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1 A-PDF Split DEMO : Purchase from to remove the watermark! "# " $I0 J 2 *' Ģ G 9 G? CD E J 0. /&, *%*, J 3, *J 3, 7&%' H(I JKL d y keb U kepld` Pe V fd S ` xl _Ô p` P d SU ae \ Y V fd S ` ³\TSya pxw i { ou f c \ i d SU ae \ Y kepld` e k Z Ÿ z c \ iep \S Xj P Z Ÿ z c \ iep \S Xj { Ww} c \ iep \S Xj Pe pw g a k \ S wa \` p [\qu V fae k v M wa kb` p f \qu keb U { oya o Z m d U \ oƒ^ g a k \ S wa \` p Û\ j P Zf \ y c \ fgbo \` Zf \ i ou c \ i d SU ae \ Y {Z Ÿ z o ke Pld` c \ id SU ae \ Y kepld` c \Ÿ z _`d U b U P Zˆfe a {«y \ fe ap Z ƒw ²a p` k \ ƒ S wa \` oyap Sm dj e ad v M wad S ` \i Pe df \ w p` Mˆ y Ww} c \ iep \S Xj Pe p o oiab z kak x \ Xy v À Z Ÿ z Z Ÿz c \ iep \S Xj { Z Ÿ z c \ iep \S Xj { Ww} c \ iep \S Xj $I JŠ $,Œ( *Ž! oi \ X U x, x À,,xn o ªO d U dfk \qu owƒ d m g a k \ S wa \` ae F(x) \ Sša _ fgbo xaboz U v o ƒ \` o dfg eb² p` B \ W qu d SU ae \ Y P A x \ U c \ id SU ae \ Y p ceb p` Mm\ f g oƒy\u Z`\ O xabo Z U vkd Z Ÿ z g [F (x) = x p A B fe ak V fad`\ƒ` A + B.g (F (x)) = x p p {Mwa _ fgbo oƒ^ _`\O k \ U g(f(x)) xl e k p { _ fgbo _`\O xpe ap { Sp _`\ O v M wa o kak x \ Xy Sp \` M[\ š e k _`\O P Mwa \ Sša p v M wad y keb U _ fgbo P d ² Zy\ U d SU ae \ Y xl e k p Mwa Z S[\ š ke aoy\s wa ou f `\Ÿ O e \U l e k Zw \ wa c \ ie \ p R u g a c \ id SU ae \ Yeb ƒ U V fo` v M w \ i kakg a cap~b U \` ªw \ ƒ U Z g ad` p oyb kepld` capybj p` of \` ou cad ` vkb p± ae a o kak g ad ` ou P \ i kak V W` coƒˆ[ou _ ap e kp \ i kakg a cdw f e \Sm e W T XO r t vkd k \ S wa \ i _ fgbo oƒ^ _àbo g a xaboz U \ ixl ou c \ i d SU ae \ Y kepld` cad` Z RSTU c \ i Pe pw g a k \ S wa \` p [\q U V fae k vkb Z U k \ S wa P Z Ÿ z e P \S Xj { Ww} c \ iep \S Xj kepld ` Pe p Zf \ i ou c \ i d SU ae \ Y Z Ÿ z Z Ÿz c \ iep \S Xj pyb y g aoya pw e k {oƒ S i Zf \ y P ˆ j _ fgbo cae ak oƒiab z kepld` oya o ow [ bo b [ e \ Myb U Pe p` p Z ƒw P Zˆfe a MW² \ z pw g a k \ S wa \` «w P o c \ id SU ae \ Y kepld` c \Ÿ z \ `d U b U P «y \ fe ap p fa kak x \ X y ke Pld ` Pe pw V fa o ke Pld ` pyb y g aoya pw e k Ww} c \ iep \S Xj kepld` Pe v oi kz U x \ Xy ae e \Sm e V fd S ` MW² \ z pw e k o k \ f L-Moments LL-MomentsÄ

2 g a Z f H J & $,Œ.Ž, Ž. * kepld ` Pe {Z`\fkePld ` c \ i Pe V fd OZ fo V w dwy e \ wbo # \ w e k p Mwa cep \S Xj cp \ U {cep \S Xj Pe e k {kepld` \ wa v o Z m d U p v M wa p U\u ab z \` ca pyb y ab z xkake ad \ š e k c \ id SU ae \ Y g a ZàbO eb² p` p U\ u ab z c \ id SU ae \ Y kaoo p` p ªWOd O V fo` v M wa kepld ` k \ U V fa š p kb Z U ² \ š ZO k \ U { b U kepld` e k v oi kz U p± ae a \ i d SU ae \ Y V fa cad` ae cdfk \qu {Z d m ou e k p U\ u ab z dj a {d SU ae \ Y k \` Z [ ou g i \` ae xl ca pyb y k \ U P Gi (θ) \` ae g ad` cad` Mwadfg p` kepld` k \ U { Wik f \ y G i (θ) = g i i =,, k Â c \ iep \S Xj Ã 0 JK&,Œ H 3 $,Œ.Ž, Ž. g a Z Ÿ z ªW d O f Z \ fe ow U ag a k \ S wa \` ae Z Ÿ z vkd pˆw \ N U ZˆWOdO c \ i e \U l {Mwa x \ wl e \ W ` Z Ÿ z c \ iep \S Xj g a k \ S wa Zf \ i g aoya xabƒ~ p` W qs U eb p` ae \ yl xaboz U adfg vkd dw O _ fgbo _`\O f P \ W qu MWb U g a p` X Z m k \ O dwœ SU g a a r pôd U Z Ÿ z c \ iep \S Xj Mwa o QfdO dfg eb² λ r = r r ( ) k r k k= E [ X r k r r =,, ªOd U pyb y g a a r k d ƒ ~ Xr k r xl e k p oƒoe \ˆ~ P a Z Ÿ z e P \S Xj Pk v M wa r g aoya p` ca o λ = E [ X λà = E [ XÀ À X À cad` Zw \ W qu xabƒ~ p` λà P A Zy\ U d SU ae \ Y k \ U λ B Zw \ W qu d SU ae \ Y k \ U P o p Sm dj d y e k Zj o ƒ ady r t v M wa g a p Ÿàe V fad`\ƒ` vkb Z U d m f \ W qu d SU ae \ Y kd Z fbyg \` dfg eb² p` xaboz U ae Q (p) = A + BSp g a p Mwa _ fgbo oƒ^ _`\O k \ U Q(p) {xl e k p ke aoy\s wa M[\ š e k _ fgb O oƒ^ _`\O xkd Z Ÿ z p Ÿàe e k kbub U B P A d SU ae \ Y Pk v M wa o ² \ š v o ƒ S i Q(p) b U c \ id SU ae \ Y! " pˆw \ N U dfg eb² p` Û\ j _ fgbo Sp \ U cad` kb Z U F X (x) = exp( e xp ξ θ ) = p lnp = e xp ξ θ ln( lnp) = xp ξ θ x p = ξ θ ln( lnp) Mf \ y e kp oiab z dfg eb² p` Û\ j _ fgbo oƒ^ _`\O V fad`\ƒ` kb` Q(p) = F (p) = ξ θ ln( lnp) ae Û\ j _ fgbo oƒ^ _`\O.θ = B P ξ = A d m \` Wƒ Z U QfdO p Ÿàe e k o QfdO d m p` Q(p) = A B ln( lnp)! "# " fe a k ae \ yl p f \qu o p c kepld ` c \ i Pe c \ iep \S Xj Pe { Ww} c \ iep \S Xj g a oƒoe \ˆ~ LL c\ iep \S Xj Pe P Z Ÿ z c \ iep \S Xj L pw V fae \ S z a p` aosàe k v Z Ÿz Z Ÿz c \ iep \S Xj Wƒ Z U Z m d U ae Pe HoskingÅ

3 λ m Lr = r r E [ X r k r+m kb Z U pˆw \ N U dfg eb² p` ( )k k= ( r k ) r =,,, oyb Z U Z m d U dfg eb² p` P aep \S Xj PkP λ m L = r E [ XÀ À +m X À +m λ m l = ( + m) λ m L À = E [ X +m λ m l À = À[( + m)( + m) Q(p)p( p)m dp oƒ S i _ fgbo Q (p)( p) m dp Â ( + m) Q(p)( p) +m dp d m ke ab U g a ce \ W ` e k xb^ h W y ke Pld` Pe V fae k P Â àpe ad Sya cad ` ca o Qfd O P p S ` e \ W ` e \ xl g a cdwj ad Sya \ f P Mm\ f xabo Z y o Z m d U àp e g a k \ S wa \` Vfad`\ƒ` {Mwa Z R X U g a k \ S wa \` \id SU ae \ Y kepld` ae dfg àpe yap wbo Mwa o x \ W` { pyb y n n i λ m m i= L = ( n + m x pi P ) s F(x) P _ fgbo _`\O P Z Ÿ z c \ iep \S Xj V W` p Ÿàe r t Mwa o kak x \ Xy d fg eb² p` E [ X J r = r! (j )!(r j)! Q (p) p j ( p) r j dp λà = λ = g a oƒoe \ˆ~ λ À P λ «Y Q (p)dp Q (p)( p )dp s! cad` cap S ` d m { \ i _ fgbo g a ce \ W ` e k p \ yl g a Z z d` e k \ f P Mm\ f xabo Z y r!t P r t àp e ad Sya y ap a [ {Mwa X U e \ W ` ad Sya pˆw \ N U ke ab U o ªO d U c \ ipyb y g a k \ S wa \` ae Z Ÿ z c \ iep \S Xj r t vkd kepld` dfg àpe wbo n g aoya p` λ Ä = λ n = x p i / n i= [( n i µ ) ( n i µ µ i= ( n ) x pi # ) H 3 3 $,Œ.Ž, Ž. kepld ` Pe \ w e k M fg \ f \` P gb yp a Z m d U Z Ÿ z Z Ÿz c \ iep \S Xj Pe \ ƒ` cd SU ae \ Y g a Z Ÿ z ªW d O Z Ÿ z Z Ÿz c \ iep \S Xj v oy kd cd S XW` xgp Pe V fae k v o ƒ S i Z Ÿ z c \ iep \S Xj ZˆWOdO c \ ipyb ye k vkb Z U kakd S ^b c \ i kak p` capyb y cad ` Z Ÿ z Z Ÿz c \ iep \S Xj n g aoya p` \` P oyb Z U QfdO (m =,, )r + m g aoya p` pyb y d² \ ƒ ~ V fd S ^b g a { \ O r g a cdwjz \ fe ow U a Wang - Önöz Bayazit

4 Z Pe {kepld` Pe V fd S ` r st v M wa Q(p) «y \ fe ap v o \` «y \ fe ap V f d S cae ak p Mwa! pw b [ e \ Myb U cg \ wpwˆ g a k \ S wa \` e \ V fae k P Zf \ y c \ i _ fgb O g a f d i cad ` Z m k \ O pyb y v M wa o ow [ bo P Â { c \ i g aoya \` Û\ j _ fgbo P k V fa c \ i g aoyag a f di w \ Y VSm\ f noi v M wa o k \ f kepld` Pe pw e k cae ak (E(λ)) Zf \ y _ fgb O H 3, +% * Mwadfg eb² p` ZfgbO oƒ^ _`\O Q (p) = A B ln ( p) Z [\ ^ _`\ O v M wa o ² \ š _ fgbo _`\ O xp e ap g a p Mwadfg eb² p` h W y Zf \ y _ fgbo oƒ^ s (p) = Q (p) p = B p oƒ^ _`\O cae ak Û\ j _ fgbo H 1', +% Mwadfg eb² p` ZfgbO Q (p) = A B ln ( lnp) Â s (p) = Q (p) p = B p lnp! "# " " "! λ m L Ä = [( n n i i= m )( i µ µ ) n + m P ( n i µ + m )x pi p ` Z Ÿ z Z Ÿ z c \ iep \S X j m = d m \` r t v oy b Z U foô Z Ÿ z c \ iep \S Xj! "# xkak WudO P xo ±\ hf \ O cad` ce \ W ` c \ Pe \` e \ V fae k vke a k kbup d fo f d` {kepld` c \ Pe e k {«y \ fe ap P Zˆfe a { \ i \Ÿ z \ `d U b U p f \qu vkb Z U ofh jd` Pe V fd S ` { Pe pw di f ˆQ(p) d j a H,, ' % ' * \ y l {o \` xp b R U\ y djke Pld` n n À eà i = ˆQ(p) xpi! i= i= V fa x kd WƒW U n oi P Mw \ i \Ÿ z \ `d U b U r t v M wae aoqu {o \` xp b R U\ y djke Pld` ˆQ(p) dj a H 3 1. * \ y l ˆQ (p) x p # B ˆQ(P) = E V f d S cae ak {kepld ` Pe V fd S ` v M wa Zˆfe a { _ fgb O i P q S U c \ i kak cad ` P Mwa Zˆ fe a fe ak E ˆQ(p) = x p p ( p)s(p) (n + )s Ã (p) H J& $,Œ.Ž, Ž. i _`\ O cae a k Zf \ y _ fgb O H 3, +% * c \ ioƒ^ dj a {Mwa eb² p` ZfgbO oƒ^ {oƒ \` xq, x p \` dàd` ªWOdO p` q = p, p ca pyb y d \ ƒ SU dfk \qu \` dfk \qu V fa xkake ad cp \ U \` \ y l fe ak Zf \ y _ fgbo g a cap U\ u cad` _ fgbo oƒ^ Z [\ ^ _`\ O k \ U s(p) xl e k p ² \ š p p ` M ˆ y S Q(p) X U g a s(p) v M wa r st vkb Z U {o \` Z [\^ _`\ O k \ U s(p) dj a H,. *Ž \ y l V ar ˆQ(p) = p ( p) (n + )s À (p)

5 9. D ½ G Ģ?@ =¼ E ¼D D ½ A Ģ ¼¹ D A <Ģ A GA ii? G? G C@D >G D ¹. ¼¹?@A 9. D»¼GA D BD¼ >@ G¹=F GA E FG G G A Ģ A <Ģ D0?@»¼GA¼@D<? G? G C@D >G A 9. D A¼@D<?@ G C@D >G D? iii G G A B 9½> = D?D ;? 9 FD< GA»¼GA D >G G? D GA E FG C@D >G? G? G ¹ < D A <Ģ =< D 9. 9 =¼D D½< ¹ D C@D >G D? F¹F C ¹< ¼¹?@A ¼¹ < 9FGC9 ¼D< A E ¼9 ¼ D > G F¹F C G9FG > +½> D?D =<?9 > D D A < ¹ < G9 <»¼A Ģ F¹F C G9FG > Ç D /FD¼ G?»¼A Ģ G@ +½> D?D ; ¼¹?@A Ç D?9 >G D?D ;? F¹F C G9FG > Ç D?9 >G = ¼ G»¼A Ģ G@ +½> Q (p) = A B ln ( µ p) = x p ( 2 ) Q ( µ p) = A B ln ( µ µ + p) = x q ¼ G@ CD E >@»¼G G [ x p x q Â = x q + ln p ln ( µ p) x p x q ˆB = ln p ln ( µ p) ln p G9 < G? =¼ E ¼D D 9 A C9 9½ * ¹ C G9FG > A G A <D D Ģ G@ <D ¼¹ D 9. A G >G Ģ ¹ º = ¼¹ G D;F xq, x p D! ½A Ģ q = µ p, p G F¹F A¼ CD E >@ { Q (p) = A B ln ( ln(p)) = xp ( ' ) Q ( µ p) = A B ln ( ln ( µ p)) = x q G@ ½ G¹ Â = x p + [ ln xp xq ln( p) ln p ˆB = [ ln. ln[ ln p xp xq ln( p) ln p

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7 k \ S wa \` ZfgbO cg \ [ ou d` cap U oqu v Â# v V W U a o N U { kag \ u P Z R ~ o N U {ZW r t vx \ ²a Z S ƒ² \ X yak v o ƒ^ _`abo g a [2 Bayazit, M. and ÖnÖz, B LL-Moments for estimating low flood quantiles. [3 Benjamin, J.R., and Cornell, C.A Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, Inc. [4 Berger, J Statistical Decision Theory, Springer Verlag. [5 Chen, E.J and Kelton, W.D. Quantile and histogram estimation. [6 Hosking, JRM L-Moments: Analysis and estimation of distributions using linear combinations of order statistics. [7 Van Gelder, P.H.A.J.M and Vrijling, J.K. A comparative study of different parameter estimation methods for Statistical disribution functions in civil engineering applications.