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P R E P R I N T
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Submi ed o S ochas ic En i onmen al Resea ch and Risk Assessmen
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The Risk o Nega i ely Biased and O e -con iden Re u n Le el
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Es ima es: A C i ique o he Me as a is ical App oach o Ex emes
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By To ben Schmi h, Ka s en A nbje g-Nielsen, and Bo Ch is iansen.
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Na ional Cen e o Clima e Resea ch, Danish Me eo ological Ins i u e, Copenhagen, Denma k
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Co esponding au ho Email: [email p o ec ed]
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ORCID TS: h ps://o cid.o g/0000-0003-3442-4381 , KAN: h ps://o cid.o g/0000-0002-6221-9505 ,
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BC: h ps://o cid.o g/0000-0003-2792-4724
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Abs ac
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Classical ex eme alue analysis (EVA) o en p o ides la ge unce ain ies on es ima ed e u n
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le els due o limi ed amoun o da a a ailable. Ma ani and Ignaccolo (2015) aim o o e come his
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by he me as a is ical ex eme alue (MEV) app oach. He e ex emes a e ea ed as la ge o dina y
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e en s desc ibed by one common, known dis ibu ion, and he e o e a much la ge pool o da a
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a e a ailable o es ima ion. They pe o med Mon e Ca lo simula ions wi h syn he ic Weibull-
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dis ibu ed ain all se ies and showed ha he MEV app oach gi es unbiased es ima es o
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ex emes wi h a smalle unce ain y han classical EVA does. Howe e , he MEV app oach neglec s
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ha many complex physical mechanisms in luence ain all. This means ha he ail beha io o he
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dis ibu ion canno be in e ed om he o dina y e en s. We he e o e eplica ed hei wo k bu
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added new Mon e Ca lo expe imen s o s udy he classical EVA and he MEV me hodologies wi h a
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sligh ly pe u bed ail o he unde lying dis ibu ion. When applying he MEV app oach, i.e. i ing
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a Weibull dis ibu ion o he pe u bed Weibull se ies, we ob ained sys ema ically nega i ely
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biased es ima es wi h oo na ow con idence in e als – MEV became o e -con iden . In con as ,
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classical EVA also he e p oduced unbiased es ima es. Finally, we showed ha goodness-o - i es s
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a e no able o p o ide guidance on whe he MEV can p o ide unbiased and con iden e u n
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le els. Fu he Mon e Ca lo simula ions showed ha hese conclusions seem o be qui e gene al
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and no dependen on he speci ic dis ibu ion. Consequen ly, he MEV app oach is unsui able o
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p o ide eliable e u n le els, and we s ongly cau ion agains using i in eal-wo ld applica ions.
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Keywo ds
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‘ex eme alue s a is ics’ , me as a is ical, MEV, GEV, Gumbel, bias
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1 In oduc ion
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Classical ex eme alue analysis (EVA) (Coles 2001) is widely used wi hin e.g. hyd ology o es ima e
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e u n le els and hei unce ain ies. The key s eng h o classical EVA is ha i p o ides a
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heo e ically ounded asymp o ic s a is ical dis ibu ion o he e u n le els. Howe e , he
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pa ame e s o his dis ibu ion a e es ima ed om he la ges alues in he da a only. In p ac ical
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applica ions, he a ailable da a is ypically limi ed, as mos eal-wo ld ime se ies a e only a ound
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100 yea s long o e en less. As a esul , he e a e e y ew alues a ailable o es ima ing he
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pa ame e s, leading o conside able unce ain ies in he es ima ed e u n le els, especially o
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la ge e u n pe iods.
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Any me hod which can educe hese la ge unce ain ies is welcomed. Ma ani and Ignaccolo
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(2015), hence o h MI2015, in oduces he me as a is ical ex eme alue (MEV) app oach wi h he
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explici aim o a oid elying on he asymp o ic dis ibu ion o ex emes in classical EVA. Ins ead,
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he MEV app oach ega ds ex emes as la ge o dina y e en s, which a e all dis ibu ed acco ding
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o a common s a is ical dis ibu ion ( he pa en dis ibu ion), which is assumed o be known. This
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seemingly sol es he p oblem o da a sho age in classical EVA by inc easing he amoun o da a
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a ailable o i ing subs an ially. In addi ion, he MEV allows o in e annual s ochas ic a ia ions
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o he s a is ical p ope ies o he dis ibu ion, hence he e m me as a is ical.
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MEV was o iginally applied o daily ain all (MI2015, Zo ze o e al. 2016) assuming a Weibull
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pa en . The MEV app oach has subsequen ly been applied o ain all (Ma a e al. 2018;
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Schellande e al. 2019; Zo ze o and Ma ani 2020; Miniussi and Ma ani 2020; Poschlod 2021;
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Falkens eine e al. 2023; Zo ze o e al. 2024; De ò e al. 2025). Besides, he MEV app oach has
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also been applied o s eam low (Miniussi e al. 2020a; Mush aq e al. 2022) , hu icane
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occu ence (Hosseini e al. 2020), ain all connec ed o opical cyclones (Miniussi e al. 2020b),
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and sea le el (Ca uso and Ma ani 2022; Boumis e al. 2024).
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The MEV me hodology assumes ha he en i e ange o ain all ollows a Weibull dis ibu ion. This
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is la gely mo i a ed by physical a gumen s ound in Wilson and Toumi (2005) e en hough hei
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conclusions e e speci ically o hea y ain all. Ma inez-Villalobos and Neelin (2019) also
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employed idealized physical a gumen s bu concluded ha he whole ange o daily ain all,
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including he ail, should be desc ibed by a gamma dis ibu ion. Papalexiou e al. (2013) i ed ou
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di e en dis ibu ions o he ail o housands o ain all s a ions wo ldwide and ound ha none
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o he ou dis ibu ions sys ema ically was be e i .
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Thus, so a , he e is no ag eed exac o m o he ail o he dis ibu ion o ain all, as assumed in
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he MEV app oach. In ou iew his is no su p ising, since ain all is a complex phenomenon,
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in ol ing e.g. he dis inc ion be ween s a i o m and con ec i e ain all, land su ace p ocesses,
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and hei in e ac ion. So a he han a simple dis ibu ion, we would expec ain all o be
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dis ibu ed acco ding o a mix u e o dis ibu ions, o which some could be unknown (Smi h e al.
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2011; Shin e al. 2015). Also i e uno is gene ally in luenced by se e al p ocesses ha will ha e
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majo impac s on he mos ex eme obse a ions (Me z and Blöschl 2008; Me z e al. 2022). Sea
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su ges could also be caused by di e en p ocesses being impo an o di e en anges o e u n
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pe iods (Su e al. 2024).
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I is impo an o ha e a ho ough unde s anding o he physical p ocesses esponsible o he
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ex emes. In pa icula , i is c ucial o conside whe he he la ges obse a ions a e ou lie s
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esul ing om e y a e physical ci cums ances. Neglec ing his could lead o unde es ima ion o
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exceedance a es o a e e en s (Me z e al. 2022) as well as hei unce ain ies (Me z e al.
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2015).
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MI2015 o e look he abo e conside a ions and he e o e he need o s udying how sensi i e he
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es ima ed e u n le els a e o he de ails o he ail o he dis ibu ion. Ins ead, MI2015 in hei
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Mon e Ca lo simula ion s udy ely exclusi ely on he Weibull dis ibu ion o bo h gene a ing he
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syn he ic se ies and subsequen es ima ing e u n le els. Mo i a ed by his we wan o in es iga e
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i he de ailed ail beha io o he dis ibu ion used o gene a e he syn he ic se ies impac s he
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esul s o he MEV app oach and he classical EVA. We will eplica e he analysis amewo k o
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MI2015 and compa e he MEV and he classical EVA me hods on Weibull-dis ibu ed syn he ic
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se ies, and in addi ion will we will do simila analysis on syn he ic se ies dis ibu ed acco ding o a
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Weibull dis ibu ion wi h a sligh ly pe u bed ail. Fu he mo e, we will in es iga e whe he
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commonly used goodness-o - i es s can dis inguish be ween he Weibull and he pe u bed
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Weibull syn he ic se ies o see i such es s could help de e mine i he MEV app oach is
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app op ia e o a gi en ime se ies.
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Recen ly, ano he c i ique o he MEV app oach (Se inaldi e al. 2025) appea ed. They showed ha
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i accoun ing o au oco ela ion, he MEV becomes s ongly biased. Ou c i ique should be
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ega ded complemen a y o hei .
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2 Me hods
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Sec ion 2.1 desc ibes he block maximum me hod o classical EVA and Sec ion 2.2 he MEV
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me hodology. Sec ion 2.3 desc ibes he Weibull and he pe u bed Weibull mix u e dis ibu ions
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used o gene a e syn he ic ime se ies. Sec ion 2.4 desc ibes he use o goodness-o - i es s.
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Finally, Sec ion 2.5 desc ibes ou p o ocol o pe o ming he Mon e Ca lo simula ions.
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2.1 Classical EVA, he block maximum me hod
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The block maximum me hod o classical ex eme alue ime se ies analysis conside s blocks o 𝑛
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independen and iden ically dis ibu ed andom a iables {𝑥𝑖}𝑖=1,𝑛 dis ibu ed acco ding o he
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pa en cumula i e dis ibu ion unc ion (CDF) 𝐹. The aim is o ind he dis ibu ion o hei
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maximum:
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𝑦𝑛=max{𝑥1,…,𝑥𝑛}. (1)
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The abo e p esump ions lead o he CDF o 𝑦𝑛 being
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𝐻𝑛(𝑦)=[𝐹(𝑦)]𝑛. (2)
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In p inciple one could es ima e 𝐹 om obse a ions and use Eq. 2 o ge 𝐻𝑛. Howe e , as Coles
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(2001) poin s ou , e y small disc epancies in he es ima e o 𝐹 can lead o subs an ial
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disc epancies o 𝐹𝑛.
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In classical EVA i is he e o e acknowledged ha he de ails o 𝐹, and in pa icula i s ail, a e
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la gely unknown. Ins ead, he Fishe –Tippe –Gnedenko heo em (Fishe and Tippe 1928;
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Gnedenko 1943) is used o ob ain a limi dis ibu ion o 𝑦𝑛 as as 𝑛→∞. This limi dis ibu ion is
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he gene alized ex eme alue (GEV) dis ibu ion:
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𝐺(𝑦)=exp[− (1+𝜉𝑦−𝜇
𝜎)−1
𝜉]. (3)
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Thus, i 𝑛 is la ge enough, hen 𝐻𝑛(𝑦)≈𝐺(𝑦) o any desi ed accu acy ega dless o 𝐹. Fo u he
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de ails, see Coles (2001).
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The pa ame e s 𝜇 and 𝜎 a e he loca ion and scale pa ame e , espec i ely, while he shape
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pa ame e 𝜉 dis inguishes be ween di e en ypes o ail beha io o 𝐺. Thus, i 𝜉=0 hen 𝐺 is
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he Gumbel o EV1 dis ibu ion wi h exponen ial ail beha io . I 𝜉>0 hen 𝐺 is he F éche o
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EV2 dis ibu ion wi h a hea y ail. Finally, i 𝜉<0 hen 𝐺 is he e e sed Weibull o EV3
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dis ibu ion, which has a bounded uppe ail.
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In p ac ical applica ions wi h ime se ies we spli he o iginal se ies o leng h 𝑁 yea s in o
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consecu i e blocks o one yea leng h o ge a se ies o annual maxima, {𝑦𝑘 }𝑘=1,𝑁. The e o e 𝑛
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om now on deno es he numbe o da a alues pe yea . The h ee pa ame e s o 𝐺, o in he
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Gumbel case wi h 𝜉=0, only he emaining wo pa ame e s, a e es ima ed using s anda d
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echniques. We will e e o GEV and Gumbel i ing o annual maxima as he GEV and Gumbel
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me hod, espec i ely.
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Fo design and isk assessmen pu poses, i is common o es ima e he e u n le el, 𝑦𝑇,
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co esponding o a e u n pe iod 𝑇, which is de ined in e ms o he exceedance p obabili y as
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1−𝐺(𝑦𝑇)=1
𝑇. (4)
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This means ha 𝑦𝑛, and he e o e also he o iginal ime se ies 𝑥𝑖, exceeds 𝑦𝑇 on a e age one ime
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du ing a ime span o 𝑇 yea s. We can isola e 𝑦𝑇 om Eq. 4 o ge
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𝑦𝑇=𝐺−1(1−1
𝑇). (5)
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2.2 The me as a is ical ex eme alue (MEV) me hodology
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In he MEV me hodology (MI2015), all da a alues, including he ex emes, a e desc ibed by a
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speci ied pa en dis ibu ion, 𝐹 - he base dis ibu ion. The co esponding ex eme alue CDF o
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annual maxima, 𝐻𝑛, is ob ained using Eq. (2), which includes i ing he base dis ibu ion o da a.
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The MEV also allows o s ochas ically a ying 𝑛,and pa ame e s 𝜗1…𝜗𝑘 o 𝐹. The e o e, he MEV
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CDF o annual maxima is de ined as he ensemble a e age o 𝐻𝑛,
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𝜁(𝑦)=∑∫𝑑𝜗1…𝑑𝜗𝑘
𝑛𝑔(𝑛,𝜗1…𝜗𝑘)𝐻𝑛(𝑦;𝑛,𝜗1…𝜗𝑘)
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=∑∫𝑑𝜗1…𝑑𝜗𝑘𝑛 𝑔(𝑛,𝜗1…𝜗𝑘)[𝐹( 𝜗1…𝜗𝑘)]𝑛, (6)
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which is he gene al o mula ion o he MEV ex eme alue dis ibu ion. He e 𝑔(𝑛,𝜗1…𝜗𝑘) is he
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join densi y dis ibu ion o he pa ame e s 𝑛 and 𝜗1…𝜗𝑘. I we assume e godici y, we can
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app oxima e he in eg a ion o e phase space by a ime a e age o e he numbe o yea s, 𝑁,
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yielding
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𝜁(𝑦)≈1
𝑁∑𝐻𝑛𝑗(𝑦;𝑛𝑗,𝜗1𝑗…𝜗𝑘𝑗)
𝑁
𝑗=1 =1
𝑁∑𝐹(𝑦; 𝜗1𝑗…𝜗𝑘𝑗)𝑛𝑗
𝑁
𝑗=1 , (7)
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whe e ϑ1j…ϑkj and nj a y among yea s. The app oxima e o mula ion in Eq. 7 elimina es he
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need o knowing 𝑔, and Eq. 7 can p o ide pe cen iles o ζ and he e o e he associa ed e u n
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le els om Eq. 5.
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2.3 The Weibull and he pe u bed Weibull pa en s
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MI2015 use a wo-pa ame e Weibull dis ibu ion wi h CDF
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𝐹𝑊(𝑥;𝑤,𝐶)=1−exp[−(𝑥
𝐶)𝑤] (8)
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as base dis ibu ion in he MEV me hodology, whe e 𝑤 and 𝐶 a e he Weibull shape and scale
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pa ame e , espec i ely. The pa ame e s can ei he be cons an o a y om yea o yea . Do no
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con use hese pa ame e s wi h he di e en pa ame e s wi h he same names used in he GEV
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and Gumbel dis ibu ions.
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We aim o in es iga e whe he he MEV me hodology and classical GEV deli e sa is ac o y esul s
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also when da a a e d own om a pa en dis ibu ion wi h a sligh ly pe u bed ail, while s ill using
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he o iginal base dis ibu ion o i ing. We he e o e cons uc a pe u bed pa en dis ibu ion,
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𝐹𝑊𝑝, whe e he o iginal (base) Weibull dis ibu ion (Eq. 8) is mixed wi h ano he Weibull
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dis ibu ion wi h a s e ched ail. Ma hema ically, his is exp essed as:
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𝐹𝑊𝑝(𝑥)= (1−𝛼)𝐹𝑊(𝑥;𝑤,𝐶)+𝛼𝐹𝑊(𝑥;𝑤,𝛽𝐶), (9)
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whe e we equi e he mixing a io 𝛼≪1 and he s e ching ac o 𝛽>1.Bo h MI2015 and he
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p esen s udy use pa ame e s de i ed om he long se ies o daily ain all om Pado a, I aly
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(Camu o 1984). Thus, in he simples se up, we use cons an pa ame e s 𝐶=7.3 mm and 𝑤=
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0.82 as in MI2015. Fu he mo e, we use 𝛼=0.02 and 𝛽=3.
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Figu e 1 PDF o Weibull ( ed line) wi h 𝐶=7.3 𝑚𝑚, 𝑤=0.82 and co esponding pe u bed
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Weibull (blue line) wi h 𝛼=0.02 and 𝛽=3. The dashed line ( igh y-axis) is he a io be ween he
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su i al unc ion (see ex o explana ion) o Weibull and pe u bed Weibull dis ibu ions.
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Figu e 1 shows he PDF o bo h he Weibull and he pe u bed Weibull as colo ed lines. This
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demons a es ha he wo PDFs o e all ha e only ha dly no iceable di e ences. As we will see in
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Sec ion 3.3, he di e ence be ween 𝐹𝑊 and 𝐹𝑊𝑝 is so small ha commonly used goodness-o - i
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es s canno sepa a e syn he ic ain all se ies gene a ed om he wo dis ibu ions. Howe e ,
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ocusing on he ails o he wo dis ibu ions by conside ing he su i al unc ion, de ined as
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Ψ=1−𝐹,
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a di e en pic u e eme ges. Thus, he a io Ψ𝑊𝑝 Ψ𝑊
⁄(do ed line in Figu e 1) is close o uni y o
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small alues o syn he ic daily ain all, bu om a ound 50 mm i inc eases exponen ially (no e
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loga i hmic scale on he igh y-axis), illus a ing ha F𝑊𝑝 has signi ican ly mo e mass in i s ail
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compa ed o F𝑊.
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We combine Eqs. (8) and (9) wi h (2) o ge an exp ession o he heo e ical CDFs o annual
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maxima, assuming 𝑛=100 we -days each yea . F om his, we calcula e he ue e u n le els o
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annual maxima, shown in Figu e 2.
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Figu e 2 T ue annual maximum e u n le els as a unc ion o e u n pe iod (loga i hmic axis) om
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Weibull pa en wi h 𝐶=7.3 𝑚𝑚, 𝑤=0.82 and 𝑛=100 and om he co esponding pe u bed
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Weibull pa en wi h 𝛼=0.02 and 𝛽=3, using Eq. 2.
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The ue e u n le els shown in in Figu e 2 beha e quali a i ely di e en . The e u n le els om
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he Weibull pa en o m a s aigh line, which means ha hey a e close o hei asymp o ic
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dis ibu ion which is a Gumbel dis ibu ion (e.g. Emb ech s e al. 1997). The ue e u n le els o
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he pe u bed Weibull do no align along a s aigh line, meaning ha hey a e a he om
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con e gence (also a Gumbel dis ibu ion).
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The ue e u n le els o he Weibull and pe u bed Weibull pa en s a e close o each o he o
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e u n pe iods a ound 10 yea and lowe e u n pe iod bu he di e ence inc eases p og essi ely
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wi h a longe e u n pe iod. Fo 100 yea s e u n pe iod he pe u bed Weibull pa en e u n le el
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is be ween 30% and 40% abo e ha o he Weibull pa en , inc easing o nea ly 80% a 1000 yea s.
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This di e ence in ail beha io is compa able in magni ude o he di e ence in ail beha io ound
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by (Papalexiou e al. 2013) when i ing di e en dis ibu ions o a daily ain all se ies. This
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sugges s ha ou cons uc ion o he Weibull and pe u bed Weibull dis ibu ions e lec s he
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unce ain y in ail beha io seen in eal da a.
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Thus, e en hough he Weibull and pe u bed Weibull pa en s a e almos iden ical o he bulk o
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he da a, he associa ed ex eme alue dis ibu ions o annual maxima a e dis inc ly di e en .
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2.4 Goodness-o - i es
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Goodness-o - i es s ha e been applied by Zo ze o e al. (2016) and Ma a e al. (2018) o jus i y
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ha obse ed ain all se ies a e dis ibu ed acco ding o he Weibull dis ibu ion, and ha he
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MEV app oach he e o e is eliable. We in es iga ed whe he such es s could de ec ha Weibull
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and pe u bed Weibull se ies a e dis ibu ed acco ding o di e en pa en dis ibu ions. I he
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es s ail o do so, hey canno be used o iden i ying i he MEV app oach is applicable.
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We used he wo-sample e sion o he es s, es ing he ( alse) null hypo hesis ha wo andom
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se ies, one dis ibu ed as a Weibull and he o he as a pe u bed Weibull dis ibu ion, come om
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he same dis ibu ion. The ou come o he es is a 𝑝- alue ha is compa ed wi h a signi icance
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le el 𝑝0, and i 𝑝<𝑝0 he null hypo hesis is (co ec ly) ejec ed. I many pai s o se ies a e es ed,
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we can calcula e a ejec ion a e. Since he wo se ies by cons uc ion a e dis ibu ed acco ding o
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di e en dis ibu ions, we would in he ideal case (neglec ing ype II e o ) ha e a ejec ion a e o
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uni y, whe eas we would ha e a ejec ion a e o 𝑝0 ( ype I e o ) i hey we e dis ibu ed
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acco ding o he same dis ibu ion.
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We applied h ee goodness-o - i es : he Kolmogo o -Smi no (Kolmogo o 1933; Smi no
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1948), he C amé - on Mises (C amé 1928; on Mises 1931) and he Ande son-Da ling (Ande son
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and Da ling 1954) es s o he pai s o se ies in he Mon e Ca lo p o ocol, gene a ed om Weibull
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and pe u bed Weibull pa en , espec i ely (see Sec ion 2.5). These es s base hei es s a is ics
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on he di e ences be ween he empi ical CDFs o he wo se ies, bu he exac o m o he es
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s a is ics di e s be ween he es s, and is e lec ed in he p ope ies o he es s (Razali and Wah
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2011).
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The Kolmogo o -Smi no es uses he maximum absolu e di e ence be ween he empi ical CDFs
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as es s a is ic, which makes i sensi i e o di e ences in he cen al pa o he dis ibu ions. Bo h
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he C amé - on Mises and he Ande son-Da ling es s use weigh ed di e ences be ween he
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empi ical CDFs, in eg a ed o e he en i e dis ibu ion as he es s a is ic. The weigh s di e
228
be ween he wo, so ha he C amé - on Mises es is mo e sensi i e he o e all di e ence o he
229
dis ibu ions, while he Ande son-Da ling es is mo e sensi i e o di e ences in he ails o he
230
dis ibu ions.
231
2.5 Mon e Ca lo p o ocol
232
We build upon he Mon e Ca lo simula ion p o ocol om MI2015 o compa e he MEV app oach
233
and he classical EVA (GEV and Gumbel dis ibu ion i s o annual maxima) wi h syn he ic se ies
234
gene a ed om a Weibull pa en , bu ex end i wi h syn he ic ain all se ies gene a ed om a
235
pe u bed Weibull pa en . The p o ocol is:
236
1. Gene a e 𝑁=50 yea s long syn he ic se ies andomly d awn om he Weibull pa en (as
237
MI2015) and om he co esponding pe u bed Weibull pa en , espec i ely.
238
9
2. Fi Weibull dis ibu ions acco ding o Eq. 7 (MEV app oach) using p obabili y weigh ed
239
momen s es ima ion o bo h Weibull and pe u bed Weibull pa en syn he ic se ies and
240
de e mine e u n le els co esponding o 10, 100, and 100 yea s e u n pe iod.
241
3. De i e annual maxima se ies om he Weibull and pe u bed Weibull pa en se ies.
242
4. Fi GEV and Gumbel dis ibu ions using maximum likelihood es ima ion (classical EVA) o
243
bo h annual maxima se ies and de e mine e u n le els.
244
5. Apply wo-sample goodness-o - i es s (Kolmogo o -Smi no , C amé - on Mises and
245
Ande son-Da ling) o he Weibull/pe u bed Weibull pai o se ies.
246
We apply, as MI2015, he abo e p o ocol o h ee di e en se ups using di e en combina ions
247
o cons an o annually a ying pa ame e s. The h ee se ups a e summa ized in Table 1 and
248
desc ibed in de ail below.
249
Table 1: Summa y o se ups. In all se ups, he ime se ies a e 𝑁=50 yea s long.
250
Se up
MEV model
Pa en wo-pa ame e Weibull dis ibu ion,
scale (𝐶) and shape (𝑤)
# we -days pe yea
(𝑛)
A
Eq. 10
Fixed a 𝐶=7.3 mm and 𝑤=0.82
Fixed a 100
B
Eq. 11
Annually a ying, andom alues ob ained
om Pado a ain all se ies
Fixed a 100
C
Eq. 12
Fixed a 𝐶=7.3 mm and 𝑤=0.82
Random, uni o mly
dis ibu ed be ween
21 and 50
251
In se up A bo h he numbe o we -days pe yea and he pa ame e s in bo h pa en dis ibu ions
252
a e cons an h ough ime. So he syn he ic se ies a e gene a ed wi h 𝑛=100 we -days each yea
253
and cons an Weibull pa ame e s o 𝐶=7.3 mm and 𝑤=0.82, as in MI2015. In his case o
254
cons an 𝐶 and 𝑤, he MEV app oach (Eq. 7) simpli ies o
255
𝜁𝐴(𝑦)=𝐻𝑛(𝑦)=𝐹𝑊(𝑦;𝑤,𝐶)𝑛=[1−exp(−(𝑦
𝐶)𝑤)]𝑛, (10)
256
and we es ima e he alues o 𝑤 and 𝐶 o he wo se ies by i ing Weibull dis ibu ions o bo h
257
he Weibull and he pe u bed Weibull syn he ic se ies.
258
MI2015 a gues ha annual a ia ions o he pa en Weibull pa ame e s a e impo an . The e o e,
259
hey applied se up B, whe e hey keep he numbe o we -days cons an a 𝑛=100 bu allow he
260
Weibull pa ame e s o a y be ween yea s as andom alues ob ained om he Pado a ain all
261
se ies (Camu o 1984). Fo se up B, he MEV app oach (Eq. 7) becomes
262
𝜁𝐵(𝑦)≈1
𝑁∑𝐹𝑊(𝑦;𝑤𝑗,𝐶𝑗)𝑛=1
𝑁∑[1−exp(−(𝑦
𝐶𝑗)𝑤𝑗)]𝑛
𝑁
𝑗=1
𝑁
𝑗=1 , (11)
263
16
4 Discussion
378
4.1 Validi y o eal-wo ld da a
379
Mon e Ca lo s udies using syn he ic ime se ies can gi e aluable insigh s in o he s a is ical
380
me hods by enabling con olled expe imen s. Tha said, such expe imen s mus be ca e ully
381
designed o p ope ly mimic he eal-wo ld examples we wan o s udy. Fu he mo e, cau ion is
382
essen ial when ans e ing conclusions om hese s udies o eal da a, as syn he ic se ies may ail
383
o cap u e he ull complexi ies inhe en in such da a. Bo h MI2015 and he p esen s udy assume
384
ha da a poin s o di e en days a e independen and iden ically dis ibu ed (s a iona y) bu
385
hese assump ions a e usually no ul illed o eal-wo ld da a.
386
Classical EVA ypically uses a block leng h o one yea and selec s one da a alue ( he la ges ) om
387
each block. Thus, since he deco ela ion ime o he ime se ies usually is in he o de o a ew
388
days, he se ies o annual maxima can be ega ded as independen . In ha case he heo y
389
desc ibed in Sec ion 2.1 is s ill alid i he numbe o da a poin s a e eplaced by an e ec i e
390
numbe o da a poin s.
391
Non-s a iona i y is ano he impo an conside a ion. Mos p ominen ly, mos hyd ological ime
392
se ies exhibi a s ong annual cycle. This means ha he p obabili y o ha ing an ex eme alue
393
a ies h ough he season, which is a majo eason o using a block leng h o one yea . Tha said,
394
he p esence o an annual cycle equi es ha he esul s om classical EVA (see Sec ion 2.1) be
395
modi ied, as discussed in Buishand (1989). Kou soyiannis (2004) conside s he illus a i e example
396
o p ecipi a ion o each mon h a e all gamma-dis ibu ed bu wi h di e en pa ame e s, which
397
leads o an o e all dis ibu ion wi h a hea y ail and hus con e gence o GEV wi h non-ze o shape
398
pa ame e .
399
In e annual a ia ions and ends a e o he ypes o non-s a iona i y. Classical EVA can
400
inco po a e ends and in luences o la ge-scale a mosphe ic ci cula ion pa e ns by in oducing
401
sui able co a ia es in o he analysis (Coles 2001).
402
The e is hus some jus i ica ion o apply classical EVA o ime se ies wi h hese elaxed
403
assump ions. On he o he hand, (Se inaldi e al. 2025) ound he MEV me hod o be biased o
404
au oco ela ed da a. Mo e discussion on he po en ial limi a ions o he MEV me hodology would
405
be welcomed be o e u he use o he me hod in p ac ical applica ions.
406
4.2 Simpli ied MEV (SMEV)
407
A a ian o MEV, simpli ied MEV o SMEV has been in oduced by Ma a e al. (2019). SMEV also
408
assumes a known pa en dis ibu ion, bu e-in oduces he ocus on he ail by applying le -
409
censo ing de ined by a h eshold. The SMEV hus ha e simila i ies o he peak-o e - h eshold om
410
EVA . The h eshold is objec i ely de e mind by a es p ocedu e (Ma a e al. 2023).
411
I may be ha he SMEV wi h i s ocus on he ail does no ha e he sho comings o MEV poin ed
412
o in his s udy. On he o he hand, he le -censo ing limi s he numbe o da a a ailable o
413
es ima ion and hus e-in oduces a p oblem, which he he o iginal MEV aimed o sol e.
414
17
5 Summa y and ou look
415
We ha e con i med he conclusion o MI2015 ha he MEV me hod p o ides unbiased es ima es
416
o Weibull pa en syn he ic se ies o all h ee e u n pe iods. The dis ibu ions o he es ima es
417
a e mo e na ow compa ed o he classical EVA (GEV and Gumbel me hods). These esul s led
418
MI2015 o conclude ha MEV was supe io o classical EVA.
419
Howe e , his conclusion changes when s udying he beha io o he pe u bed Weibull pa en
420
se ies. Now he MEV me hodology se e ely unde es ima es he e u n le els, wi h he ue alue
421
being la ge han he 95 h pe cen ile o he dis ibu ion. Consequen ly, he MEV me hod gene ally
422
ca ies he isk o being o e -con iden . The GEV me hod, on he o he hand, p o ides unbiased
423
es ima es wi h he ue alue inside he 5-95 pe cen ile in e al o he dis ibu ion. The es ima es
424
o he Gumbel me hod a e unbiased o 10 yea e u n pe iod, bu becomes inc easingly
425
nega i ely biased o la ge e u n pe iods. This is a consequence o incomple e con e gence (see
426
Figu e 2) which means ha a GEV i is supe io o a Gumbel i , which leads o unde es ima ion o
427
he highe e u n le els (Kou soyiannis and Balou sos 2000).
428
Se ups B and C we e o mula ed o accoun o a ia ion o he Weibull pa ame e s and/o
429
numbe o we -days be ween yea s. Ou analysis shows ha in gene al his lexibili y does no in
430
gene al make he MEV me hod mo e capable o ca ching he ex emes o he Weibull mix u e
431
pa en se ies, so conclusions ob ained o se up A hold also o se ups B and C. This is despi e he
432
assump ion o s a iona i y, which gua an ees he GEV me hod o be alid, is iola ed in se up B.
433
Ano he impo an esul om he simula ion s udy is ha h ee commonly used goodness-o - i
434
es s a e unable o dis inguish e ec i ely be ween ime se ies d awn om he Weibull and
435
Weibull mix u e pa en dis ibu ions. The e o e, his does no p o ide a way o de ec se ies
436
whe e he MEV me hod can be used and whe e i should be a oided.
437
We epea ed ou Mon e Ca lo ekspe imen s (se up A) wi h di e en combina ions o base and
438
pe u ba ion dis ibu ions. These con i med he conclusion ha he MEV me hodology is o e -
439
con iden . We he e o e ind e easonable o assume ha he indings o he p esen s udy
440
gene alize o a la ge class o dis ibu ions. This implies ha he isk p oducing biased and o e -
441
con iden e u n le el es ima es is a gene al ca ea o he MEV me hodology.
442
Acknowledgemen s
443
The au ho s would like o acknowledge he suppo o he Danish Go e nmen h ough he
444
Na ional Cen e o Clima e Resea ch (NCKF) and he Danish Clima e A las.
445
Au ho con ibu ions: CRediT
446
To ben Schmi h: Concep ualiza ion; analysis; w i ing manusc ip . Ka s en A nbje g-Nielsen, Bo
447
Ch is iansen: Concep ualiza ion; con ibu ing o manusc ip .
448
Funding sou ces
449
18
Funding was p o ided by he Danish S a e h ough he Na ional Cen e o Clima e Resea ch
450
(NCKF) and he Danish Clima e A las.
451
452
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