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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  ( + 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  `ape 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 `ap 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 `ape 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 Ÿ`ae 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 `ap 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 `ape 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

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  { c \ i g aoya \` ˆU\ 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 ˆU\ j _ fgbo H 1', +% Mwadfg eb² p` ZfgbO Q (p) = A B ln ( lnp)  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 foˆo 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 à (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 \` d`ad` 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)

9. D Ģ @A ½ G Ģ?@ =¼ E ¼D D ½ =< @? A Ģ <D @ ¼¹ D A <Ģ D0 @? A GA ii? G? G C@D >G D ¹. ¼¹?@A 9. D»¼GA D BD¼ >@ G¹=F GA E FG G G AB @? A Ģ A <Ģ D0?@»¼GA¼@D<? G? G C@D >G D @¹ =< @? A 9. D A¼@D<?@ G C@D >G D? ½F?»¼G @ iii G G A B 9½> = D?D ;? 9 FD< GA»¼GA D >G G¹ @? G? D GA E FG C@D >G? G? G ¹ < D A <Ģ D0 @? A @¹ =< D 9. 9 D½ @ =¼D D½< ¹ D C@D >G D? F¹F C G9FG > @AF @ ¹< ¼¹?@A ¼¹ < 9FGC9 ¼D< A E ¼9 ¼ D C9 @D¼? > G F¹F C G9FG > A @ +½> D?D ;? @¹ =<?9 > D D A < ¹ < G9 <»¼A Ģ G@ @? C9 @D¼ F¹F C G9FG > A @»½ Ç D /FD¼ G?»¼A Ģ G@ +½> D?D ; ¼¹?@A Ç D?9 >G D?D ;? F¹F C G9FG > A @? >»¼G»½ @ 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  = x q + ln p ln ( µ p) x p x q ˆB = ln p ln ( µ p) ln p G9 < Ģ @? G? =¼ E ¼D D 9 G¹ @? A C9 9½ * ¹ C G9FG > A G A <D 9. 9 9. D Ģ G@ <D ¼¹ D 9. A G >G Ģ ¹ º = ¼¹ G ½ @ 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  = x p + [ ln xp xq ln( p) ln p ˆB = [ ln. ln[ ln p xp xq ln( p) ln p

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. 2002. LL-Moments for estimating low flood quantiles. [3 Benjamin, J.R., and Cornell, C.A. 1970. Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, Inc. [4 Berger, J. 1980. Statistical Decision Theory, Springer Verlag. [5 Chen, E.J and Kelton, W.D. Quantile and histogram estimation. [6 Hosking, JRM. 1990. 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.