Estimation of Quantiles and Confidence Intervals for the Frechet Distribution

7/24/2019 Estimation of Quantiles and Confidence Intervals for the Frechet Distribution

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Stochastic Hydrology and Hydraulics 10:187 -207 Springer-Verlag 1996

E s t i m a t i o n o f Q u a n t il e s a n d C o n f i d e n c e I n t e r v a ls

f o r th e L o g - G u m b e l D is tr ib u t io n

J . -H . H e o

Department of Civil Engineering, Yonsei University, Seoul, K o r e a

J . 1 . S a l a s

Hy drolic Scien ce and Engineering Program, Department of Civil Engineering,

Co lo rado

St a t e U n i v e r s i t y , F or t C o l l i ns ,

CO 80523 , US A

A b s t r a c t : The log-Gumbel distribution is one of the extreme value distributions wh ich has been

widely used in flood frequency analysis. This distribution h as bee n exam ined in this paper regarding

quantile estimation and confidence intervals of quantiles. Specific estimation M gorithms ba sed on

the methods o f mom ents (MO M), probability weighted mom ents (PWM ) and maximum likelihood

(ML) are presen ted. The applicability of the estimation procedures and comparison amon g the

methods have been illustrated based on an application example considering the flood data of the St.

Mary's River.

K ey w or ds : log-Gumbel distribution, flood frequency analysis, quantile estimation, confidence in-

tervals.

1 I n t r o d u c t i o n

T h e l o g - G u m b e l d i s tr i b u t i o n is o n e o f t h e m a n y d i s t r i b u t io n s c o m m o n l y u s ed f o r

f r e q u e n c y a n a l y s i s i n h y d r o l o g y . I n t h e h y d r o l o g i c l i t e r a t u r e , t h e l o g - G u m b e l di s-

t r i b u t i o n is a ls o k n o w n a s t h e F r e c h e t d i s t r ib u t i o n ( N E R C , 1 9 7 5 ) . T h e i m p o r t a n c e

o f u s i n g t h e G u m b e l a n d l o g - G u m b e l d i s t r ib u t i o n s f o r f l oo d f re q u e n c y a n a l y s i s w a s

e x a m i n e d b y S h e n e t a l . ( 1 9 8 0) a n d O c h o a e t a t. ( 1 9 80 ) . T h e y s t u d i e d t h e ef f ec t

o f t h e t a i l b e h a v i o r a s s u m p t i o n s o f t h e s e d i s t r i b u t i o n s f o r f i t t in g t h e a n n u a l f l o o d s o f

m o r e t h a n 2 00 s ta t i o n s i n T ex a s , N e w M e x ic o , a n d C o l o r a d o . T h e y c o n c l u d e d t h a t

t h e l o g - G u m b e l d i s t r i b u t i o n p r o v i d e d a b e t t e r f i t f o r m o r e t h a n t w o - t h i r d s o f a ll t h e

s t a ti o n s . T h e y a l so n o t e d t h a t t h e l o g - G u m b e l d i s tr i b u t i o n g e n e r a ll y g a v e g r e a t e r

e s t i m a t e s o f e x t r e m e f l oo d m a g n i t u d e s t h a n t h e G u m b e l d i s t r ib u t i o n .

T h e l o g - G u m b e l d i s t r ib u t i o n is r e l a t e d t o t h e t y p e I I G e n e r a l E x t r e m e V a l u e ( G E V )

d i s t r i b u t io n o r G E V - 2 . P r e s c o t t a n d W a l d e n (1 9 80 ) d e r iv e d t h e e x p e c t e d v a l ue s

o f t h e s e c o n d o r d e r p a r t i a l d e r i v a t i v e s o f t h e l o g - l i k el i h o od f u n c t i o n o f t h e G E V

d i s t r ib u t i o n w i t h r e sp e c t t o t h e p a r a m e t e r s . T h e s e e x p e c t e d va l ue s a r e e l e m e n t s

o f t h e w e ll k n o w n F i s h e r ' s i n f o r m a t i o n m a t r i x w h i c h g iv e s t h e a s y m p t o t i c v a r ia n c e -

c o v a r i a n c e m a t r i x o f t h e m a x i m u m l i ke l ih o o d e s t i m a t o r s ( K e n d a l l a n d S t u a r t , 1 97 9).

S u b s e q u e n t l y , P r e s c o t t a n d W a l d e n ( 1 9 8 3 ) g a v e a n i t e r a t i v e p r o c e d u r e f o r o b t a i n i n g


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188

m a x i m u m l i k el ih o o d e s t i m a t e s o f t h e p a r a m e t e r s o f t h e G E V d i s tr i b u ti o n a n d d e ri ve d

t h e o b s e r v e d in f o r m a t i o n m a t r i x o f t h e c e n s o r e d / c o m p l e t e s a m p l e s a s a r e a so n a b l e

a p p r o x i m a t i o n f o r t h e m a x i m u m l ik e li ho o d e s t im a t e s . T h e y c o m p a r e d t h e o b se r v e d

i n f o r m a t i o n m a t r i x w i t h F i s h e r ' s i n f o r m a t i o n m a t r i x u s i n g s i m u l a t i o n e x p e r i m e n t s

a n d c o n c l u d e d t h a t t h e o b s e r v e d i n f o r m a t i o n m a t r i x i s p r e f e r a b l e t o e s t i m a t e t h e

v a r ia n c e s a n d c o v a r ia n c e s o f t h e m a x i m u m l ik e l ih o o d e s t i m a t o r s .

H o s k i n g e t a l. (1 98 5) s h o w e d h ow t o e s t i m a t e p a r a m e t e r s o f t h e G E V d i s tr i b u -

t io n b a s e d o n t h e m e t h o d o f p r o b a b i l it y w e ig h te d m o m e n t s ( P W M ) a n d g a v e th e

a s y m p t o t i c v a r i a n c e s o f t h e p a r a m e t e r s a n d a t a b l e w h ic h c a n b e u s e d to o b t a i n t h e

a s y m p t o t i c v a ri a nc e s o f t h e P W M q u a n t il e e s t im a t o r s . T h e y a ls o c o m p a r e d t h e m e t h -

o d s o f P W M , m a x i m u m l ik e l ih o o d , a n d a e n k i n s o n ' s p r o c e d u r e b y u s in g s im u l a t i o n

e x p e r i m e n t s a n d c o n c l u d e d t h a t P W M e s t i m a t o r s a r e g o o d fo r s m a l l s a m p l e si ze ( N

< 1 0 0 ). R e c e n t l y , L i u a n d S t e d i n g e r ( 1 99 2 ) c o m p a r e d a l t e r n a t i v e v a r i a n c e s o f t h e

P W M q u a n t i l e e s t i m a t o r f o r t h e t w o - a n d t h r e e - p a r a m e t e r G E V d i s t r i b u t i o n s .

T h i s p a p e r d i sc u s se s t h r e e m e t h o d s o f q u a n t i le e s t i m a t i o n f o r th e l o g - G u m b e t d is t ri -

b u t io n , n a m e l y , th e m e t h o d o f m o m e n t s ( M O M ) , t h e p r o b a b i l i t y w e i g h te d m o m e n t s

( P W M ) m e t h o d , a n d t h e m a x i m u m l ik e l ih o o d (M L ) m e t h o d . I n a d d i t i o n , th e c o nf i-

d e n c e l i m i ts o n p o p u l a t i o n q u a n t i l e s b a s e d o n t h e M O M , P W M a n d M L m e t h o d s a r e

d e r iv e d b y u s in g t h e c o r r e s p o n d i n g a s y m p t o t i c v a r ia n c e s . F i n al ly , t h e a p p l i c a b i li t y

o f t h e p r o p o s e d e s t i m a t i o n p r o c e d u r e s a r e i ll u s t r a te d b y u s i n g o b s e r v e d f lo o d d a t a .

2 M o d e l d e f i n i t i o n

C o n s i d e r t h a t r a n d o m v a r i a b l e s X a n d Y a r e r e l a t e d a s Y =g n ( X- x o ) , in w h i c h Xo i s

a lo w e r b o u n d p a r a m e t e r . I t m a y s h ow n t h a t Y i s G u r n b e l d i s t r i b u t e d w i t h l o c a ti o n

p a r a m e t e r y o a n d s c a le p a r a m e t e r c t, if X i s l o g - G u m b e l d i s tr i b u t e d w i t h p a r a m e -

t e r s Xo, y o a n d c~. T h u s , t h e c u m u l a t i v e d i s t r i b u t i o n f l m c t i o n o f t h e l o g - G u m b e l

d i s t r i b u t i o n i s g i v e n b y

F (x ) = e x p ( - e x p [ - n ( X - X o ) - - y o ] ) (1)

for x > Xo an d c~ > 0. A ss um ing th at c~ = 1//3 a n d y o = g n ( O - x o ) , i t m a y b e

s h o w n t h a t E q . ( 1) t a k e s t h e f o r m

xp r0 xol )

L X - X o J ( 2 )

in w h ic h X o < X < e C , 0 > X o a n d /3 > 0. E q u a t i o n ( 2) is a n o t h e r f o r m o f t h e l o g -

G u m b e l d i s t r i b u t i o n .

T h e t o g - G u m b e t d i s t r i b u t i o n i s r e l a t e d t o t h e G E V - 2 d i s t r ib u t i o n . F o r i n s t a n c e , b y

a s s u m i n g t h a t x o = x 'o + ce'/[3', c~ = - f l ' a n d y o = g n ( - a " / / 3 ' ) , i t m a y b e s h o w n t h a t

t h e C D F g i v e n b y E q . (1 ) c a n b e w r i t t e n in t h e f o r m o f t h e C D F o f t h e G E V - 2 d i s tr i-

' a ' a n d / 3 '

u t i o n in w h i c h x o , a r e r e s p e c t i v e l y t h e l o c a t i o n , s c a l e, a n d s h a p e p a r a m e -

t e r s o f s u c h G E V - 2 d i s t r i b u t i o n a n d t h e s h a p e p a r a m e t e r / 3 ' is n e g a t i v e. I n a d d it i o n ,

i t m ay be a l so show n tha t by ass um ing /3 = -1 / /3 ' , 0 = X'o and Xo = X'o + a'//3', t h e

/ Cr

D F o f E q . ( 2) t a k e s t h e f o r m o f t h e C D F o f t h e G E V - 2 d i s t r ib u t i o n i n w h i ch Xo,

a n d / 3' a r e t h e G E V - 2 p a r a m e t e r s a s a b o v e d e fi n ed . I n th e r e m a i n d e r o f t h i s p a p e r ,

w e w il l u s e t h e I o g - G u m b e l m o d e l g i v e n b y E q . ( 2 ).

T h e d e r i v a t i v e o f F ( x ) o f E q. (2 ) g iv e s t h e p r o b a b i l i t y d e n s i ty f u n c t i o n ( P D F ) a s


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, IO-xol, (_fxo1%

f ( x ) - ( x - X o ) t x - X o j e x p t x - X o j / ( 3 )

F i g u r e 1 s h o w s s o m e t y p i c a l sh a p e s o f t h e P D F o f t h e l o g - G u m b e l d i s tr i b u t i o n . L i ke -

w i se , t h e r - t h m o m e n t o f X a r o u n d Xo c a n b e s h o w n t o b e

E [(X - Xo) ] = (O - Xo) F(1 - r//3 ) (4)

w h e r e 12 (.) d e n o t e s t h e c o m p l e t e g a m m a f u n c t i o n . N o t e t h a t s u c h r - t h m o m e n t e x i s t s

o n l y i f /3 > r . T h e m e a n a n d v a r i a n c e c a n b e o b t a i n e d f r o m E q . ( 4) r e s p e c t i v e l y a s

# = x o + ( 0 - X o ) r ( 1 - 1 // 3) ( 5 )

f o r 3 > 1 , a n d

au

= (0 - xo) ~ IF(1 - 2 /3 ) - F2(1 - 1//3)] (6)

f o r / 3 > 2 . L ik e w i s e , t h e s k e w n e s s co e f f i ci e n t i s g iv e n b y

r (X - 3 /~) - 3 r (1 - 2 /3 ) r (1 - 1 /3 ) + 2 ra ( 1 - 1 //3 )

h t = [ F ( x - 2 / 3 ) - r ~ ( 1 - 1 / / 3 ) ] a / ~ ( 7 )

f o r /3 > 3 . I t m a y b e s h o w n th a t t h e s k e w n e s s c o e ff i c ie n t o f t h e l o g - G u m b e l d is -

t r i b u t i o n i s g r e a t e r t h a n 1 .1 3 96 ( r e c a l l t h a t t h e s k e w n e s s c o e ff i c ie n t o f t h e G u m b e l

d i s t r i b u t i o n i s e x a c t l y e q u a l t o 1 .1 39 6 ). A l s o , t h e e x p r e s s io n o f t h e s k e w n e s s c oe f -

f i ci e nt o f E q . ( 7) i s t h e s a m e as t h a t o f t h e G E V - 2 d i s t r i b u t i o n i f t h e p a r a m e t e r / 3

i n E q . ( 7 ) i s r e p l a c e d b y - 1 // 3 ' i n w h i c h / 3' i s t h e s h a p e p a r a m e t e r o f t h e G E V - 2

d i s t r i b u t i o n a n d h a s a n e g a t i v e v a l u e . A d d i t i o n a l ly , t h e m o d e i s g i v e n b y

m o d e ( x ) X o + ( 0 X o ) [ 1 + ~ / 3 ] - 1 /0

= - - - (8)

[ P J

3 E s t i m a t i o n o f q u a n t i l e s

T h e q u a n t i l e e s t i m a t o r :K T o f t h e l o g - G u m b e l d i s t r ib u t i o n c a n b e o b t a i n e d f r o m E q .

( 2) b y r e p l a ci n g F ( x ) b y ( t - l / T ) a s

~:T = ~o + (0 - %) [- &( 1 - l / T) ] -1 /a (9 )

w h e r e xo , ~ a n d / ~ a r e t h e e s t i m a t o r s o f t h e p a r a m e t e r s . A l so , t h e e s t i m a t o r ) ( T m a y

b e g e n e r a l l y w r i t t e n i n t e r m s o f t h e s a m p l e m e a n f i, t h e s a m p l e s t a n d a r d d e v i a t i o n

3 -, a n d t h e f r e q u e n c y f a c to r I ( T ( Ch o w , 1 9 51 ) a s

)(w = /2 + KT3 - (1 0)

in w h ic h I ~w m a y b e o b t a in e d f r o m g q s . ( 5 ) , (6 ) a n d ( 9 ) a s

14T = [--gn(1 -- l / T )] -1 /a - F(1 - 1/ /3) (11)

[ r ( 1 - 2 / ~ ) - r ~ ( 1 - 2 / ~ ) 1 1 n

N o t e t h a t 9 o m E q s . ( 7 ) a n d ( 1 1) , t h e f r e q u e n c y f a c t o r I 4T i s a f u n c ti o n o f t h e s k e w n e s s

c o e m c i e n t a n d t h e r e t u r n p e r i o d . N u m e r i c a l v al u e s f o r s u c h a f u n c t i o n a r e s h o w n in

Tab te 1 .


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T a b l e 1 . F r e q u e n c y f a c to r s f o r t h e l o g - G u m b e l d i s t r i b u t i o n

C o e f f. N o n e x c e e d a n c e P r o b a b i l i t y ( q )

o f 0 , 0 1 0 . 2 0 , 5 0 . 8 0 , 9 0 . 9 5 0 , 9 8 0 . 9 9 0 . 9 9 9

S k n e w n e s s

"y C o r r e s p o n d i n g R e t u r n P e r i o d ( T )

1,01 1 .25 2 5 10 20 50 100 1000

1 .14 -1~6192 - .817 5 - .1692 .7115 1 ,2999 1 .8684 2 .6101 3 .170 4 5 .0486

1 . 20 - I . 6 1 0 7 - . 8 1 5 8 - , 1 7 0 9 . 7 0 8 3 1 , 2 9 7 7 1 . 8 6 8 6 2 . 6 1 5 7 3 . 1 8 1 7 5 . 0 8 9 4

1 . 30 - 1 . 5 8 5 9 - . 8 1 1 2 - , 1 7 6 1 . 6 9 8 8 1 . 2 9 1 3 1 . 8 6 9 8 2 . 6 3 3 6 3 . 2 1 7 2 5 . 2 1 6 9

1 . 40 - 1 . 5 4 5 0 - . 8 0 3 0 - . 1 8 4 6 . 6 8 2 6 1 . 2 7 9 8 1 . 8 7 0 4 2 . 6 6 1 8 3 . 2 7 5 3 5 . 4 3 4 4

1 . 5 0 - -1 .5 15 2 - , 7 9 6 7 - , 1 9 0 5 , 6 7 0 5 1 . 2 7 0 5 1 . 8 6 9 7 2 . 6 8 1 0 3 . 3 1 6 6 5 . 5 9 7 8

1 . 6 0 - 1 . 4 9 1 9 - . 7 9 1 5 - , 1 9 4 9 . 6 6 0 8 1 ~2 62 8 1 . 8 6 8 4 2 , 6 9 5 2 3 . 3 4 8 3 5 , 7 2 8 4

1 . 7 0 - 1 . 4 6 6 0 - . 7 8 5 5 - . 1 9 9 8 , 6 4 9 7 t , 2 5 3 6 1. 8 6 6 1 2 . 7 1 0 0 3 . 3 8 2 7 5 . 8 7 6 1

1 . 8 0 - 1 A 3 8 2 - . 7 7 8 8 - . 2 0 4 8 . 6 3 7 6 1 , 2 43 1 1 . 8 6 2 6 2 . 7 2 4 8 3 . 4 1 8 6 6 . 0 3 7 9

1 . 9 0 - 1 . 4 0 9 6 - . 7 7 1 5 - , 2 0 9 7 . 6 2 48 1 , 2 3 1 6 1 , 8 5 8 0 2 . 7 3 8 6 3 . 4 5 4 2 6 . 2 0 7 2

2 . 0 0 - 1 , 3 8 1 6 - . 7 6 4 1 - , 2 1 4 3 . 6 12 1 1 , 2 1 9 7 1 . 8 5 2 3 2 , 7 5 0 6 3 . 4 8 7 6 6 . 3 7 5 4

2 . 1 0 - 1 . 3 5 5 7 - . 7 5 7 0 - , 2 1 8 3 . 6 00 1 1 . 2 0 8 0 1 . 8 46 1 2 . 7 6 0 3 3 . 5 1 7 1 6 . 5 3 3 1

2 . 2 0 - 1 . 3 3 3 1 - . 7 5 0 6 - . 2 2 1 7 . 5 8 9 4 1 . 1 9 7 2 1 . 8 3 9 8 2 . 7 6 7 6 3 . 5 4 1 8 6 . 6 7 2 5

2 . 3 0 - 1 , 3 1 4 3 - . 7 4 5 1 - , 2 2 4 4 . 5 8 0 4 1 . 1 8 8 0 1 . 8 3 4 0 2 . 7 7 2 9 3 . 5 6 1 2 6 . 7 8 8 9

2 ~40 - i , 2 9 9 4 - . 7 4 0 6 - . 2 2 6 4 . 5 7 3 2 1 . 1 8 0 4 1 . 8 2 9 0 2 , 7 7 6 5 3 . 5 7 6 1 6 . 8 8 1 6

2 . 5 0 - 1 , 2 8 8 0 - . 7 3 7 1 - . 2 2 8 0 . 5 6 7 6 1 1 7 4 5 1 . 8 2 4 9 2 . 7 7 8 9 3 . 5 8 7 1 6 . 9 5 3 4

2 . 6 0 - 1 , 2 7 9 0 - . 7 3 4 3 - . 2 2 9 1 . 5 6 3 2 1 . 1 6 9 7 1 . 8 2 1 5 2 . 7 8 0 5 3 . 5 9 5 4 7 . 0 0 9 3

2 . 7 0 - 1 , 2 7 1 7 - . 7 3 2 0 - , 2 3 0 1 . 5 5 9 6 1 . 1 6 5 8 1 . 8 1 8 7 2 . 7 8 1 7 3 . 6 0 2 1 7 , 0 5 5 2

2 . 8 0 - 1 , 2 6 5 1 - . 7 2 9 9 - 2 3 0 9 , 5 5 6 4 1 , 1 6 2 2 1 . 8 1 6 0 2 . 7 8 2 7 3 . 6 0 8 0 7 . 0 9 6 7

2 9 0 - 1 , 2 5 8 4 - . 7 2 7 8 - . 2 3 1 7 . 5 5 3 0 1 , 1 5 8 5 1 . 8 1 3 3 2 . 7 8 3 6 3 . 6 1 3 8 7 . 1 3 8 9

3 . 0 0 - 1 . 2 5 0 9 - . 7 2 5 3 - , 2 3 2 6 . 5 4 9 3 1 . 1 5 4 3 1 . 8 1 0 2 2 . 7 8 4 4 3 . 6 2 0 2 7 . 1 8 5 8

3 . 1 0 - 1 . 2 4 2 2 - . 7 2 2 5 - , 2 3 3 6 , 5 4 5 0 1 . 1 4 9 4 1 . 8 0 6 4 2 , 7 8 5 2 3 . 6 2 7 3 7 . 2 4 0 4

3 . 2 0 - 1 . 2 3 1 9 - . 7 1 9 1 - . 2 3 4 8 . 5 3 9 8 1 , 1 4 3 5 1 . 8 0 1 7 2 , 7 8 5 9 3 . 6 3 5 5 7 . 3 0 5 2

3 .30 -1 .21 98 - .7151 ~ .2361 .5337 1 .1365 1 .7960 2 ,786 4 3 .6447 7 .3814

3 . 4 0 - 1 , 2 0 5 7 - . 7 1 0 3 - , 2 3 7 6 . 5 2 6 5 1 , 1 28 1 1 . 7 8 91 2 . 7 8 6 4 3 . 6 5 4 8 7 . 4 6 9 8

3 . 5 0 - 1 . 1 8 9 7 - . 7 0 4 7 - . 2 3 9 2 . 5 1 8 3 1 . 1 1 8 3 1 . 7 8 0 8 2 . 7 8 5 7 3 . 6 6 5 4 7 . 5 6 9 8

3 . 6 0 - 1 . 1 7 2 0 - . 6 9 8 5 - . 2 4 0 8 . 5 0 9 2 1 . 1 0 7 3 1 . 7 7 11 2 . 7 8 4 1 3 . 6 7 6 0 7 . 6 8 0 0

3 . 7 0 - 1 . 1 5 3 1 - . 6 9 1 6 - . 2 4 2 4 . 4 99 4 1 . 0 9 5 1 1 . 7 60 1 2 . 7 8 1 3 3 . 6 8 6 1 7 . 7 9 7 4

3 . 8 0 - 1 , 1 3 3 4 - . 6 8 4 4 - , 2 4 3 9 A 8 9 1 1 . 0 8 2 2 1 . 7 4 8 0 2 , 7 7 7 3 3 . 6 9 5 0 7 , 9 1 7 9

3 . 9 0 ~ 4 .I 1 4 1 - . 6 7 7 0 - . 2 4 5 3 . 4 7 8 9 1 . 0 6 91 1 . 7 3 5 4 2 , 7 7 2 1 3 . 7 0 2 3 8 . 0 3 5 5

4 . 0 0 - 1 . 0 9 6 0 - . 6 7 0 0 - , 2 4 6 4 . 4 6 94 1 . 0 5 6 5 1 . 7 2 3 0 2 , 7 6 6 2 3 . 7 0 7 6 8 . 1 4 3 5

4 . 1 0 - 1 . 0 8 0 6 - . 6 6 4 0 - , 2 4 7 2 . 4 6 12 1 , 0 4 5 6 1 . 7 1 1 9 2 . 7 6 0 2 3 . 7 1 0 9 8 . 2 3 4 6

4 . 2 0 - 1 , 0 6 8 9 - . 6 5 9 3 - , 2 4 7 8 . 4 5 50 1 , 0 3 7 2 1 . 7 0 3 3 2 . 7 5 5 2 3 . 7 1 2 7 8 . 3 0 2 4

4 . 3 0 - 1 . 0 6 2 0 - . 6 5 6 5 - , 2 4 8 1 . 4 5 13 1 . 0 32 1 1 . 6 9 8 0 2 . 7 5 1 9 3 . 7 1 3 5 8 . 3 4 2 3

4 . 40 - 1 . 0 6 0 2 - , 6 5 5 8 - , 2 4 8 1 . 4 5 03 1 . 0 3 0 8 1 . 6 9 6 6 2 . 7 5 1 1 3 . 7 1 3 6 8 . 3 5 2 4

4 .50 -1 .06 34 - .6571 - .2480 .4520 1 ,0332 1 .6991 2~7526 3 .71 33 8 .3344

4 . 6 0 - 1 , 0 7 0 5 - . 6 5 9 9 - , 2 4 7 7 , 4 5 5 8 1 . 0 3 8 3 t . 7 0 4 4 2 , 7 5 5 9 3 . 7 1 2 5 8 . 2 9 3 3

4 . 7 0 - 1 . 0 8 0 0 - . 6 6 3 7 - , 2 4 7 2 . 4 6 09 1 . 0 4 5 2 1 . 7 1 1 5 2 . 7 6 0 0 3 . 7 1 1 0 8 . 2 3 7 8

4 . 8 0 - 1 . 0 9 0 0 - . 6 6 7 7 - . 2 4 6 7 . 4 6 62 1 . 0 5 2 3 1 . 7 1 8 7 2 . 7 6 3 9 3 . 7 0 9 0 8 , 1 7 9 3

4 . 9 0 - 1 . 0 9 8 0 - . 6 7 0 8 - . 2 4 6 2 . 4 7 04 1 . 0 5 7 9 1 . 7 2 4 4 2 , 7 6 6 9 3 . 7 0 7 1 8 . 1 3 1 8

5 . 0 0 - 1 . 1 0 1 3 - . 6 7 2 1 - , 2 4 6 1 . 4 7 22 1 . 0 6 0 3 1 . 7 2 6 7 2 . 7 6 8 0 3 . 7 0 6 2 8 . 1 1 1 9


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191

f x)

1 . 0

0 . 5

0 . 0

/ , LG( Xo , '0 , # )

l , . L G ( I . 0 , 2 . 0 , I . 0 )

l ',, L G ( 1 . 0 , 2 . 0 , 2 . 0 )

. . . . . L G ( 1 . 0 , 2 . 0 , 3 , 0 )

, %

. - Z / ] , , i

,2,'.,.,,

I

1

x

F ig ur e 1. So m e example of the probability density function of the log-Gumbel distribution of Eq.

(a)

T h e r e f o r e , t h e e s t i m a t i o n o f q u a n t i le s r e q u ir e s e s t i m a t i n g t h e p a r a m e t e r s o f t h e

l o g - G u m b e l m o d e l . T h r e e e s ti m a t i o n p r o c e d u r e s a r e p r e s e n t e d f or t h e lo g - G u m b e l

d i s t r ib u t i o n ; t h e m e t h o d s o f m o m e n t s , p r o b a b i l i t y w e i g h t ed m o m e n t s , a n d m a x i m u m

l ike l ihood .

3. I Method of moments MOM)

By s u b s t i t u t i n g # , ~ a n d "~ i n E q s. ( 5 ), ( 6 ) a n d ( 7 ) b y th e i r c o r r e s p o n d i n g s a m p l e

e s t i m a t o r s ,5 , ff a n d "~ t h e m o m e n t e s t i m a t o r s 0 , f l a n d i o c a n b e o b t a i n e d . T h e

s k e w n e s s co e f f i c ie n t i n E q . ( 7 ) is o n l y a f u n c t i o n o f t h e s h a p e p a r a m e t e r f t. T h u s , t h e

m o m e n t e s t i m a t o r o f t h e s h a p e p a r a m e t e r , / ) c a n b e o b t a i ne d f r o m t h e a p p r o x i m a t e

r e g r e s s i o n e q u a t i o n s g i v e n b y

~) = 222.52 22 - 313.1 802 ~/+ 179 .5053 ~2 _ 50.605 8 ;/a

+ 6 .9785 42 ~4 _ 0.376 228 ~5

wh ich is val id for 1 .48 < ~ < 5,4 an d

= 1731,6756 - 23 42.8143 ~ + 802.1566 ~2

(12a)

(12h)

val id for 1 .1396 < ~ ~ 1 .48, in wh ich "~ is th e sam ple skewn ess coeff ic ient . For a

m o r e p r e c i s e s o l u t i o n o f f l, E q . ( 7 ) c a n b e s o l v e d n u m e r i c a l l y b y t h e N e w t o n - R a p h s o n

m e t h o d . F o r t h i s p u r p o s e , E q . ( 7 ) i s r e w r i t t e n a s


6/21

192

0 ( 8 ) = r ( 1 - a / / } ) - a t 0 - 2 / / } ) r ( ~ - 1 / / } ) + 2 I ' 3 ( 1 - 1 / / } ) _ #

[ P ( 1 - 2 / ; ? ) - r ~ ( 1 - t / / } ) l , / ~

a n d t h e f i r s t d e r i v a t i v e o f G ( / ) ) w i th r e s p e c t t o ) i s g iv e n b y

1

G'( /}) = /)2IF( 1 _ 2//) ) - P2(1 - 1//))] s/2 x

{ j a r ' ( 1 - a / / } ) - 6 r ' ( 1 - 2 / / } ) r ( 1 - 1 / / ) ) - a t ' 0 - t / / ) ) r ( 1 - 2 / / } )

( 1 3 )

+ 6 1 2 ' ( t - 1 / / ) ) I ~ 2 ( 1

- 1 / / ) ) ] [ r 1 - 2 / / ) ) -

F 2 ( 1 - t / / ) ) ] - [ P ( t - a / / ) )

- a p ( 1 - 2 / / } ) r ( 1 - 1 / / ) ) + 2 r ~ ( 1 - 1 / / } ) ] [ a t ' ( 1 - 2 / / ) )

- a p ' ( 1 - 1 / / ) ) r ( 1 - 1 / / } ) ] } ( 1 4 )

w h e r e F ' ( . ) is t h e f i rs t d e r i v a t i v e o f t h e g a m m a f u n c ti o n . T h e r e f o r e , t h e r e c u r s iv e

e q u a t i o n t o e s t i m a t e / } i n t h e i te r a t i o n i + l i s

/ )i+, = / )~ - G( / ) i ) /G ' ( / ) i )

T h e i t e r a t i o n p r o c e e d s u n t i l t h e e r r o r c r i t e r i o n

/}i+1 -/ ) i

Estimation of Quantiles and Confidence Intervals for the Frechet Distribution

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