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Hydrol. Earth Syst. Sci., 17, 851–862, 2013

www.hydrol-earth-syst-sci.net/17/851/2013/

doi:10.5194/hess-17-851-2013

© Author(s) 2013. CC Attribution 3.0 License.

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How extreme is extreme? An assessment of daily rainfall

distribution tails

S. M. Papalexiou, D. Koutsoyiannis, and C. Makropoulos

Department of Water Resources, Faculty of Civil Engineering, National Technical University of Athens,

Heroon Polytechneiou 5, 157 80 Zographou, Greece

Correspondence to: S. M. Papalexiou (smp@itia.ntua.gr)

Received: 6 April 2012 – Published in Hydrol. Earth Syst. Sci. Discuss.: 2 May 2012

Revised: 6 February 2013 – Accepted: 6 February 2013 – Published: 28 February 2013

Abstract. The upper part of a probability distribution, usu-

ally known as the tail, governs both the magnitude and the

frequency of extreme events. The tail behaviour of all prob-

ability distributions may be, loosely speaking, categorized

into two families: heavy-tailed and light-tailed distributions,

with the latter generating “milder” and less frequent extremes

compared to the former. This emphasizes how important for

hydrological design it is to assess the tail behaviour correctly.

Traditionally, the wet-day daily rainfall has been described

by light-tailed distributions like the Gamma distribution, al-

though heavier-tailed distributions have also been proposed

and used, e.g., the Lognormal, the Pareto, the Kappa, and

other distributions. Here we investigate the distribution tails

for daily rainfall by comparing the upper part of empirical

distributions of thousands of records with four common the-

oretical tails: those of the Pareto, Lognormal, Weibull and

Gamma distributions. Speciﬁcally, we use 15 029 daily rain-

fall records from around the world with record lengths from

50 to 172yr. The analysis shows that heavier-tailed distribu-

tions are in better agreement with the observed rainfall ex-

tremes than the more often used lighter tailed distributions.

This result has clear implications on extreme event modelling

and engineering design.

1 Introduction

Heavy rainfall may induce serious infrastructure failures and

may even result in loss of human lives. It is common then

to characterize such rainfall with adjectives like “abnormal”,

“rare” or “extreme”. But what can be considered “extreme”

rainfall? Behind any discussion on the subjective nature of

such pronouncements, there lies the fundamental issue of in-

frastructure design, and the crucial question of the threshold

beyond which events need not be taken into account as they

are considered too rare for practical purposes. This question

is all the more pertinent in view of the EU Flooding Direc-

tive’s requirement to consider “extreme (ﬂood) event scenar-

ios” (European Commission, 2007).

Although short-term prediction of rainfall is possible to a

degree (and useful for operational purposes), long-term pre-

diction, on which infrastructure design is based, is infeasible

in deterministic terms. We thus treat rainfall in a probabilistic

manner, i.e., we consider rainfall as a random variable (RV)

governed by a distribution law. Such a distribution law en-

ables us to assign a return period to any rainfall amount, so

that we can then reasonably argue that a rainfall event, e.g.,

with return period 1000 yr or more, is indeed an extreme. Yet,

which distribution law we should choose is still a matter of

debate.

The typical procedure for selecting a distribution law for

rainfall is to (a) try some of many, a priori chosen, parametric

families of distributions, (b) estimate the parameters accord-

ing to one of many existing ﬁtting methods, and (c) choose

the one best ﬁtted according to some metric or ﬁtting test.

Nevertheless, this procedure does not guarantee that the se-

lected distribution will model adequately the tail, which is

the upper part of the distribution that controls both the mag-

nitude and frequency of extreme events. On the contrary, as

only a very small portion of the empirical data belongs to the

tail (unless a very large sample is available), all ﬁtting meth-

ods will be “biased” against the tail, since the estimated ﬁt-

ting parameters will point towards the distribution that best

describes the largest portion of the data (by deﬁnition not

Published by Copernicus Publications on behalf of the European Geosciences Union.

852 S. M. Papalexiou et al.: How extreme is extreme?

belonging to the tail). Clearly, an ill-ﬁtted tail may result

in serious errors in terms of extreme event modelling with

potentially severe consequences for hydrological design. For

example, in Fig. 1 where four different distributions are ﬁtted

to the empirical distribution tail, it can be observed that the

predicted magnitude of the 1000-yr event varies signiﬁcantly.

The distributions can be classiﬁed according to the asymp-

totic behaviour of their tail into two general classes: (a) the

subexponential class with tails tending to zero less rapidly

than an exponential tail (here the term “exponential tail” is

used to describe the tail of the exponential distribution), and

(b) the hyperexponential or the superexponential class, with

tails approaching zero more rapidly than an exponential tail

(Teugels, 1975; Kl¨

uppelberg, 1988, 1989). Mathematically,

this “intuitive” deﬁnition of the subexponential class for a

distribution function Fis expressed as

lim

x→∞

1−F (x)

exp(−x/β ) = ∞ ∀β > 0,(1)

while several equivalent mathematical conditions, in order to

classify a distribution as subexponential, have been proposed

(see, e.g., Embrechts et al., 1997; Goldie and Kl¨

uppelberg,

1998). Furthermore, this is not the only classiﬁcation, as sev-

eral other exist (see, e.g., El Adlouni et al., 2008, and refer-

ences therein). In addition, many different terms have been

used in the literature to refer to tails “heavier” than the expo-

nential, e.g., “heavy tails”, “fat tails”, “thick tails”, or, “long

tails”, that may lead to some ambiguity: see for example the

various deﬁnitions that exist for the class of heavy-tailed dis-

tributions discussed by Werner and Upper (2004). Here, we

use the term “heavy tail” in an intuitive and general way, i.e.,

to refer to tails approaching zero less rapidly than an expo-

nential tail.

The practical implication of a heavy tail is that it predicts

more frequent larger magnitude rainfall compared to light

tails. Hence, if heavy tails are more suitable for modelling

extreme events, the usual approach of adopting light-tailed

models (e.g., the Gamma distribution) and ﬁtting them on

the whole sample of empirical data would result in a signif-

icant underestimation of risk with potential implications for

human lives. However, there are signiﬁcant indications that

heavy tailed distributions may be more suitable. For exam-

ple, in a pioneering study Mielke (1973) proposed the use

of the Kappa distribution, a power-type distribution, to de-

scribe daily rainfall. Today there are large databases of rain-

fall records that allow us to investigate the appropriateness of

light or heavy tails for modelling extreme events. This is the

subject in which this paper aims to contribute.

2 The dataset

The data used in this study are daily rainfall records from

the Global Historical Climatology Network-Daily database

(version 2.60, www.ncdc.noaa.gov/oa/climate/ghcn-daily),

Fig. 1. Four different distribution tails ﬁtted to an empirical tail (P,

LN, W and G stands for the Pareto, the Lognormal, the Weibull

and the Gamma distribution). A wrong choice may lead to severely

underestimated or overestimated rainfall for large return periods.

which includes over 40000 stations worldwide. Many of the

records, however, are too short, have many missing data, or

contain data that are suspect in terms of quality (for details

regarding the quality ﬂags refer to the Network’s website

above).

Thus, only records fulﬁlling the following criteria were se-

lected for the analysis: (a) record length greater or equal than

50yr, (b) missing data less than 20%, and (c) data assigned

with “quality ﬂags” less than 0.1%. Among the several dif-

ferent quality ﬂags assigned to measurements, we screened

against two: values with quality ﬂags “G” (failed gap check)

or “X” (failed bounds check). These were used to ﬂag sus-

piciously large values, i.e., a sample value that is orders of

magnitude larger than the second larger value in the sample.

Whenever such a value existed in the records it was deleted

(this, however, occurred in only 594 records in total, and in

each of these records typically one or two values had to be

deleted). Screening with these criteria resulted in 15137 sta-

tions. The locations of these stations as well as their record

lengths can be seen in Fig. 2, while Table 1 presents some ba-

sic summary statistics of the nonzero daily rainfall of those

records.

We note that we did not ﬁll any missing values as we

deemed it meaningless for this study, focusing on extreme

rainfall, because any regression-type technique would under-

estimate the real values. Missing values only affect the effec-

tive record length and, given the relatively high lower limit of

record length we set (50yr, while much smaller records are

often used in hydrology, e.g., 10–30yr), the resulting prob-

lem was not serious. Additionally, the percentage 20% of

missing daily values refers to the worst case and is actually

much smaller in the majority of the records; thus, missing

values would not alter or modify the conclusions drawn.

Finally, we note that the statistical procedure we describe

next failed in a few records, for reasons of algorithmic con-

vergence or time limits. Excluding these records, the total

number of records where the analysis was applied is 15029.

Hydrol. Earth Syst. Sci., 17, 851–862, 2013 www.hydrol-earth-syst-sci.net/17/851/2013/

S. M. Papalexiou et al.: How extreme is extreme? 853

Fig. 2. Locations of the stations studied (a total of 15137 daily rainfall records with time series length greater than 50 yr). Note that there are

overlaps with points corresponding to high record lengths shadowing (being plotted in front of) points of lower record lengths.

3 Deﬁning and ﬁtting the tail

The marginal distribution of rainfall, particularly at small

time scales like the daily, belongs to the so-called mixed type

distributions, with a discrete part describing the probability

of zero rainfall, or the probability dry, and a continuous part

expressing the magnitude of the nonzero (wet-day) rainfall.

As suggested earlier, studying extreme rainfall requires fo-

cusing on the behaviour of the distribution’s right tail, which

governs the frequency and the magnitude of extremes.

If we denote the rainfall with X, and the nonzero rain-

fall with X|X > 0, then the exceedence probability function

(EPF; also known as survival function, complementary dis-

tribution function, or tail function) of the nonzero rainfall,

using common notation, is deﬁned as

P(X > x|X > 0)=¯

FX|X>0(x) =1−FX|X>0(x), (2)

where FX|X>0(x) is any valid probability distribution func-

tion chosen to describe nonzero rainfall. It should be clear

that the unconditional EPF is easily derived if the probabil-

ity dry p0is known: ¯

FX(x) =(1−p0)¯

FX|X>0(x). Since we

focus on the continuous part of the distribution, and more

speciﬁcally on the right tail, from this point on, for notational

simplicity we omit the subscript in ¯

FX|X>0(x) denoting the

conditional EPF function simply as ¯

F (x). To avoid ambigu-

ity due to the term “tail function” for EPF, we clarify that we

Table 1. Some basic statistics of the 15 137 records of daily rain-

fall. Apart from probability dry (Pdry), these statistics are for the

nonzero daily rainfall.

No. of nonzero Median Mean SD

Pdry (%) values (mm) (mm) (mm) Skew

min 15.11 320 0.40 1.00 1.76 1.37

Q553.92 2121 1.70 3.61 5.01 2.36

Q25 68.55 4038 3.00 6.18 8.28 2.85

Q50 76.35 5973 4.80 9.27 12.08 3.28

(Median)

Q75 83.65 8497 6.90 12.65 16.42 3.94

Q95 91.36 13060 10.20 17.75 24.25 5.38

max 98.25 27867 25.70 83.96 158.02 26.31

Mean 75.13 6604 5.18 9.77 12.97 3.56

SD 11.46 3508 2.70 4.60 6.20 1.31

Skew −0.74 1.12 1.03 1.16 1.88 5.58

use the term “tail” to refer only to the upper part of the EPF,

i.e., the part that describes the extremes.

At this point, however, we need to deﬁne what we con-

sider as the upper part. A common practice is to set a lower

threshold value xL(see, e.g., Cunnane, 1973; Tavares and

Da Silva, 1983; Ben-Zvi, 2009) and study the behaviour for

values greater than xL. Yet, there is no universally accepted

method to choose this lower value. A commonly accepted

method (known as partial duration series method) is to de-

termine the threshold indirectly based on the empirical dis-

tribution, in such a way that the number of values above the

www.hydrol-earth-syst-sci.net/17/851/2013/ Hydrol. Earth Syst. Sci., 17, 851–862, 2013

854 S. M. Papalexiou et al.: How extreme is extreme?

threshold equals the number of years Nof the record (see,

e.g., Cunnane, 1973). The resulting series, deﬁned in this

way, is known in the literature as annual exceedance series

and is a standard method for studying extremes in hydrology

(see, e.g., Chow, 1964; Gupta, 2011).

This may look similar to another common method in

which the Nannual maxima of the Nyears are extracted

and studied. However, the method of annual maxima, by se-

lecting the maximum value of each year, may distort the tail

behaviour (e.g., when the three largest daily values occur

within a single year, it only takes into account the largest

of them). For this reason, instead of studying the Ndaily an-

nual maxima, here we focus on the Nlargest daily values of

the record, assuming that these values are representative of

the distribution’s tail and can provide information for its be-

haviour. Thus, the method adopted here has the advantage of

better representing the exact tail of the parent distribution.

It is worth noting that a common method of studying se-

ries above a threshold value is based on the results obtained

by Balkema and de Haan (1974) and Pickands III (1975).

Loosely speaking, according to these results the conditional

distribution above the threshold converges to the General-

ized Pareto as the threshold tends to inﬁnity. The latter in-

cludes, as a special case, the Exponential distribution. We

note, though, that these results are asymptotic results, i.e.,

valid (or providing a good approximation) if this threshold

value tends to inﬁnity (or if it is very large). In the case

where the parent distribution is of power type or of expo-

nential type, the theory is applicable even for not so large

threshold values because the convergence of the tail is fast.

In other cases, e.g., Lognormal or Stretched Exponential dis-

tributions, the convergence is very slow. The same applies to

the classical extreme value theory (EVT), which predicts that

the distribution of maxima converges to one of the three ex-

treme value distributions. For some examples illustrating the

slow convergence to the asymptotic distributions of EVT (the

same philosophy applies for Balkema–de Haan–Pickands

theorem), see, e.g., Papalexiou and Koutsoyiannis (2013) and

Koutsoyiannis (2004a).

Given that each station has an N-year record of daily val-

ues and a total number nof nonzero values, we deﬁne the

empirical EPF ¯

FN(xi), conditional on nonzero rainfall, as

the empirical probability of exceedence (according to the

Weibull plotting position):

¯

FN(xi)=1−r(xi)

n+1,(3)

where r(xi)is the rank of the value xi, i.e., the position of

xiin the ordered sample x(1)≤... ≤x(n) of the nonzero val-

ues. Thus, the empirical tail is determined by the Nlargest

nonzero rainfall values of ¯

FN(xi)with n−N+1≤i≤n

(note that xL=x(n−N+1)). Some basic summary statistics of

the series of the Nlargest nonzero rainfall values are pre-

sented in Table 2.

Obviously the number of nonzero daily rainfall values is

n=(1−p0)ndNwhere nd=365.25 is the average number

of the days in a year. According to the Weibull plotting po-

sition given in Eq. (3), the exceedence probability ¯p(xL)of

xLwill be

¯p(xL)=1−n−N+1

n+1=N

(1−p0)ndN+1≈1

(1−p0)nd.(4)

This shows that the exceedence probability of the threshold

xLdepends only on the probability dry p0. Interestingly, the

average p0of the records analysed in this study is approxi-

mately 0.75, which implies that the exceedence probability

of xLis on average as low as 0.01, while even for p0=0.95

its value is 0.055. We deem that values above this threshold

can be assumed to belong to the tail of the distribution. We

note that there are studies (see e.g., Beguer´

ıa et al., 2009)

in which the threshold value was chosen to correspond to

the 90th percentile, a value much smaller than the one cor-

responding to our choice of threshold. In Sect. 6 we discuss

further the selection of the threshold, also in comparison with

different methods of selection.

The ﬁtting method we follow here is straightforward, i.e.,

we directly ﬁt and compare the performance of different the-

oretical distribution tails to the empirical tails estimated from

the daily rainfall records previously described. The theoreti-

cal tails are ﬁtted to the empirical ones by minimizing numer-

ically a modiﬁed mean square error (MSE) norm N1 deﬁned

as

N1 =1

N

n

X

i=n−N+1¯

F (x(i) )

¯

FN(x(i) )−12

.(5)

A complete veriﬁcation of the method and a comparison with

other norms is presented in Sect. 6. Here we only note that its

rationale (and advantage over classical square error norms)

is that it properly “weights” each point that contributes in

the sum. Namely, it considers the relative error between the

theoretical and the empirical values rather than using the x

values themselves. For example, if we consider the classical

square error, i.e., (xi−xu)2, with xudenoting the quantile

value for probability uequal to the empirical probability of

the value xi, then large values would contribute much more

to the total error than the smaller ones. This may be a prob-

lem especially for rainfall records where the values usually

differ more than one order of magnitude, e.g., from 0.1mm

to more than 100mm. Obviously, the best ﬁtted tail for a

speciﬁc record is considered to be the one with the smallest

MSE.

The proposed approach, which ﬁts the theoretical distribu-

tion only to the largest points of each dataset, ensures that

the ﬁtted distribution provides the best possible description

of the tail and is not affected by lower values. As an example

of the ﬁtting method, Fig. 3 depicts the Weibull distribution

ﬁtted to an empirical sample (the station was randomly se-

lected and has code IN00121070) by minimizing the norm

Hydrol. Earth Syst. Sci., 17, 851–862, 2013 www.hydrol-earth-syst-sci.net/17/851/2013/

S. M. Papalexiou et al.: How extreme is extreme? 855

Fig. 3. Explanatory diagram of the ﬁtting approach followed. The

dashed line depicts a Weibull distribution ﬁtted to the whole empir-

ical distribution points, while the solid red line depicts the distribu-

tion ﬁtted only to the tail points.

given by Eq. (5) in two ways: (a) in all the points of the em-

pirical distribution, and (b) in only the largest Npoints. It is

clear that the ﬁrst approach (dashed line) does not adequately

describe the tail.

It is well known that several other methods have been ex-

tensively used to estimate the parameters of candidate dis-

tributions, e.g., the lognormal maximum likelihood and the

log-probability plot regression (Kroll and Stedinger, 1996),

and more recently the log partial probability weighted mo-

ments and the partial L-moments (Wang, 1996; Bhattarai,

2004; Moisello, 2007). Yet, the advantage of the proposed

method is that any tail can be ﬁtted in the same manner and

can be directly compared with other ﬁtted tails since the re-

sulting MSE value can clearly indicate the best ﬁtted; in the

aforementioned methods an additional measure has to be es-

timated in order to compare the performance of the ﬁtted dis-

tributions.

4 The ﬁtted distribution tails

It is clear from the previous section that any tail can be ﬁtted

to the empirical ones. Nevertheless, in this study we ﬁt and

compare the performance of four different and common dis-

tribution tails, i.e., the tails of the Pareto type II (PII) the Log-

normal (LN), the Weibull (W), and the Gamma (G) distribu-

tions. These distributions were chosen for their simplicity,

popularity, as well as for being tail-equivalent (or for having

similar asymptotic behaviour) with many other more compli-

cated distributions. It is reminded that two distribution func-

tions Fand Gwith support unbounded to the right are called

tail-equivalent if limx→∞ ¯

F (x)/ ¯

G(x) =cwith 0 < c < ∞.

The Pareto and the Lognormal distributions belong to

the subexponential class and are considered heavy-tailed

Table 2. Some basic statistics of the 15 137 tail samples deﬁned for

an N-year record as the Nlargest nonzero values.

No. of tail Median Mean SD Max

values (mm) (mm) (mm) (mm)

min 50 8.90 10.42 3.01 21.50

Q552 28.30 31.71 8.61 68.60

Q25 61 43.55 48.24 13.85 110.00

Q50 70 62.75 69.12 19.01 152.40

(Median)

Q75 97 85.30 93.72 27.59 218.40

Q95 122 130.30 144.70 47.48 357.60

max 172 977.00 1041.02 395.96 1750.00

Mean 79 68.78 76.01 22.50 175.06

SD 23 34.84 38.20 13.21 93.42

Skew 0.80 2.73 2.58 3.55 1.79

distributions; the Weibull can belong to both classes, depend-

ing on the values of its shape parameter, while the gamma

distribution has essentially an exponential tail but not pre-

cisely (see below). From a practical point of view, the or-

dering of these distributions, from heavier to lighter tail,

is Pareto, Lognormal, Weibull with shape parameter <1,

Gamma and Weibull with shape parameter >1 (see, e.g., El

Adlouni et al., 2008). Note that Pareto is the only power-type

distribution while the other three are of exponential form.

Speciﬁcally, the Pareto type II distribution is the simplest

power-type distribution deﬁned in [0,∞). Its probability den-

sity function (PDF) and EPF are given, respectively, by

fPII(x) =1

β1+γx

β−1

γ−1(6)

¯

FPII (x)=1+γx

β−1

γ,(7)

and it is deﬁned by the scale parameter β > 0 and the shape

parameter γ≥0 that controls the asymptotic behaviour of the

tail. Namely, as the value of γincreases, the tail becomes

heavier and consequently extreme values occur more fre-

quently. For γ=0 it degenerates to the exponential tail while

for γ≥0.5 the distribution has inﬁnite variance. Many other

power-type distributions are tail-equivalent, i.e., exhibiting

asymptotic behaviour similar to x−1/γ with the Pareto type II

tail, e.g., the Burr type XII (Burr, 1942; Tadikamalla, 1980),

the two- and three-parameter Kappa (Mielke, 1973), the Log-

Logistic (e.g., Ahmad et al., 1988) and the Generalized Beta

of the second kind (Mielke Jr. and Johnson, 1974).

Another very common distribution used in hydrology is

the Lognormal with PDF and EPF, respectively,

fLN(x) =1

√π γ x exp −ln2x

β1/γ !(8)

¯

FLN(x) =1

2erfc lnx

β1/γ !(9)

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856 S. M. Papalexiou et al.: How extreme is extreme?

where erfc(x) =2π−1/2R∞

xe−t2dt. The distribution com-

prises the scale parameter β > 0 and the parameter γ > 0 that

controls the shape and the behaviour of the tail. Lognormal

is also considered a heavy-tailed distribution (it belongs to

the subexponential family) and can approximate power-law

distributions for a large portion of the distribution’s body

(Mitzenmacher, 2004). Notice that the notation in Eqs. (8)

and (9) differs from the common one and illustrates more

clearly the kind of the two parameters (scale and shape).

The Weibull distribution, which can be considered as

a generalization of the exponential distribution, is another

common model in hydrology (Heo et al., 2001a, b) and its

PDF and EPF are given, respectively, by

fW(x) =γ

βx

βγ−1exp−x

βγ(10)

¯

FW(x) =exp−x

βγ.(11)

The parameter β > 0 is a scale parameter, while the shape

parameter γ > 0 governs also the tail’s asymptotic behaviour.

For γ < 1 the distribution belongs to the subexponential fam-

ily with a tail heavier than the exponential one, while for

γ > 1 the distribution is characterized as hyperexponential

with a tail thinner than the exponential. Many distributions

can be assumed tail-equivalent with the Weibull for a speciﬁc

value of the parameter γ, e.g., the Generalized Exponential,

the Logistic and the Normal.

Finally, one of the most popular models for describing

daily rainfall is the Gamma distribution (e.g., Buishand,

1978), which, like the Weibull distribution, belongs to the

exponential family. Its PDF and EPF are given, respectively,

by

fG(x) =1

β0 (γ ) x

βγ−1exp−x

β(12)

¯

FG(x) =0γ , x

β/ 0(γ ) (13)

with 0(s , x) =R∞

xts−1e−tdtand 0(s) =R∞

0ts−1e−tdt.

Generally, we can assume that the Gamma tail behaves

similar to the exponential tail. Yet, this is only approxi-

mately correct as the Gamma distribution belongs to a class

of distributions (denoted as S(γ ); see, e.g., Embrechts and

Goldie, 1982; Kl¨

uppelberg, 1989; Alsmeyer and Sgibnev,

1998) that irrespective of its parameter values cannot be

classiﬁed as subexponential, while it is not tail-equivalent

with the exponential. This can be seen from the fact that

the limx→∞ ¯

FG(x)/ ¯

G(x) is 0 for β < βEand ∞for β > βE,

where ¯

G(x) =exp(−x /βE)is the exponential tail. Yet it is

noted that if compared with an exponential tail with β=βE,

then

lim

x→∞

¯

F (x)

¯

G(x) =

0 0 < γ < 1

1γ=1

∞γ > 1.(14)

Therefore, in this case and practically speaking, for 0 < γ <

1 the Gamma distribution has a “slightly lighter” tail than

the exponential tail as it decreases faster, while for γ > 1 it

exhibits a “slightly heavier” tail as it decreases more slowly

than the exponential tail.

All four distributions we compare here, and consequently

their tails, have similarities in their structure as all have

two parameters and speciﬁcally one scale parameter and one

shape parameter. Nevertheless, among the various distribu-

tions with the same parameter structure, inevitably some are

more ﬂexible than others. One way to quantify this ﬂexibil-

ity is by comparing them in terms of various shape mea-

sures (e.g., skewness, kurtosis, etc.). For example, the fea-

sible ranges of skewness for the Pareto, Lognormal, Weibull

and Gamma are, respectively, (2, ∞), (0, ∞), (−1.14, ∞)

and (0, ∞). Therefore, the Weibull distribution seems to be

the most “ﬂexible” distribution among them and the Pareto

the least. Yet this argument is not valid when we focus on the

tail because the general shape of the tail is basically similar

and what differs is the rate at which the tail approaches zero.

5 Results and discussion

The basic statistical results from ﬁtting the four distribution

tails, following the methodology described, to the 15029

daily rainfall records are given in Table 3. In order to assess

which tail has the best ﬁt, the four tails were compared in

couples in terms of the resulting MSE, i.e., the tail with the

smaller MSE is considered better ﬁtted. As shown in Fig. 4,

the Pareto tail, when compared with the other three distribu-

tions, was better ﬁtted in about 60% of the stations. Interest-

ingly, the distribution with the heavier tail of each couple, in

all cases, was better ﬁtted in a higher percentage of the sta-

tions, which implies a rule of thumb of the type “the heavier,

the better”!

Another comparison revealing the overall performance of

the ﬁtted tails was based on their average rank. That is, the ﬁt-

ted tails in each record were ranked according to their MSE,

i.e., the tail with the smaller MSE was ranked as 1 and the

one with the largest as 4. Figure 5 depicts the average rank

of each tail for all stations. Again, the Pareto performed best,

while the most popular model for rainfall, the Gamma distri-

bution, performed the worst. The percentages of each distri-

bution tail that was best ﬁtted are 30.7% for Pareto, 29.8 %

for Lognormal, 13.6% for Weibull and 25.8 % for Gamma.

Again, the Pareto distribution is best according to these per-

centages; interestingly, however, the Gamma distribution has

a relatively high percentage, higher than the Weibull. This

does not contradict the conclusion derived by the average

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S. M. Papalexiou et al.: How extreme is extreme? 857

Table 3. Summary statistics from the ﬁtting of the four distribution

tails into the 15 029 tail-samples of daily rainfall (expressed in mm).

Pareto Lognormal

MSE β γ MSE β γ

Min 0.002 0.42 0.001 0.002 1.22 0.531

Mode∗0.011 7.54 0.134 0.012 8.78 1.060

Mean 0.017 8.80 0.140 0.018 9.46 1.087

Median 0.021 9.51 0.145 0.022 10.59 1.107

Max 0.336 54.79 0.797 0.322 76.74 2.284

SD 0.015 4.92 0.076 0.015 6.44 0.214

Skew 2.910 1.23 0.495 2.755 1.73 0.561

Weibull Gamma

MSE β γ MSE β γ

Min 0.002 0.02 0.230 0.002 3.79 0.010

Mode 0.013 4.33 0.661 0.015 17.50 0.092

Mean 0.019 5.91 0.678 0.023 23.15 0.219

Median 0.022 6.88 0.692 0.032 28.18 0.294

Max 0.298 52.72 1.491 0.482 120.00 2.433

SD 0.015 4.69 0.139 0.034 17.30 0.269

Skew 2.151 1.82 0.668 4.377 1.65 2.567

∗The mode was estimated from the empirical density function (histogram) after

smoothing.

rank. The explanation is that the Gamma distribution was

ranked as best in some cases, but when it was not the best

ﬁtted, it was probably the worst ﬁtted.

Figure 6 depicts the empirical distributions of the shape

parameters of the ﬁtted tails. It is well-known that the most

probable values are the ones around the mode, which for the

Pareto shape parameter is 0.134. Interestingly, this value is

close to the one determined in a different context by Kout-

soyiannis (1999) using Hershﬁeld’s (1961) dataset. This im-

plies that power-type distributions, which asymptotically be-

have like the Pareto, will not have ﬁnite power moments of

order greater than 1/0.134 ≈7.5. Moreover, as the empirical

distribution of the Pareto shape parameter in Fig. 6 attests,

values around 0.2 are also common, implying non-existence

of moments greater than the ﬁfth order. We should thus bear

in mind that sample moments of that or higher order (some-

times appearing in research papers) may not exist. Regarding

the Weibull tail, the estimated mode of its shape parameter

is 0.661, implying a much heavier tail compared to the ex-

ponential one. Finally, it is worth noting that the estimated

mode of the Gamma shape parameter is as low as 0.092. The

shape parameter of the Gamma distribution controls mainly

the behaviour of the left tail, resulting in J- or bell-shaped

densities (loosely speaking, the right tail is dominated by

the exponential function and thus behaves like an exponen-

tial tail). A value that low corresponds to an extraordinarily

J-shaped density, which would be unrealistic for describing

the whole distribution body of daily rainfall. In other words,

Pareto

60%

Lognormal

40%

Pareto

59%

Weibull

41%

Pareto

67%

Gamma

33%

Lognormal

58%

Weibull

42%

Lognormal

66%

Gamma

34%

Weibull

73%

Gamma

27%

0

20

40

60

80

100

Pareto

vs.

Lognormal

Pareto

vs.

Weibull

Pareto

vs.

Gamma

Lognormal

vs.

Weibull

Lognormal

vs.

Gamma

Weibull

vs.

Gamma

Records better fitted H%L

Fig. 4. Comparison of the ﬁtted tails in couples in terms of the re-

sulting MSE. The heavier tail of each couple is better ﬁtted to the

empirical points in a higher percentage of the records.

a Gamma distribution ﬁtted to the whole set of points would

most probably underestimate the behaviour of extremes.

We searched for the existence of any geographical pat-

terns, potentially deﬁning climatic zones, in the best ﬁtted

tails, i.e., the existence of zones in the world where the ma-

jority of the records were better described by one of the stud-

ied distribution tails. The maps in Fig. 7, which depict the

locations of the stations where each distribution tail was best

ﬁtted, did not unveil any regular patterns in terms of the best

ﬁtted distribution but rather seem to follow a random varia-

tion.

Another way to investigate for geographical patterns, as

the previous map did not reveal any useful information, is

to study the ﬁtted tails grouped into two coarser groups: the

subexponential group and the exponential-hyperexponential

group. The former includes the Pareto, the Lognormal and

the Weibull with γ < 1 tails, while the latter includes the

Gamma and the Weibull with γ≥1 tails. Among the 15029

records, subexponential tails were best ﬁtted in 10911 cases

or in 72.6% while exponential-hyperexponential tails were

best ﬁtted in 4118 or in 27.4%. Further, in order to get a

clearer picture instead of constructing maps with the loca-

tions where the ﬁrst-group or the second-group tails were

best ﬁtted, we studied the percentage of subexponential tails

that were best ﬁtted in large regions. Speciﬁcally, we con-

structed a grid covering the entire earth using a latitude

difference 1ϕ =2.5◦and longitude difference 1λ =5◦. In

each grid cell we estimated the percentage of the best ﬁtted

subexponential tails simply by counting the number of the

best ﬁtted subexponential tails divided by the total number

of records within the cell. We present these percentages in

the form of a map in Fig. 8, using a colour scale as shown

in the map’s legend. The cells plotted in the map are those

containing at least two records, so that the calculation of per-

centages have some meaning.

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858 S. M. Papalexiou et al.: How extreme is extreme?

Fig. 5. Mean ranks of the tails for all records. The best-ﬁtted tail

is ranked as 1 while the worst-ﬁtted as 4. A lower average rank

indicates a better performance.

The map of Fig. 8 clearly shows that in the vast majority

of cells subexponential tails dominate (percentage>60 %).

Particularly, out of 532 cells having at least two records, 255

and 163 have percentages of subexponential tails 60–80%

and >80%, respectively. In contrast, in only 35 and 79 cells

are the percentage values in the ranges 0–40% and 40–60 %,

respectively.

6 Veriﬁcation of the ﬁtting method

The use of a different norm for ﬁtting the tail into the em-

pirical data could potentially modify the conclusions drawn.

Nevertheless, this argument is pointless in the sense that the

main concern should be the efﬁciency of the norm used, i.e.,

if it possesses desired properties, e.g., if it is unbiased and has

lower variance in comparison to other candidates. Usually,

the error is expressed in terms of random variable values,

e.g., rainfall values, and not in terms of probability. However,

a literature search did not reveal or verify that the commonly

used norms, e.g., the classical MSE norm, are better than the

norm N1 used here (see Eq. 5).

For this reason, we implemented a Monte Carlo scheme,

which actually replicates the method we followed, where we

evaluate the performance of the norm N1 and also compare

it with the more common norms N2 and N3 deﬁned as

N2 =1

N

n

X

i=n−N+1xu

x(i) −12(15)

N3 =1

N

n

X

i=n−N+1xu−x(i) 2.(16)

Here, xu=Q(u) is the value predicted by the quantile func-

tion Qof the distribution under study for uequal to the em-

pirical probability of x(i) (the ith element the sample ranked

Fig. 6. Histograms of the shape parameters of the ﬁtted tails.

in ascending order) according to the Weibull plotting posi-

tion. The norm N2 has the same rationale as the one we used

but the error is estimated in terms of rainfall values, rather

than in terms of probability, while the norm N3 is the classi-

cal and most commonly used MSE norm.

The Monte Carlo scheme we performed can be summa-

rized in the following steps: (a) we generated 1000 random

samples from each one of the four distributions we studied

with sample size equal to 6600 values, which is approxi-

mately the average number of nonzero daily rainfall values

per record; (b) we selected the scale and the shape parameter

values to be approximately equal with the median values re-

sulted from the analysis of the real world dataset (see Table 3)

in order for the generated random samples to be representa-

tive of the real data; and (c) we ﬁtted each distribution to

its corresponding random sample and estimated the parame-

ters by applying our method for each one of the three norms,

while we set Nequal to 80yr, which is approximately the

average record length.

The results are presented in Fig. 9. The whiskers of the

box plots express the 95% Monte Carlo conﬁdence interval

of the parameters while the dashed lines show the true param-

eter values. It is clear that the norm N1 we used results in al-

most unbiased estimation of the parameters while, especially

for the Pareto and the Lognormal distributions, it results in

markedly smaller variance compared to the classical norm

N3. The norm N2 seems to perform very well for the Pareto,

Lognormal and Weibull distributions (although somewhat bi-

ased) but the results are poor for the Gamma distribution.

The classical and the most commonly used norm N3 is by far

the worst in term of bias except for the Gamma distribution,

for which it performs equally well as N1. In particular, for

the subexponential distributions of this simulation, i.e., the

Pareto, the Lognormal and the Weibull, the classical norm

N3 fails to provide good results. This may point to a more

general conclusion, i.e., that the classical MSE, which is in-

spired based on properties of the normal distribution, is not

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S. M. Papalexiou et al.: How extreme is extreme? 859

Fig. 7. Geographical depiction of the 15029 stations where the best ﬁtted tail is (a) Pareto in 4621, (b) Lognormal in 4486, (c) Weibull in

2051, and (d) Gamma in 3871.

Fig. 8. Geographical variation of the percentage of best ﬁtted subexponential tails in cells deﬁned by latitude difference 1ϕ =2.5◦and

longitude difference 1λ =5◦. In total, in 72.6% of the 15 029 records analysed, the subexponential tails were the best ﬁtted.

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860 S. M. Papalexiou et al.: How extreme is extreme?

Fig. 9. Results of a Monte Carlo scheme implemented to evaluate

the performance of the norm N1 used in ﬁtting of tails in this study,

in comparison to commonly used ones (N2, N3).

a good choice for subexponential distributions. This needs to

be further investigated; however, we deem that there is a ra-

tionale supporting the following conclusion: subexponential

distributions can generate “extremely” extreme values com-

pared to the main “body” of values, and thus, in the classical

norm these values will contribute “extremely” to the total er-

ror heavily affecting the ﬁtting results.

Another issue of potential concern for the validity of the

conclusions drawn is the impact of the sample size, i.e., the

number of the largest events N, or equivalently the threshold

xL, for which the four distribution tails are ﬁtted. As men-

tioned before, we used the annual exceedance series, a stan-

dard method in hydrology in which Nequals the number of

the record’s years. Obviously, Ncan be deﬁned in many dif-

ferent ways, either with reference to record length or as a

ﬁxed number for every record studied.

In order to assess the impact of number of events in the

performance of the four ﬁtted distribution tails, we selected

randomly 2000 records among the 15029 analysed and we

ﬁtted the four distribution tails using six different meth-

ods for deﬁning N. The ﬁrst method (M1) is the one we

used for all above analyses, in which Nequals the number

of the record’s years. In the second (M2) and third (M3)

Fig. 10. Performance results of the four ﬁtted tails in 2000 randomly

selected records using six different methods for selecting the sample

size: (top panel) percentage of records in which each distribution

tail was best ﬁtted; (bottom panel) average ranks of the ﬁtted tails

(lower average rank indicates better performance).

methods we deﬁned the threshold xLas the 90th- and the

95th-percentiles, respectively, so that Nequals the number

of events included in the upper 10% and 5 %, respectively,

of the nonzero values. Obviously, in these two methods N

varies from record to record depending on the total number

of nonzero values, and on the average it equals 667 and 333

values for M2 and M3, respectively. In the rest three methods

(M4, M5 and M6) Nis deﬁned as a ﬁxed number for every

record, i.e., 50, 100 and 200 values, respectively.

The performance results comparing the six methods are

summarized in Fig. 10, which depicts (a) the percentage of

cases in which each distribution was best ﬁtted and (b) the

average rank of each distribution tail. Again the Pareto II

tail was best ﬁtted in a higher percentage of records in all

cases (M1–M6) with the percentage values varying in a nar-

row range. The results are essentially the same with those

obtained from the analysis of the whole database. The only

noticeable difference regards the method M2, in which the

Weibull tail sometimes seems to “gain ground” over the

Gamma and the Lognormal tails. In general it seems that

the Weibull tail improves its performance as Nincreases.

Thus, in M4 where Nhas the lowest value, i.e., 50 values,

it performs the worst, while in M2 where Nis maximum

(667 values on the average), it performs the best. The average

Hydrol. Earth Syst. Sci., 17, 851–862, 2013 www.hydrol-earth-syst-sci.net/17/851/2013/

S. M. Papalexiou et al.: How extreme is extreme? 861

rank, which is a better measure of the overall performance of

the distribution tails, remains essentially the same for each

distribution in all methods. An exception is observed again

in M2 where the Weibull tail performs better than the Log-

normal tail. Apart from this exception the general conclusion

is again that the Pareto II performs the best, followed by the

Lognormal and the Weibull tails, while the Gamma tail per-

forms the worst in all cases.

7 Summary and conclusions

Daily rainfall records from 15029 stations are used to inves-

tigate the performance of four common tails that correspond

to the Pareto, the Weibull, the Lognormal and the Gamma

distributions. These theoretical tails were ﬁtted to the empir-

ical tails of the records and their ability to adequately capture

the behaviour of extreme events was quantiﬁed by comparing

the resulting MSE. The ranking from best to worst in terms

of their performance is (a) the Pareto, (b) the Lognormal,

(c) the Weibull, and (d) the Gamma distributions. The anal-

ysis suggests that heavier-tailed distributions in general per-

formed better than their lighter-tailed counterparts. Particu-

larly, in 72.6% of the records subexponential tails were better

ﬁtted while the exponential-hyperexponential tails were bet-

ter ﬁtted is only 27.4 %. It is instructive that the most popular

model used in practice, the Gamma distribution, performed

the worst, revealing that the use of this distribution under-

estimates in general the frequency and the magnitude of ex-

treme events. Nevertheless, we must not neglect the fact that

the Gamma distribution was the best ﬁtted in 25.8% of the

records.

Additionally, we note that heavy tails tend to be hidden

(see, e.g., Koutsoyiannis, 2004a, b; Papalexiou and Kout-

soyiannis, 2013), especially when the sample size is small.

Thus, we believe that even in the cases where the Gamma tail

performed well, the true underlying distribution tail may be

heavier. This leads to the recommendation that heavy-tailed

distributions are preferable as a means to model extreme rain-

fall events worldwide. We also note that the tails studied

here are as simple as possible, i.e., only one shape parame-

ter controls their asymptotic behaviour. Yet there are many

distributions with more than one shape parameters which

may affect their tail behaviour. Particularly, the Generalized

Gamma (Stacy, 1962) and the Burr type XII distributions

were compared as candidates for the daily rainfall (based on

L-moments) in anonther study, using thousands of empirical

daily records and the former performed better (Papalexiou

and Koutsoyiannis, 2012).

The key implication of this analysis is that the frequency

and the magnitude of extreme events have generally been un-

derestimated in the past. Engineering practice needs to ac-

knowledge that extreme events are not as rare as previously

thought and to shift toward more heavy-tailed probability dis-

tributions.

Acknowledgements. Four eponymous reviewers, Aaron Clauset,

Roberto Deidda, Salvatore Grimaldi and Francesco Laio, and four

commenters, Santiago Beguer´

ıa, Federico Lombardo, Chris Onof

and Patrick Willems, are acknowledged for their public review

comments, as is the editor Peter Molnar for his personal comments.

Most of the comments helped us to improve the original manuscript.

Edited by: P. Molnar

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