d. GEV data {Zi}, iZ1,.,n, where the sample size n is fixed. 4.1. Calculation of ix In the following, it will be of primary importance to be able to calculate the index ix ðnÞ 2 f1; .; ng of the o.s. Zðix Þ ‘closest’ to x, in a way independent of the parameters of the GEV distribution—which, then, need not be estimated in advance. In fact, on the one hand, some of the approximations presented below are valid in a neighborhood of x; on the other hand, Zðix Þ would represent an estimator of x. First of all let us state the following result (Salvadori, 2002b).

Result 2. The integer ix ðnÞ given by  n C1 ix ðnÞ Z e

(14)

is the index of the order statistic Zðix Þ closest to x in probability. Thus, Zðix Þ represents the o.s. best approximating x in probability (or, mathematically speaking, in a weak sense), i.e. having the largest probability of being the o.s. closest to x. The calculation of ix is interesting in itself: in fact, for a sample of size n, we find that Zðix Þ always occupies the same relative position with respect to the other o.s.’ (roughly, around n/3), independently of the values of the GEV parameters. Clearly, such a result may be of great relevance in hydrologic applications, since no preestimates of k and a are required to make an estimate of x (as in the case, for instance, when using MM and LM). A further important result is as follows (Salvadori, 2002b). Result 3. The (sequence of) order statistic(s) Zðix ðnÞÞ converges in probability to x as n/N. Therefore, the r.v. x^ix Z Zðix Þ

(15)

is an (asymptotically) unbiased weakly consistent estimator of x, whose calculation does not require the knowledge of the parameters k and a. Mathematically this means that, for all 3O0, Prfjx^ix K xj % 3g/ 1 as n/N. As we shall see later in the application, Section 6, x^ix effectively provides valuable estimates of x as compared to those calculated via standard techniques (which, however, do require the knowledge of k and a). Noting this weak estimator of x, let us now turn to the problem of more strongly estimating the position parameter x and the scale parameter a, and the quantities zq.

4.2. Calculation of x, a, and zq Let us define the coefficients, for iZ1,.,nK1,
 ln Fi lnðKln Fi Þ : 4a;i Z ln and 4x;i Z lnðKln FiC1 Þ ln FiC1 (16)

C. De Michele, G. Salvadori / Journal of Hydrology 301 (2005) 37–53

Considering the parameters x and a, and the quantile zq we may state the following result (Salvadori, 2002b, 2003). Result 4. The r.v. x^ given by x^ Z

4x;ix ZðixC1Þ K Zðix Þ 4x;ix K 1

(17)

1 ðZ K Zðix Þ Þ 4a;ix ðixC1Þ

(18)

is an (asymptotically) unbiased MSE-consistent estimator of a, provided that k>K1/2. Let q2(0,1), and let k^ be an (asymptotically) unbiased MSE-consistent estimator of k. The r.v. z^q given by ^ Z ðq; kÞ ^ z^q Z x^ C aj

(19)

is an (asymptotically) unbiased MSE-consistent estimator of zq. The constraint k>K1/2 is a mathematical requirement which guarantees the existence of the variance of the corresponding LOS estimators. It is important to stress that neither a^ nor x^ require the knowledge of any of the remaining GEV parameters (respectively, {k, x} or {k, a}), which may represent a considerable advantage in the applications. Indeed, no preestimates of k (the most difficult parameter to estimate) are needed; thus, as a difference with, e.g. the MM or the LM techniques (where the calculations form a chain estimating, sequentially, k/a/x), the estimates of all the relevant parameters are independent (uncorrelated) of one another, and possible errors in the estimate of one parameter do not affect the calculation of the others. For practical purposes, we redefine a^ and x^ as follows: a^ Z

Zðix Þ K ZðixK1Þ ZðixC1Þ K Zðix Þ C 34a;ixK1 34a;ix C

and

ZðixC2Þ K ZðixC1Þ 34a;ixC1

" 4x;ixK1 Zðix Þ K ZðixK1Þ 4x;ix ZðixC1Þ K Zðix Þ 1 C x^ Z 3 4x;ixK1 K 1 4x;ix K 1 # 4x;ixC1 ZðixC2Þ K ZðixC1Þ C ; 4x;ixC1 K 1 (21)

is an (asymptotically) unbiased MSE-consistent estimator of x, provided that k>K1/2. The r.v. a^ given by a^ Z

43

(20)

i.e. as the arithmetic average of three estimators of a and x calculated using the four o.s.’s closest to x. In fact, while on the one hand such an empirical trick does not affect in a significant way the distortion of the estimators, on the other hand it greatly contributes to a reduction in their variance. Therefore, a neighborhood of the weak estimator Zðix Þ turns out to be that part of the sample containing the most relevant information for the estimate of the quantities of interest. In addition, it must be noted that, on the one hand, the ‘external’ estimator of the shape parameter k^ can be freely chosen among many competitive estimators; on the other hand, the goodness of z^q will strongly ^ which determines the estimated depend upon that of k; value of the function jZ. As for the GP case, the estimate of k could be carried out using a robust QQ-plot technique, which uses only a limited portion of the sample, the same philosophy adopted by the GEV–LOS method illustrated here.

5. Weak estimators of the quantiles Besides a point estimate of a given quantile, it is also possible to provide an estimate by intervals. Although these estimators generally only converge in probability (i.e. weakly), nevertheless they may provide distribution free confidence intervals, and thus none of the parameters of the probability distribution involved need to be estimated in advance. Here, the general philosophy is to combine results of non-parametric statistics and specific results derived in the previous Sections for GP–GEV distributions, in order to provide new estimators of the GP–GEV quantiles which do not require the estimates of all the parameters. Let Y be a continuous r.v. with distribution function FY, and let yq Z FK1 be the Y ðqÞ quantile of order q of Y, where q2(0,1). Also, let

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{Yi}, iZ1,.,n, be a sample of size n extracted from FY, and let Y(1)!/!Y(n) be the o.s.’s associated with the sample. The o.s.’s Y(1),.,Y(n) generate a partition of the real axis in nC1 disjoint random intervals (Y(i), Y(iC1)), iZ0,.,n, where conventionally Yð0Þ ZKN: and YðnC1Þ ZCN: Clearly, any of such intervals has a positive probability of containing the quantile yq, which can easily be calculated when Y has a GP–GEV distribution. Most importantly, it is possible to show that there exists a particular interval which is expected to contain yq with a probability larger than for any other interval, as stated in the following result (Salvadori, 2003). Result 5. The index iq(n) given by

empirical strategy to by-pass the problem as follows (Salvadori, j2003). k q s ; and let rZ C 1; so that q! sC1 Let sZ 1Kq 1 i b ðsK nC 1Þ nC1 c and Fi Z sC1 ; iZ1,.,s. Then, considering as the law of Y either a GP or a GEV distribution, it is possible to derive the following (approximate) result. Result 7. Let q2(0,1); the r.v. y^q given by

iq ðnÞ Z bðn C 1Þqc

where RwU(0,1) is a uniform r.v. independent of the sample, is an estimator of yq, the calculation of which does not require the knowledge of any of the GP–GEV parameters. The theoretical analysis of the properties of y^q is difficult, and also it makes no sense studying its asymptotic behavior (since, for n large enough, y^iq can be used instead). Thus, at present, its performance has been investigated only via numerical tests, and below it will be tested using hydrological data.

(22)

identifies the random interval ðYðiq Þ ; YðiqC1Þ Þ which has the highest probability of including the quantile yq. Clearly, the privileged interval ðYðiq Þ ; YðiqC1Þ Þ also provides a confidence interval for yq, and confidence intervals at any desired level can be easily generated. However, we do not investigate this point in the present paper. Instead, we introduce a weak point estimator as follows (Salvadori, 2003). Result 6. Let q2(0,1); the (sequence of) order statistic(s) Yðiq ðnÞÞ converges in probability to yq as n/N. Therefore, the r.v.

y^ q ZYðnÞ  8
rC1 > > C^cRlnð2Þ; GPcase; > :aln ^ ^ CaRln ; GEVcase; lnFsK1 lnFs (24)

6. Practical case studies y^iq Z Yðiq Þ

(23)

is an (asymptotically) unbiased weakly consistent estimator of yq, whose calculation does not require the knowledge of the parameters of the law of Y. As a consequence, we note that the o.s. Yðiq Þ best approximating yq in probability always occupies the same relative position within the given sample around a fraction q of the sample size. Therefore, in case all the parameters are unknown, a neighborhood of the weak estimator Yðiq Þ turns out to be that part of the sample containing the most relevant information for the estimate of the quantile of interest. If 1O qO n=ðnC 1Þ then iqZn, and therefore the interval (Y(n),CN) has the highest probability of including the quantile yq. Because such an interval is semi-infinite, no practical inference could be made on yq. However, even in such a case, we may propose an

In this section, we present applications of the techniques illustrated above to hydrological processes. About the assumption of a negative shape parameter, Meigh et al. (1997) analyzed the variability of the regional flood frequency distribution with climate, using a GEV model: they considered 22 regions throughout the world and showed how in subtropical regions the variability of peak flood is high, and rare floods can be extremely large; incidentally, these results support the use of a negative shape parameter in our regions. Indeed, in north-western Italy (the region pertaining to the data analyzed below), De Michele and Rosso (2000) estimated the GEV parameters of maximum annual rainfall depth for the temporal durations of 1, 3, 6, 12, and 24 h: for 58 rainfall stations with a sample dimension larger than 30 years they estimated a

C. De Michele, G. Salvadori / Journal of Hydrology 301 (2005) 37–53

negative shape parameter in 86% of the cases, while for the remaining 14% the hypothesis of a null shape parameter cannot be rejected. Furthermore, considering the regionalization of maximum annual peak flood in north-western Italy, De Michele and Rosso (2000) estimated GEV shape parameters in the range (K0.32, K0.09). As illustrated in Salvadori (2002a,b, 2003) on simulated samples, some standard techniques, such as the LM or the MM, do not perform well when the shape parameter takes on values in specific intervals (e.g. when k, kz0 in the LM case, or when k, kzK1/3 in the MM case). This means that the results provided by these techniques must be considered with care when the shape parameters range in such regions. Below we consider three case studies involving hydrological data in which the LOS techniques provide reliable results, whereas those derived via other methods apparently are either suspicious or unreliable. The first case study of interest concerns the Bisagno drainage basin, located in Thyrrhenian Liguria (north-western Italy), where hourly recorded rainfall depth data collected by five gauges are available for a period of seven years. Assuming homogenous climatic conditions within the basin, we may legitimately consider the data collected by these gauges as samples extracted from the same statistical population; thus, this is essentially equivalent to analyzing a data set of 35 years of measurements (for more detail, see Salvadori and De Michele (2001)). It must be pointed out that the rainfall values, at the temporal scale Dt, are in effect calculated as the maximum rainfall depth in a time interval Dt within any single storm. In the analysis below, we consider four increasing temporal scales: namely, Dt1Zl h, Dt2Z2 h, Dt3Z 3 h, and Dt4Z6 h. Since the digital sampling of rainfall introduced series of identical data, we only consider rainfall values larger than 30 mm, a threshold above which such repetitions disappear: clearly this corresponds to a low-censoring of the sample but, as is well known (see, for example, Davison (1984) and Smith (1984)), the GP distribution is stable with respect to excesses over thresholds operations. Furthermore, without loss of generality, the position parameter b is arbitrarily put equal to zero: actually, this is the natural lower bound

45

of the rainfall depth distribution (and, indeed, the experimental measurements are as small as 0.2 mm). Most importantly, not only the robust QQ-plot technique always estimates a negative shape parameter k for all durations, but also such estimates are quite close to one another: therefore, given such an empirical evidence and the conditions for the simple scaling derived in Eq. (5), we may assume a common value k^QQ (namely, the sample average) to be used by the LOS technique when estimating the quantiles; we recall that no estimates of k are needed to calculate c in the LOS case. On the contrary, the MM always provides positive estimates of k, which are also more scattered than the QQ-plot ones: this is definitely in contrast to some refined results derived from the QQ-plot analysis (not shown), which clearly indicate that the shape parameter should be negative; incidentally, such a conclusion is also supported by the LM estimates. In addition, a GP model based on a positive k from the MM results would imply that the variables of interest were bounded above (but unbounded towards KN), which is not physically meaningful. In Table 2, we report the estimates of the quantities of interest for all the four temporal scales considered: we show the sample averages k^ of the shape parameters, and the errors indicated correspond to one sample standard deviation. The estimates of the parameter c are calculated using, as a common value of the shape parameter k, respectively, k^MM and k^LM ; instead, we recall that no estimates of k are needed in the LOS case. Furthermore, the estimates of the three standard quantiles x0.9, x0.99, and x0.999 are also reported. Finally, the parameter d is calculated via linear regressions in the log–log plane, considering the scaling of both c and the quantiles: shown are the estimated values, as well as the errors corresponding to 95% confidence intervals. In Fig. 1, we show the observed rainfall data and the corresponding fitted GP conditional c.d.f. curves for the duration of 1 h. Evidently, the LOS fit is good, whereas the MM one is more and more biased for larger and larger values, due to positive estimate of the shape parameter (as shown below, such a behavior introduces underestimates of the upper quantiles). Note that the available sample contains extreme events: for instance, rainfall depths as large as z90 mm are found. In Fig. 2, we show the scaling

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Table 2 Analysis of the GP data of the Bisagno basin: estimates of the quantities of interest using different techniques (units are in mm); n indicates the sample size

n

d

1h

2h

3h

6h

45

96

129

157

20.6 52.0 114.7 190.1

25.4 64.2 141.5 234.6

32.9 83.3 183.6 304.4

0.49G0.17 0.49G0.17 0.49G0.17 0.49G0.17

26.3 55.4 103.5 145.4

33.4 69.8 130.6 183.4

47.2 103.9 194.3 272.9

0.49G0.04 0.57G0.01 0.57G0.01 0.57G0.01

20.3 52.4 117.9 199.7

24.8 64.0 144.0 243.9

34.0 87.7 197.3 334.2

0.50G0.08 0.50G0.08 0.50G0.08 0.50G0.08

LOS—kQQ ZK0:081G0:011 c 13.7 x0.9 34.6 76.3 x0.99 x0.999 126.4 MM—kMM ZC0:061G0:092 c 16.9 x0.9 37.2 x0.99 69.6 x0.999 97.7 LM—kLM ZK0:097G0:068 c 13.9 x0.9 35.8 x0.99 80.5 136.4 x0.999

of the quantiles x0.9, x0.99, and x0.999; evidently, a scaling behavior is present. As already mentioned, the MM estimates are larger than the (reliable) LOS ones for small q’s, and are progressively smaller and smaller for larger and larger q’s. As a general comment, it is clear that the LOS estimates are to be considered as valuable, whereas the MM ones both fail to provide a reliable fit of the

data and correct estimates of the extreme quantiles. In addition, the LM estimates reported in Table 1 support the results provided by the LOS technique (for the sake of clarity, the LM estimates are not shown in Figs. 1 and 2). The second case study of interest concerns rainfall data collected at Viu` Fucine station, located in Stura di Viu` river basin, a tributary of the Po river (north-

Fig. 1. Observed rainfall data (diamonds) at 1 h temporal scale, and corresponding fitted GP conditional c.d.f. curves: the reduced variate yZKlnð1K pÞ is plotted.

C. De Michele, G. Salvadori / Journal of Hydrology 301 (2005) 37–53

47

Fig. 2. Scaling of the GP quantiles xq using the LOS technique (circles), and the Method of Moments (triangles). The straight lines are fitted via linear regressions.

western Italy). The measurements consist of maximum annual rainfall depths for a period of 44 years. Here, we show that a GEV distribution with a (strongly) negative shape parameter k is compatible with the measurements, whereas the MM fails to

provide a reliable fit of the data. However, as already mentioned, this is not surprising, since the MM technique does not perform well when k zK1=3: In the analysis below, we consider five increasing temporal scales: namely, Dt1Z1 h, Dt2Z3 h,

Table 3 Analysis of the GEV data of the Viu` Fucine station: estimates of the quantities of interest using different techniques (units are in mm); n indicates the sample size

n

d

1h

3h

6h

12 h

24 h

32

39

40

42

44

7.8 40.6 40.0 65.8 118.6 223.1

14.1 59.1 59.0 104.4 199.0 386.3

29.5 82.9 81.0 177.8 376.5 769.7

43.0 110.4 111.0 248.7 538.1 1110.8

0.55G0.44 0.50G0.04 0.50G0.03 0.52G0.20 0.54G0.32 0.55G0.38

11.4 41.2 68.7 101.9 139.8

24.6 57.0 116.6 188.6 270.6

34.6 82.1 165.9 267.2 382.6

57.1 110.9 249.1 416.3 606.6

0.67G0.18 0.49G0.03 0.57G0.08 0.61G0.11 0.63G0.14

9.6 39.9 68.9 122.1 214.8

18.7 56.1 112.4 216.0 396.3

27.0 80.3 161.3 310.4 569.9

42.3 109.8 237.0 470.9 877.9

0.61G0.16 0.50G0.02 0.55G0.07 0.58G0.11 0.60G0.13

LOS—kQQ ZK0:30G0:12 a 8.9 x 23.1 xix 23.0 51.7 z0.9 z0.99 111.4 z0.999 229.7 MM—kMM ZK0:06G0:06 a 7.3 x 23.9 z0.9 41.5 62.7 z0.99 z0.999 87.0 LM—kLM ZK0:25G0:04 a 6.4 x 22.8 z0.9 42.1 z0.99 77.8 139.7 z0.999

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Dt3Z6 h, Dt4Z12 h, and Dt5Z24 h. In Table 3, we report the estimates of the quantities of interest for all the five temporal scales considered. The shape parameter k turns out to be always negative: its sample average k^ is shown, and the error indicated corresponds to one sample standard deviation. In the LOS case, k is estimated via a robust QQ-plot technique, and it turns out to be quite different from zero; incidentally, such a conclusion is also supported by the LM estimates. On the contrary, the estimates provided by the MM are suspicious, being close to zero: however, this is not surprising, as the MM often returns upward biased negative estimates of k when k is not close to zero; clearly, such a behavior introduces underestimates of the upper quantiles. The estimates of the parameters x and a are calculated using, as a common value of the shape parameter k, respectively, k^MM and k^LM ;we recall that no estimates of k are needed in the LOS case. In addition, for the sake of comparison, also reported are the weak estimates x^ix ; which evidently show a striking agreement with the other estimators of x. Furthermore, the estimates of the three standard quantiles z0.9, z0.99, and z0.999 are also reported. Finally, again, the parameter d is calculated via linear regressions in the log–log plane, considering the scaling of both the position and scale parameters, and the

quantiles: shown are the estimated values, as well as the errors corresponding to 95% confidence intervals. In Fig. 3, we show the observed rainfall data and the corresponding fitted GEV c.d.f. curves for the duration of 24 h. Evidently, the LOS fit is good, whereas the MM one is clearly unable to provide a reliable interpolation, presumably as a consequence of a wrong estimate of the shape parameter. Indeed, for short temporal scales, all the techniques considered apparently provide fairly good fits. On the contrary, for large durations involving more extreme events (the relevant ones in the hydrologic practice), the MM has problems, whereas the LOS estimates show a good performance, as it is evident considering the plots of Fig. 3; as a matter of facts, also this sample contains extreme events: for instance, maximum annual rainfall depths as large as z460 mm are found. In addition, note that the fits are calculated using, as GEV parameters for the duration of 24 h, those estimated via the scaling relations provided by Eq. (8), instead of those directly fitted on the available data. In Fig. 4, we show the scaling of the quantiles z0.9, z0.99 and z0.999: note how a scaling behavior is present. As already mentioned, the MM estimates are more and more biased downward as q increases, and such a behavior is even more evident than in the previous case study (however, such a drawback was

Fig. 3. Observed rainfall data (diamonds) at 24 h temporal scale, and corresponding fitted GEV c.d.f. curves: the reduced variate yZ KlnðKlnðpÞÞ is plotted.

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49

Fig. 4. Scaling of the GEV quantiles zq using the LOS technique (circles), and the Method of Moments (triangles). The straight lines are fitted via linear regressions.

already clear from experiments on simulated samples, as shown in Salvadori (2002b, 2003)). Also in this case, it is clear that the LOS estimates are to be considered as valuable, whereas the MM ones fail to provide both a fit of the data and correct estimates of the extreme quantiles. The third case study concerns rainfall data collected at Rimasco station, located in Sesia river basin, a tributary of the Po river (north-western Italy). The measurements consist of maximum annual rainfall depths of various durations for a period of 42 years. Here, the statistics of the available sample is well modeled by a GEV distribution with a shape parameter quite close to zero, i.e. a Gumbel distribution: the interesting point is that many of the LOS approximations become exact as k/0 K (see Salvadori (2002b)), and therefore we test whether the LOS techniques might be used even in a quasiGumbel case. In the analysis below, we consider five increasing temporal scales: namely, Dt1Z1 h, Dt2Z3 h, Dt3Z 6 h, Dt4Z12 h, and Dt5Z24 h. In Table 4, we report the estimates of the quantities of interest for all the five temporal scales considered. The shape parameter k turns out to be quite close to zero: shown is its ^ and the error indicated corresponds sample average k; to one sample standard deviation. In the LOS case, k is estimated via a robust QQ-plot technique (not shown);

note that the estimates provided by the MM are reliable, since the MM generally returns valuable estimates of the shape parameter when kz0. On the contrary, the LM technique may provide suspicious estimates in such a situation (see Salvadori (2002b, 2003)); indeed, in the present case, the LM estimates of k are always positive: we shall see below how this may affect the fit of the data; note that this would also imply that the variables of interest were bounded above (but unbounded towards KN), which is not physically meaningful. The estimates of the parameters x and a are calculated using, respectively, k^MM and k^LM as a common value of k; instead, we recall that no estimates of k are needed in the LOS case. In addition, for the sake of comparison, also reported are the weak estimates x^ix ; which again show a striking agreement with the other estimators of x. Furthermore, the estimates of the three standard quantiles z0.9, z0.99, and z0.999 are also reported; note that, in the LOS case, the weak estimates y^q of the quantiles are shown, being interested in testing the reliability of such estimators in a quasi-Gumbel context (incidentally, the results are quite close to the ones provided by the other LOS estimators illustrated in this work). Finally, again the parameter d is calculated via linear regressions in the log–log plane, considering the scaling of both the position and scale parameters, and the quantiles: shown are the

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Table 4 Analysis of the GEV data of the Rimasco station: estimates of the quantities of interest using different techniques (units are in mm); n indicates the sample size

n

d

1h

3h

6h

12 h

24 h

34

38

41

42

42

10.3 38.3 38.0 66.8 104.6 124.2

18.5 59.5 59.0 103.5 133.8 171.4

23.3 88.8 87.6 144.8 168.6 261.6

37.8 122.6 122.0 210.2 263.3 345.6

0.51G0.17 0.53G0.06 0.53G0.06 0.52G0.02 0.49G0.04 0.53G0.02

13.5 38.3 68.6 100.3 131.4

18.7 58.6 100.7 144.7 187.8

23.3 85.8 138.3 193.1 246.9

38.3 119.6 205.7 295.6 383.8

0.49G0.09 0.52G0.05 0.51G0.03 0.50G0.04 0.50G0.05

14.8 38.5 67.5 90.0 106.3

22.1 59.0 102.1 135.6 160.0

29.0 87.3 143.9 187.8 219.8

46.2 121.5 211.8 282.0 333.0

0.52G0.05 0.53G0.05 0.52G0.07 0.52G0.07 0.52G0.07

LOS—kQQ ZK0:09G0:03 a 7.6 x 23.6 xix 23.4 y0.9 41.6 y0.99 51.8 64.8 y0.999 MM—kMM ZK0:0001G0:00001 a 7.6 x 23.5 z0.9 40.5 z0.99 58.3 z0.999 75.7 LM—kLM ZC0:13G0:05 a 8.6 x 23.6 z0.9 40.5 53.5 z0.99 z0.999 63.0

estimated values, as well as the errors corresponding to 95% confidence intervals. In Fig. 5, we show the observed rainfall data and the corresponding fitted GEV c.d.f. curves for the duration of 1 h. Evidently, the LOS fit is good. For the

sake of comparison, we also plot the fitted Gumbel curve which, incidentally, is practically identical to the MM one (being k^MM z0). In addition, it is evident how the ability of the LM estimates to fit the data becomes weaker when increasing the strength of the

Fig. 5. Observed rainfall data (diamonds) at 1 h temporal scale, and corresponding fitted GEV c.d.f. curves: the reduced variate yZKlnðKlnðpÞÞ is plotted. Also shown is the corresponding fitted Gumbel c.d.f.

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Fig. 6. Scaling of the GEV quantiles zq using the LOS weak estimators y^q (circles), and the Method of Moments (triangles). The straight lines are fitted via linear regressions.

events considered, a consequence of the positive estimate of the shape parameter. In Fig. 6, we show the scaling of the quantiles. Since the LOS weak estimators of the quantiles are particularly reliable when kz0, in this case we plot the estimates provided by y^q : Indeed, the LOS and the MM estimates are empirically compatible and consistent, and the LOS weak estimators seem to provide valuable estimates: however, it is worth recalling that no estimate of k was used for calculating the weak estimates. Overall, the results indicate that the LOS estimators (using no pre-estimates of the shape parameter) always provide estimates of the position and the scale parameters consistent with those of the MM or the LM techniques at the top of their performances (which depends upon the value of the shape parameter). In addition, the LOS estimates of the quantiles are reliable. Furthermore, the estimates provided by the weak estimators (using no information at all about the parameters) should be considered as valuable. Such a conclusion is particularly true if we take into proper account the drawbacks of each technique, as put in evidence by the results on simulated samples (see Salvadori (2002a,b, 2003)). Overall, these examples may show the practical validity and usefulness of the LOS approach outlined in the present paper.

7. Conclusions In this paper, we emphasize several important facets of hydrologic modeling and estimation of parameters, quantiles, and scaling behavior of Pareto-like probability distributions. In particular, we consider the possibility of estimating the position and the scale parameters of both the GP and the GEV distributions, using direct techniques (LOS). In contrast to conventional methods, these techniques may provide valuable estimates considering only a proper subset of the sample; in turn; this may prevent the use of anomalous values, which may affect standard methods. Essentially, the LOS estimators are linear combinations of ‘selected’ order statistics, and provide estimates of parameters and quantiles either in case the shape parameter is known or not. In some cases, the constraints to be imposed on the shape parameter for the techniques to work are less strict than for other methods; furthermore, some of the new estimators proposed may estimate a given parameter without using any information about the others. The LOS estimators may be coupled with robust techniques for the estimation of the shape parameter in order to calculate the quantiles. In addition, weakly consistent estimators are

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introduced, whose calculation does not require the knowledge of any parameter. Furthermore, we show how to investigate the temporal scaling features of GP and GEV distributed data using the LOS techniques. Case studies involving hydrological processes were presented, which illustrate the applicability of the LOS techniques in hydrologic practice. The results show that the new estimators proposed may provide a reasonable alternative to standard methods, and may serve, at least, as a methodology to cross-check the estimates resulting from the application of other techniques.

Acknowledgements The research was partially supported as a part of the project ‘FRAMEWORK’ (Flash-flood Risk Assessment under the iMpact of land use changes and river Engineering WORKs) granted by the European Commission, D.G. XII, Contract no. ENV4-CT97-0529. The support of CNR-GNDCI via the RIVERS-project is also gratefully acknowledged by C. De Michele. The support of ‘Progetto Giovani Ricercatori’ is gratefully acknowledged by G. Salvadori.

References Anderson, P.L., Meerschaert, M.M., 1998. Modeling river flows with heavy tails. Water Resour. Res. 34 (9), 2271–2280. Beirlant, J., Teugels, J.L., Vynckier, P., 1996. Practical Analysis of Extreme Values. Leuven University Press, Leuven. Davison, A.C., 1984. Modeling excesses over high thresholds, with an application, in: Tiago de Oliveira, J. (Ed.), Statistical Extremes and Applications. Reidel, Dordrecht, pp. 461–482. De Michele, C., Rosso, R., 2000. Valutazione delle piene nell’Italia nordoccidentale: bacino padano e Liguria tirrenica La Valutazione delle Piene in Italia. Publ. CNR-GNDCI, Roma pp. 4.1–4.26 (in Italian). De Michele, C., Salvadori, G., 2002. On the derived flood frequency distribution: analytical formulation and the influence of antecedent soil moisture condition. J. Hydrol. 262, 245–258. Falk, M., Hu¨sler, J., Reiss, R., 1994. Laws of Small Numbers: Extremes and Rare Events. Birkha¨user, Basel. Feller, W., 1971., second ed An Introduction to Probability and Its Applications, vol 2. Wiley, New York. Gnedenko, B.V., 1943. Sur la distribution limite du terme maximum d’une se´rie ale´atoire. Ann. Math. 44, 423–453.

Gumbel, E.J., 1954. Statistical theory of extremes and some practical applications. Nat. Bur. Standards Appl. Math. Ser. 33. Gunasekara, T.A.G., Cunanne, C., 1991. Expected probabilities of exceedance for non-normal flood distributions. J. Hydrol. 128, 101–113. Gupta, V.K., Waymire, E., 1990. Multiscaling properties of spatial rainfall and river flow distributions. J. Geophys. Res. 95, 1999–2009. Hill, B.M., 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat. 3 (5), 1163–1174. Hosking, J.R.M., 1990. L-moments: analysis and estimation of distributions using linear combinations of order statistics. J. R. Stat. Soc., Ser. B 52 (2), 105–124. Hosking, J.R.M., Wallis, J.R., 1987. Parameter and quantile estimation for the Generalized Pareto distribution. Technometrics 29 (3), 339–349. Hosking, J.R.M., Wallis, J.R., Wood, E.F., 1985. Estimation the Generalized Extreme Value distribution by the method of probability weighted moments. Technometrics 27 (3), 251–261. Jenkinson, A.F., 1955. The frequency distribution of the annual maximum (or minimum) value of meteorological elements. Q. J. R. Meteorol. Soc. 81, 158–171. Kottegoda, N.T., Rosso, R., 1997. Probability, Statistics, and Reliability for Civil and Environmental Engineers. McGrawHill, New York. Madsen, H., Rasmussen, P.F., Rojsberg, D., 1997a. Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events 1: at-site modeling. Water Resour. Res. 33 (4), 747–758. Madsen, H., Rasmussen, P.F., Rojsberg, D., 1997b. Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events 2: regional modeling. Water Resour. Res. 33 (4), 759–770. Martins, E.S., Stedinger, J.R., 2000. Generalized maximumlikelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour. Res. 36 (3), 737–744. Meigh, J.R., Farquharson, F.A.K., Sutcliffe, J.V., 1997. A worldwide comparison of regional flood estimation methods and climate. Hydrol. Sci. J., Ser. B 42 (2), 225–244. National Environmental Research Council, 1975. Flood Studies Report, 1, Hydrological Studies. NERC Publication, London p. 550. ¨ no¨z, B., Bayazit, M., 1995. Best-fit distributions of largest O available flood samples. J. Hydrol. 167, 195–208. Pickands, J., 1975. Statistical inference using extreme order statistics. Ann. Stat. 3 (1), 119–131. Reiss, R., Thomas, M., 2001. Statistical Analysis of Extreme Values. Birkha¨user, Basel. Salvadori, G., 2002a. Linear combinations of order statistics to estimate the position and scale parameters of the Generalized Pareto distribution. Stoch. Environ. Res. Risk Assess. 16, 1–17. Salvadori, G., 2002b. Linear combinations of order statistics to estimate the position and scale parameters of the Generalized Extreme Values distribution. Stoch. Environ. Res. Risk Assess. 16, 18–42. Salvadori, G., 2003. Linear combinations of order statistics to estimate the quantiles of Generalized Pareto and Extreme Values laws. Stoch. Environ. Res. Risk Assess. 17 (1/2), 18–42.

C. De Michele, G. Salvadori / Journal of Hydrology 301 (2005) 37–53 Salvadori, G., De Michele, C., 2001. From Generalized Pareto to Extreme Values laws: scaling properties and derived features. J. Geophys. Res. 106 (D20), 24063–24070. Samorodnitsky, G., Taqqu, M.S., 1994. Stable Non-Gaussian Random Processes. Chapman & Hall, New York. Schertzer, D., Lovejoy, S., 1987. Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes. J. Geophys. Res. 92 (D8), 9693–9714.

53

Smith, R.L., 1984. Threshold methods for sample extremes, in: Tiago de Oliveira, J. (Ed.), Statistical Extremes and Applications. Reidel, Dordrecht, pp. 621–638. Wang, Q.J., 1997. LH moments for statistical analysis of extreme events. Water Resour. Res. 33 (12), 2841–2848. Willeke, G.E., Hosking, J.R.M., Wallis, J.R., Guttman, N.B., 1995. The national drought atlas (draft). IWR Report 94-NDS-4. US Army Corps of Engineering, Fort Belvoir, VA.