A comparison of methods of fitting the double exponential distribution

A comparison of methods of fitting the double exponential distribution

Iournal o! Hvdrolog I, I 0 (1970) '2 '~t)-'27'~ ; ,, '~ North- Holland Puhh.s'hm~,, (~,., ,.Inlsterdam Nol hl he reprlliJlJuelJ hV pllillilprlrll l~...

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.Iournal o! Hvdrolog I, I 0 (1970) '2 '~t)-'27'~ ; ,, '~ North- Holland Puhh.s'hm~,, (~,., ,.Inlsterdam Nol hl he reprlliJlJuelJ hV pllillilprlrll

l~r illlerifl111n WlllllllJl

Wrlllell pVrlTll%llm hOrll II1¢ plll'lll,dlel"

A COMPARISON OF METHODS OF F l l " I I N G T H E DOUBLE EXPONENTIAL

DISTRIB[fflON

M D L O W E R Y and .I mE' NASH In,~ttlute o f Hl'drolog I', II'ullmHbrd, U. k'

Abstracl: A number u r m e l h o d s o r f i l l m g l h e d o u h l e e x p o n e n h a l d i s l r i h u l i o n l u a s a m p l e ol dala are c o m p a r e d 1 h e m e l h o d s are expressed m a c o m m o n n . l a l , o n and c o m p a r e d in h.as and efficiency For each fill.ng m e l h o d , forrrtulae I'or Ihe variance ol eslimale,~ ate nblained and s o m e in c u n e n l use ate corTecled. II i.~ s h o w n Ihal nell Io Ihe m e l h u d ol m a x | m u m h k d i h o . d Ihe m e l h o d c~l' momenl~, ~s Ihe mosl accutale II i~ also virluallv unbiased and ,,implesl h~ apply

Nolal,ion The principle symbols used are hsled. ()lhers which are necessary I'rom time to t~n~e are defined in l,he relevanl, par[ o r l,he l,cxl p

probabilil,v (~I" nun exceedence. return peri(~d in years. = paramelers of lhe d4~uble exponenllal dlsl,rlbulJun. = magnil,ude of a fll.~d evenl. = mean or lhe ,I' populal.i~n. = slandard devlalinn of lhe ,I' pi~pulal,i~iL = standard devialJt)n or a sample of lhe Y's. = mean nr a sample of lhe Y's. = esl.mal,e of ,l., of rel,urn period T see,and cenlral momenl uf l,he ,i' p(~pulaliun, lhlrd cenlral m(~menl of lhe ,~.' pnpulallon. = l'ourl,h t:entral momenl or l,he ,~f popul:]l,lo||. = Gumbel's paran|el,er free varial,e. mean ur l,he Y pupulati~n. = = ,~l,andard devial,lon or l,he )' pupulatiun = mean or l,he Y's I'rom a sample of size n. =

T I~, I I

,i'

/I rl' ¢'],¥

I['

,~' ( r ) It: 11~ 11,4 )' tl I, O' I,

}'~

=

=

=

s t a n d a r d deviall~n of lhe )"'s I'r(~m a sample o r size n.

fin

=

A',K(T)

= C h o w ' s frequency I'acl,~r.

K

=

mean ~1" l,he

K's

from a sample ,)1' s~z,en. 259

201)

M D. LI(.)WERY A N / ) I. E'. NASH

=

.,.I It

N Zi

Z,, Z~ Z 2

= = = = = = = =

= = =

standard deviation of the K's from a sample (,1" size n. coefficient o f a'.,,. Jn G u m b e r s predich,m equation. sample Mze number ,,f samples ,~f size n mean or the X populallon ( =10. standard deviation t,f Ihe X populatl,m ( = a ) magmlude or an evenl wtth transf,~rmed return period K = 5 refers Io any one or z~, z , ,.,r Z~ esttmate ~,I"Z. standard deviation ,,f ;~ estimaled I'r,,m N samples. ~wer a symbol =ndicates a sample estimate corrected I',,r b=as. over a symbol tndtcates the mean.

InIToduclion 'The use ,,1' Ihe d,,uble exponenttal distribution in the, I'requencv analysts ,,r annual maxtma events tn hydr,,Ir, gy, wtule not capable o1" rtgorous iust~hcahon, is widely accepted and is to s,,me extent lUSttfied empirtcally by su,.'.h acceptance. Despite rls exlenstve use, however, there ~s n,, generally accepted fitltng method. G u m b e l l ) states that the use ,,f the maximum I,kel,h,',od meth,.,d is "very c.omphcated, and requtres numerical work to an extent wh,ch is proh,b,ttve for routine w(,rk" In pracltce Ihe parameters are esttmated by the method uf moments or by regresstons or the c,bserved magnitudes ,,n a prior~ eshmates ,.,1" the probability associated wtlh each magnitude. 'l-'hese methods however are not all equally easy to apply nor are they equally effictent and with,~ut bin.,.,. To compare the fitltng methnds in c, mnt.stency and bias we require estimators for each meth,~d, t e. algebratc expressions m sample terms, wh,ch are used I,., esllmate the p,,pulatJon parameters. 'T',, compare the fitting meth,,ds tn relative el-ficiencv we requtre, error/ornndae for each method, i.e. expresst,,ns, Jn p,,pulatt~,n lerms, ,,1' the samphng vartance of esltmates or the magmlude c,,rrespondmg to gtven probabihttes ,,r return pert,,ds. In addltl,,n, tr an eshmale from a stngle sample rs Io be ,,f practtcal use we must have s,,me ~dea o1" Ihe err,,r to which t t t s subject, and th~s ~n lurn requires an error/brmttla in samph' terms, because, of c~,urse, the p,~pulaht, n IS

unknown. In addtUon t,, the sampl,ng variance or the esltrnates there is, tn hydro

Iogtcal practtce, annther ,source nf err,~r due h, the posstbthly or the p~,pulation radtng t~, conlorm t,, Ihe assumed I',,rm - the d,,uble exp,,nenttal dmmbuli,m. 'I'hJs source or err~,r cannot be analysed stat=slically and =s m,t treated ~n tl'us paper.

MEq"H(.IDS I'IF' FI'T'TING THE D(')LFBLE,', E,:XP(.)NEN'T'IAL DI,~TRIBI.J'T'I(.)N

'2hi

The double exponential distribution

Annual maxlm;a values (d" hydrolog.::al var=ables are often assumed Io be distribuled in accordance wllh the double exponenttul dlslrfbution. p(.~")

=

exp(-

exp - ,x(X - , ) )

(la)

where p ( X ) ts the probabihlv or an event not exceedtng A' and 0c and . are parameters of the d=strtbuttun. Equali(m (la) may be wrttlen cLmvenientlv in lerms u f a reduc, ed vartale )' p(X) = exp(-

e~p(-

)'))

(Ib)

where |' = .'~t(,Y - u)

(Ic)

By tnvertton or Eq. f i b ) the relaUunship may be wrtllen in terms (d' the return perrod T' (the reciprocal or the probabdily of exceedence). T

)' = - In In

T-

(Id)

I

or X

=

I

-

u

T In I n - - - . T - I

(le)

Gumbel=) showed that the mean p~, and slandard devmt=on o"v or Ihe parameter flee distrlbutmn descrtbed by Eq. ( I b) are, respecltvely,

lh, = '~ (Euler's number = (I '~'T72

)

/7

nv = - . - .

(2)

v'h

If/.~ and o" are lhe mean ;and standard devlalion of .Y lhen rrom Eqs. (Ic) and (2) It = u + i,/~ ] n ], O ":-

(3a)

or It

-'-

/I

-

-

}'a

IT

/i'

(7,b)

0 \lib

which tnserted in E q . ( l e ) yields another form of the double exponential

262

M.D.

L,I.)VVERY A N D I. E,'. NASH

dtstrtbulion

X

=

i/

p-

'~'6

1' +

r)

In In---T_ I

o

whtch ts tn the form ,~t' =/.~ + K (T) a

(4a)

with

K(T) = - v"6('), + In.ln. n

-T'- - ) 'T'- I

(4b)

a function oi" the return period cmly ((."how"). C'how's I'ac,lor K = K ( T ) Is analogous t,~ G u m b e l ' s reduced vurtate )' as can be seen by companng Eq. (Id) and E'q (4b) Clearly

K =

V'6 (r

-

-

-

-

('~)

)')

,,/

K and Y are transl'ormatlons o1" the return period which are linearly related to .i' the magnitude of the exceeded event.

Fitting

the double

exponential

distribution

F'Hting Ihe d,mblc expunential dtslrtbultun h'~ a sample of data tnwdves esttrnat,ng, by sample slaUsttcs, Ihe parameters t~ and n ~1" Eq. (I) ~r'/.t and o o1" Eq. (4). It is only a mailer ~1" conven,ence whJt~h pair uf parameters is ch~sen l'or estimation by any one fitUng method. Replacement of Ihe p~pulattun parameters ~n the d~slr~button equaltun by Ihetr sample estimates conslttutes the pred~ctton equation which may he tn Ihe I'orrn ,~1"F,'.q. (I) or E,'q. (4). (a) 'THE' ME,T H O D

(iF'M(.IIMEN'TS

'The m~st ~bvi~us and direct method ~1' estimalion - Ihuugh nol pupular tn hydrology -esttmates by comparing firsl and second moments of the data and the dtslribuU~m. 'Thus It ts esltmated by ,~ =

I

ZX

(6a)

tl

and c~ by

•/ i

t~lt' =

z(x-,i')"

I1~

(d' x so defined is an almust unbiased estimate uf o).

(Oh)

MET'HODS OF FrT'T'ING THE DOLrBLE EXPONENTIAL DIS'T'RJBLII'I()N

263

The pred~ctmn equations becomes ,~'(T) = ,1" + K ( T ) a x .

(71

( b ) THE: MEI'H(.)D (;)F' REGR.E'SSll)N

F'ltllng by regressmn on a plollmg posllion as Iradihonally used in hydroh~gy (Chow'.;)), inw.~lves estimalmn of the parameters It and o of Eq. (4a) by a linear regress,m of X on K. 'Thus the Interprelation o f / t and a as a mean and standard dewallon is not used; they are trealed merely as the parameters of a linear relatJ~mshlp between X and K. The method reqmres the assignment of "plotllng poslllons" o r a prior~ estimates of T (and hence ~1' K by Eq. (4b)) to each event in the sample. ThJs is usually done by arranging the sample in descending order or magmlude and assigning a return period T = ( n + I)/r to each event, r is the rank of the event and n the sample size When the T's have been transformed to K's by Eq.(4b) a linear regressl~m (~1".V ~n K estlmate,~ p and a by i~ = 5' - g Z',~,/~,_ ,,k,:

(Sa)

d'-

(Sb)

, ,,

2k,

where ~ = X - ,~' and k = g - ,¢. 'The predlc, hun equation f.r this method bec~mes

,~'(h') =

,¢ +

(K

-

R)

z--~;'

(,))

Note that as Ihe plolhng posllmn K's are obtained from the rank and sample size only, g' and #K = x/S(K - / ¢ ) " / ( n - / ) also are functions of sample size 4"rely. They are analogous to P and o,, as used and tabulaled by Gumbell). 'The values of ,¢ and d~ (which will be required laler) can be got from G u m b e r s table through Eq. (5) A seleclicm of values ts given in 'Table I. (C) GI.IMBEL'S FI'n'ING METHf)D

Oumbel's litlmg method is similar to Ihe regression meth~)d ~n that for each event the same a p r i o r i estimates ~)1"T are made and these Iransformed into values ~1" Ihe reduced varlate. Gumbel uses the reduced varmte )", Eq. (Id) and like Chow evaluates the plotting position by T = ( n + I)/r. This

204

M, D

LOWE'RY

AND I. E' NA,~H

TABLE I ,~' u n d #t~ u~ I ' u n c l l o n s ~ll' ~/

14

-

U724

'7'~ ti4

*lO

--.022,1

~142

q

-

l)~'71'1

'/hl'll

fill

-

q~tfl

Olllh

10

- ,()~',40

'7t0~

'70

-

I l l '7~

~ t08

I ",

- ,()5(l~

12 t7

N0

-

Ill ~8

q Itl'7

21.)

()41 ~

fl ~()2

ql)

- 014 ~

q4 I

Uthl

14hl4'=i

liiO

- ()1 14

q,.l~,l

1( I

- 01 ~'iJ

14II 21

2 '~l)

- (X)hh

q '710

1~

- (i,?N7

Nq2'7

'~Oli

- 110211'1

083'~

40

- 02t~2

~-~012

IIX)0

4'~

-

q()~?

2'i

-

0?41

- IX)21

9~q'~ I O(XXI

differs only in n l l l a l l l l n rrom Chow's uNe ~1' plotling pqilSlllllns K's. H~wc'ver in,..lead or ~lblaimng Ihe paramelers n and ~ ~il )l' = . +

I

)'

(1())

'~t

by regression or ,Jr on )' (which would be iderliil.'.al wiih Ch(lw's regresslcln melhod) Gumbel adapts as the ,i,' ),, relaUonship the ge~lmeirl¢ mean o1' the, regrcssllln,'., nr )' i'ln ,ll' and ,i,,,cln )'. 'This I,s equivalent to estimating 0t and u by
(II) (12)

F "'~'

where ,t' =

~-I Z ' ( , \ ' ,

-

and

ft. =

\

It

'

N~lie: n,t, and rr. as used by Gumbel are sample,, quantities which thcrerore du not reqmr= c~rrc,,cll~m I'or bias, though c~)rreci~.m ~il bmh would m~t change Eq.(12) 'The bias In (3umbers melh~d discussed beh~w has a ditTerent clrig~n. Insertk)n or the estlmah'~rs I ' ~ r . and = m E q . ( l a ) ~lr ( l e ) provide,',, G u m b e r s predicilun equation l'('r) - ; ,~' ( T ) = ,~' + - o,t, On

( I .'~a)

METHODSt)F FITTING T'HE' DIiI.IBLE EXPONENTIAL DI~TRIBIrTIC)N

26.~

In Chow's m~tatlon d'us bec,r~mes X('T') = ,tl" +

K('T')

-

K

n,~

( I th)

f7 k

which is o1' Ihe I'~rm ,~'('T') = ,r + ,4 (,I, T) n,~,

(13c)

The equation may however be expressed in lerms or the, unbiased estimate 8,~, by K ( ' T ) - #7

,¢ (r)

= ,r + --

-

a,~

( I ~d

)

which is or the rorm ,V ('T') = ,1" + ,4' (., 'T') ,"$',~, l[ IN

(13e)

necessary t(i remember t o USe /7)

l

%,

n-

I

(14a)

wllh Eq.(13d)and iT&

nK = \I -

-

li

(14h)

in Eq. (13b) ( d ) THE MAXIMLIM L,IKELIHi.)OC) ME,'TH()D

The maximum llkehho(ld method ehr~use,s thi~,,.,e value,,,, or the parameters ¢x and . in Eq. (la) which maximise, Ihe hke,hhood or oblalning the given sample I'rom the population so defined. Kimball :l) shows Ihal the maximum likelihood esiimalors are Imphcilly defined by J",l' e

-

I,il'

I/~,= ,~"- >_,e_,,,~,

(1~)

and # =

-',~' In

Z'e ,,x

(16)

'These, 4.~annot easily be, made, exphcll "l'he predlchl)n equahon is obtained hv numerical s~duiion or Eqs. (15) and (16) and insert.m rur ,'x and u in Eq (le).

~)06

M, D, L,I)WERY AND I. E', NASH

Eqs. (15)and (16) may be wr=llen in Chow's notat=on as

.t' =

n

,£ _

"

\ n ,~,"~,,I I v' [ ~'\' '\ l ,., e x p / . l / \ ~ V,'6] J

(17)

and li =

';-

In

('

exp

::" i]

(iX)

C o m p a r i ~ n of estimators

'The several methods uf fittmg may he crmipared in terms ~l" the estmmtes ur p, n and .k'(K) derrved above ('Table 2) Bias in the estimators

The btas attrtbutable t~ each method o1" filling may he Found by corn paring the populatmn quantity wtth the expected values of the ¢sttmat4~rs o1' Table 2. The results, m so Far as the authors can dertve the expected values, are .shown in Table 3. A very slight bias tn Ihe moment,~ estimator of a Js neglected in 'Table 3 'T'hts bias could be removed by usm,e I

E(~,~)=r,

I

4(.-

I)

,]

- I()n''

((,)uenoullle ='-',). Nnmerical tests of bias

~,()(H) random numhers rec,tangularly distributed between zer~ and umty were generated and treated as probablhttes p(X). T h r o u g h inverh~m ~.~1" Eq. (la), with arbitrary values or ~( = I/3())and u( = It.)()) these hl)(H) values were converted tnh~ hI.XH) values of ,V, a random variable dlstrtbuted ac.curd,ng tu F;.q. tla). These were g-rouped m N = 500 samples each or size n = It)and by each sample Ihe p~)pulatmn mean It, Lhe standard devtati~m o, and I . ' ( T ) = p + ha, were esttmated using each fithng melhod tn turn. Thus '~()0 eshmates Z ~1" each o1' the three population quanttttes (Z I =/t; Z, = n, Z~ = p + %) were , b l a t n e d by ea~:h melhod If the ¢stimah~r Js unbiased the mean ol'the N values ol'~, should approach the populatton value Z as N increases tndefimtely II' 8 z ts the observed slandard denvat,un (ablaut Z) ~1" Ihe N ,ndivldual e.sUmates then Z, the

ME"TH()C),',; (.)F,' F'IT'T'IN(]

r.~ r..) .~: .._-:

I':-i

,'

,L,

I-

I

~'

'~ 't::

~JJ

THE,, D(.)IIBL~, EXP()NE, NT'IAL, C)I,',;TBIBI.ITI()N

20"7

~, ..t::,--:

~L~

e-

I'i

--

I

I

,"

I

I

r-:

I

3 B

I i

.ijl rlJ ..£)

I

b

I--1

.=1

,b

~-,

c

.'r'J ,.-:

r~ B

I~ I,~

~,

I"

+

'2 I:: ..~

e s~

~

el t-:

,,~ff ,

L-I

, I ,

,

i-,

~.~..

,11

,~,

i~

l'#

P.~l ~,

I. I

i~

,'i./

I

I

I i--I

I

I I

I~'l I

÷ I

6k.

~'I

~

,~

I

+ II ,-..)

II

q-'.~

÷ II

,,~ g, e.) O' I

:< ÷ ,:1

'~(")~

M.D.

L,OW['RY

AND

I. E'. NASH

mean ul'the Nest,males, should be. asymplohcally normally d,stributed about Z with slandard dewamm 8z/~,,/N and =

(2

-

Z) v''N

(Iq)

r~z should be d=strlbuted as students / with N - 2 shows the I values obtained TABLE

degrees ,~1"freedom. 'Table 4

,.I

Value,., ol t I'or S(')('I sample', ol ,,ize I(I m

PopulaUon

p ~"~ u ~ %t

Momenl,,

-

-I

L,ea,~I square',,

Gun'lbel

I ~tl

h.h",0

I 81'~

12 ,-I'~g

17 'Thl

- ,-I,Iql

12218

1'7 I1'~

- ~HX

11,4

7 12~1

Ma~irrluni [Ikellh~.~d

I '~'79

The hypothesis ,~1"zero bias must he rejected I'~r all filling methud.,,, except m~ments. In the maximum hkehhc, od method the. bins may he in the e.',,hmate or a only. In the ph~ttmg posit=on mc'thods the .nd=c.ations or bias are very strong and occur m both parameter.,,. No doubt an adlustmcnt or the ploHing p~.~.sih,m cl.~uld be made which would ehminate the bias, hut as we shall see thai these melhods are al.s~ less el-ficient than filling by moments the lalter method is prel'erahle on bolh gr~.~unds. 'Table 5 shows a repelJli~n or Ihls lesl using the same data grouped in I()() samples ~l' size 50 'TABLE

Value'., id I, I'~r I(.)(l ,.,ample',,, ol ,,Jze ~O Pl~pulahon

/.1 11 ~ %1

Monlenl~

L,eaM square'~

Gumhe]

Maximum hkehhlw~d

I '~42

'4 22X

~ 272

I (-~

I (X)q

h 1112

7 '78=-)

I O80

The indlcahons ~1" bins are weaker reflecting the smaller bias in the larger samples

ME, THODS (IF' F'IT'T'ING THE', DLII.IBLE E"XPONE,:N'TIAL DISTRIBLI'TION

260

Accuracy

From the practl,.'ul hydrological point of v,ew we may ludge the result ol a method ~1' tilting by the expected value of the err~r in the estimale. 'The best meth~d is clearly that for which the expected error I'rl~m all sources is minimal. We shall use lhe term accurm.'y for the properly of havln~, a small expe~.'ted err~r. Stat~sl~cmns use the concept of e/fir/e/','r I(~r a s~mdar purpose. Fitt,ng method.,, are compared in elt:iclen~.'.V according t~ the varmnce ~1" the estimates I~btalned. I

var ,~'('r')= - /I--

I

z[,~,'('r)-

e,v(r)]"

(2o)

where E,~'(T) is the expected value or mean of all ,~'(T). E'.t-]ic~c,n~.;y, ~n tfus sense, .s a measure or precis=on rather than 4~1'accuracy a.,, it d~es not reflect any bias to wh,ch a fitting method may be SUhlect but merely the scatter ol'estlmutes about their own mean. 'The mean square error (m.se), defined by

mse =

I

~[,~;'('r)-

x('r)]"

(21)

II

(where X(T) is Ihe population value) includes the effccls of b,~th bias and el'ficlency and provides, at least in the pre.senl context, the best crilerion by which t~ judge the relallve merlls of a number or fithng methods. The I'olh.~wmg numerK'al tesl was des,gned for this purpo,se. Numerical lesl of mean square error of estimales

Each of the 5()(I samples II1' ,~lze I() drawn from a double exponential p~pulatl,m p r o v i d e d an e s l l m a t e ~ ol'each of Ihe three population quantities Z~ =p, Z , = a and Z~ =/.l-I-5o'. The rool mean square error I'(~r each set of '~()O estimates was 4:onlputed and appears in 'Table

'T'ABtt h Comparinon,~ or Ihe tool mean ~quure error,~ Momenls

Regre,,,,ion

(3umbel

Ma.xlmurn likelihood

12 2'7 If 47

I t t0 I~ 0'~

I t lC-~ 17 ,.10

,,) .) I..~., IO.X2

~ , S4

fix ~

q~ ,.1~

f,2 41

M D. LOWERY AND I. [;. NASH

270

'The several entr=es =n Table 6 are slandard dewati,ms and are subiecl h~ s a m p h n g variance The srgmficance o1" the difference between each pair o1" e n l r t e s m a y besludted u,~nglhe F ( v a r , a n c e ralto) lesl The values ~l' F For Z =1~+ 5n are given tn Table 7 'T'ABt e 7

Value,, o1' F for Z = p + ~n Momenln

Momenln Regres~Jun Gumbel

Regrcn,aon

Gurnhel

Ma~.lmum Itkehhond

I I 89h

I

2.21,,~ I (]6q

Ma~ L.~kellhood

I 179 2 027

I

2. t~0

'The value o1' F a t the ~",, level ts 1.15 I'or N = 5()(). 'T'hus ,~n the hvpc~thests that the I'our melhods are equally el-~ctent Ihe ddTerence between any pair ,~1' enlries, excephng momenls and max=mum hkehho~d ts stgmficanl. The larger rmse's ~1' the p l o l l m g postlic, n methods c.annol be aItrtbuted to chance alone but to a real difference =n the accuracy hi" the method',;. It has Ihus been eslabhshed thai the smlplesl tilting mclhod (vt;.:., moments) ~s better both m b=as and accuracy Ihan the methods whl,.'h depend ~)n a prwrz plolllng

posfltons. II could be argued thai Ihe bmsed rnelhod could

be correcled and perhaps [hen excell lhe melhod o1' moments In accuracy The accuracy could, in the absence o1' bia,g be expressed in lerms ~1" relallve

eh'fictency. F~r Ihts reas, m the relaltve el-'ficlency ~l the several melhods was also tested using the same numerical data an bel',lre. Table 7 shows Ihe observed standard dewalinn o1' the 5IX) estimates ol'Ihe p,~pulal,'m quantities p, a and .u + ~,o. DilTerences between corresp, mdJng entries in Tables ¢) and 8 are due h) bias. 'T'aate 8

(."omparison ol relalnv¢ t'hmlL'nL*rieyin IIle ab.'.,¢nL'eul bla'.., - s.d ol Z.. Z

Mnmenl,,

lJ /~ -

'~rt

Regre',,.s,on

(.';umbel

Maximum hkehhuod 12hi)

12 2,~ II ,II

12 '7",

12 7~

I ~.~1,

1,4 ~1

I0 h~

¢'v..I '~(.)

'T7 ~h

81 2?-)

hl.81

METHODS (.IF F'I'T'TIN(."; 'THE, DL)LIBLE" E'XPONEN'T'IAL DISTRIBLIT'ION

271

As wtth Table 6, an F'test can be apphed t~ these entrtes and the, values are given in 'Table 9 I'or Z = p + 5a 'TABLe, q Values ~1' F for Z - # + %rt

Momenl~ R egr e;~,,don Gumb~l Max Likelihood

Momenls

Re~s~,,ion

(_';umbel

I I 4hi I ~SH I,OH9

I IOH7 I ~L-H

I I 7h)

M a ~lml.lm

hkeliho.d

Even apart frum bins the methuds of regressl(m and Gumbel are le.ss el~ctent than momcnls ur maximum hkehhood, Ihe ddTercnce belween Ihe lalter two being small. I1 would seem Iherefi~re, that Ih¢ deveh~prnenl ~1 a method c,r correctmg for bias tn the meth(~d ur ret;,ressmn, Gumbers method, or ma~,rmum hkeliho(~d, would n~l be worlh while as the method ~1"nw, ments ts ,simplest, unbiased and subiecl to a .sampl,ng variance only very slighlly grealer than thai ~1" maximum hkehhood and lens Ihan that or the others. Formulae for variance of estimates Kaczmarek 4) studied the variance of ¢,sttmates ~1" X(T') obtained by Oumbel's fitting meth(~d. He used the principle Ihat tl" H ts a I'uneti(m of sample moments such as H = H(,J~, o~)

(22)

it is asvmptotically m~rrnallv dislributed wtth variance given

by

(hHY' (,'H ' ' ,,/-I ,')H v a r H = \ , L ¢ , ] v;.lr.r+~,mt-~ ) y a r n S + ,'~--h,~,oo~.,,cov(.r,o~)

(2t)

where the partral denvat,ve.s are evaluated at the expected values (~1",E and 2 O,k •

Applying the name principle t(~ Eq. ('1) writlen as X ( T ) = ,r + K ~,/6~

(24)

we may oblam Ih¢ varmnce or )['(T) for moments fitlmg. From Eq. (24) ,'~2 (T)

~,~,

0 ,£ ('r) - I,

...,

=

2n

(2~)

.'1' i 7 . '1

M. C). LI(.)WEIRY A N [ )

). E:. N A S H

The variance and covarlance terms are

var..~ = /Idn

(20a)

vat a~ : (P4 - .'~n,,)/" : + -U:,/(" '~ ' - I) coy

(,~,

~2

~,,,, ) = u ),i.

( F"sherZ'))

(2~b)

(Kendall and Stuarl ~))

(2he)

where p:, ~l) and ~4 are, respectively, the 2nd, 3rd and 4lh central m~ments (~I" the p~pulat,on. The moment rall~)s ll~//.r~''2 and /.14//.1~ arc the coe~clents ur skew and kurtosis which for the double exponential dmtrlbut,.m are 1.14 and 5.40 respectively. Insert,m of these valuer in Eq. (23) yields =

+ n

k n

+

K:

+ ',,

n

(27a) n-

I

/nr mnments fit/rag. For reasonably large samples we may take n = , wrHe

I and

0- 2

vat ,~'(T')= - [ I

+ I.I,-IK + I I()K 2]

(appru~.).

(2.7b)

II

'To assess the accurac.y of an est,mal¢ ,~'(T) obtained I'r,,m a s=ngle sample we can (rely substflule #~. fc, r a ~n Eqs. (27a) ur (27b) oblain,ng var X ( T ' ) = ~''~' [I + 1.14K + 1 I()K"].

(27¢)

II

Comparison ~)I" the prediction equallons for lhe method of moments and Gumhel's fi[llng melhod, (Eqs. (7) and (13d)) shows that /( in the former Is replaced by , 4 ' = ( K I - ~ ' ) / # K in the latter. A ¢orresponding change in Eq. (27a) welds var ,~i' ('r)

I + I.I,4,4' + ( , 4 ' ) 2 t ) 6 l ) ÷ l ) ~ t ) - -

= II

H-

or in terms o["

/ ,4 = (K,

var,~'('T') =

o[ II

I + 1.14A

-

g)Io~

= ,4'

,/

,/,, , ( --

II

II

+ ,4" (,l.~,()+ l).O('l

,,,)] II

,

ror lhe varianc, e ~I" ¢slimates oblalned by Gumbel's filling melh~d

(28a)

ME'T'HOr)s (.IF' FIT'TIN(.3 THE' [)(.IIIBLE, E X P U N E N I I A L

F)I,~'TRJBI.I'T'ION

27.'t

For large n this approaches var ,~'('T')=

r/?

[I + I 14,4 + 1.1(),4"]

(28b)

It

which Is equivalent tu Eq.(27b) wllh ,4 replacing K. Kaczmarek derived Eq. (28b) hut apparently made a numer,.'al err~r in lakmg Ihe c~eh~cienl uf skew iJ~/p~/:= 1.298 instead ul' Ihe square ro~l i~l' this quanlily el:, 1.14. (O'Brlen i'.l)). 'This error altered (he coel~clent of K E'q. (28b) to 1.298. Eq. (2•b) wflh the eoelTielent ~1" K as 1.298 was subsequently recommended by W. M.O. '~). Nash and Am~r,~eh~* H) indepei|denlly ~1" Kaez.marek derwed an exprcssi~m I'~r ear ,~'(T') I'~r munlenls filUng. 'They used a Munte-(.Tarh~ process to evaluale the e~e~clenl ~1" K and ~blained I. I~. Later;~), ~n bc,,¢~mHng aware ~1' Kaczmarek's work they accepted Ihe superior,re of the analyUeal development and altered the ¢~etficient of K to 1.29X - Kaezmarek's analylieal hut zncorrect value - thus compounding the conl'us~on. The, ¢4~rrect expressl~mS

I'or n|oments and Gumhel's filling melhods, respectzvely, are. Eqs. (27,',) and (21~a) and the approximate equivalenls Eqs. (27b) and (214b) In an elabt~rate algvhrale analysm Kimhall :~) obtazned an expressi~m I'~r Ihe variance uf esUmates oblained thr~ugh max~murn h k e h h ~ d . ,2

vur ,~'(T) = a.,,. [()~7~ + ()'-),..IgK + ()~()~K:]

(2~)

II

T'h~s expression ~s ~n terms uf 8.~ a sample ¢sUmal¢. In th~s fi~rm the equauon ,s mare useful in eslimaUng the,, suscephbflily to err~r ~1" a single eslimate ,~'(T) bul ~s less usel'ul in sludying relative el"fic.~eney. Again I'ollowing Ihe hnes ~1" Kaezmarek's work ~t ~s p,.~ssJble h~ find an un analysed expressJun for the varmnce uf estimate,.; ~btained by regressi~m filing. A more general I'ormulaliun ~1" Eq. (2 t) g~ven by Kendall and Stuarl. *~)

can be apphed to obtain the ear=ante o1" ,~'(T) I'~r regression filling varX(T)=

vzlr,t"+\

varZ'~k +

.k °' cuv(,£,E,~/,)

(t())

A c~mplex express,in Fur ear (,.T,v'vA) ~s l-.,,iven by Kendall and Stuart but no expl,c~l expression rut the eovar~ance term is km>wn h~ Ihe authurs and Iherel'ore Eq. (30) cannot be develuped. The usual fi~rmula for Ihe varmnee or mdwidual esl,males appropriate h', a s,mple hnear ret.,',ress,m of ),' ur K where, the err~rs arc,, umformly d~slr~buted along the regress,.>n hn¢ ~s gjven by Ezekiel and F'~s [") as

[,

ear .¢¢(T)= \~ + Z'(K - K,)"]

(.tl)

274

M. D. LI(.IWE'RY AND ,I, E', NASH

where S" ~s the residual variance (,1' the )."s. E.q (31) expresses var ,~'(7') as the sum or the variance ~1" the mean ,~" and ( A - ~ ) " by the varmnce ~,1"the shape or the regression hne. H~wever, the assignment of K values to the sample X's tn acc~rdance w~th their observed rank d(~e,,; not comply with the normal requtrement~ or least :,quares fitting where the X's and the A"s u~e the values or s,multaneouslv ~_~bserved attributes ~1' the ~nd~wduals m the sample ~r the X's are values at preselected K values. In the present c~mtext, the mean ~l' ),' ~s suhiecl h~ the same varmnce as if a sample of the k"s ahme were drawnt,~, v a r , ~ = a 2 / , Nashtt)allempledt~alh~wfi~rthtsbvm~,d~l'v Ing Eq. (~1)h, vat .~'(T)

=

~" ,,

+

(K

- / ~ ) " , ., s, .

'T'h~s man,festlv glves the ct~rrect varmnce ~l" the mean or .'f but even tl' the mtttul variance is als~ used as Ihe c~.~el-~cient of ( K - R ) " / ( S ( K - R ) '~) ~t ~s t~hvt~us rrom Eq. (31)) thai the formula ts st,II deficienl ,n :1 c~.wariance term c o v ( , t , S ~ k ) It seems, Iherefi~re, thai there ~s n~e~press~on avadable I'~r the samphng variance ~>1'least squares estimates C o m p a r i s o n o f the error formulae

Bearing in mind that ,4 ts greatel than A' I~r ;.111practical values (~1"h' and n, and c, mlpartng Eqs. (2'7aL (28a), and (29) is ,s clear thai vur ,('(T) increases I'n~m E q. (2q) t~ E"q. (2ga) to Eq. (2'7a) c~nfirmnng the pr~gresslvely h',wer eh~ctencv (~l" the melhods or maximum hkehh(~t~d, moments and G u m b e r s rneth~d Conclusions

Of the several fitting methods the .simplest h~ apply is that ~1' mL~mertls. This melhod is virtually unbiased and is m~re el~ctent and m~re accurate (m the rose sense) than either of the melhc)ds which depend ~n an a priori ph~tllng p~s,tlon. In additr~m these melh~ds are bmsed. 'The metht~d ~1' m a x m m m hkehh~t~d ts shghtlv m~re el~ctent than rn~rnenls but tt may he shghtlv hmsed. It ,s also extremely di~cull t~ apply Eq. (27a) ts an expression Jn p~pulalton terms and Eq. (27c) an upprt~xl mate I'~rmula in sample terms I'~r the variance t)l" eslilTlalen, obtained by moments, or the magmtude corresponding to a given probubihty or return perr~d, assuming that the sample is rand~mlv drawn I'r,~m a dc~uble e~tp~ nenttal p(~pulali~m.

MET'HODS (.)F F,rT'TING 'THE DI.')UBLE EXP()NEN'T'IAL DISTRIBIrT'II)N

27.~

Acknowledgemenl 'T'h=s paper =.s p u b h s h e d by pcrrn=.~.~ion (~1" the D=reclor, Institute o r H y dr(~h~gv, W a l l i n g h ~ r d , Berk,s, E n g l a n d .

References I) E J Gumbel, Slah,~llC'~o1' eMrerne~, (Columbia LlnlV Pre'.,.~, 19hi)) 2) V T (."huw, Frequency analvsi.~ o1' hydrolo~c dala (Univ. Ilhnoi,, En~.; Exp. Sla Bull ,-114 (19'~1)) ~) B. F' Kimball, SuI-Ticlent E'~lunallon F"un~'ll~)n,~ P 229 of Slall.sll¢!~ ol E'xlreme,, by E J (.3umbel (Clflumh=a LlmverMly Pre~.s 196t)) 4) Z Kacz.marek. Eh~clency ~1 Ihe e~,llmall~m o1' fl,~od~.~wHh a b,lven relurn perlud, lASH l'¢~r~mlo, vol III (19~,7) ~) R A F'~her, Momenls and producl m~menls (~1'samphng dl.~lrlbullons Proc. L.o,d Malh. So~' 30 199-21H f~) Kendall and Sluarl, The advanced Ihe,~rv ,~1'.slall.sll~..,~ Vol. I (Gril-T-in and C u , 19'~8) 7) W M.() Gu=de h~ Hydromeleorolog,¢'al PraL'hce~ W.M (.) - No. Its8 'T'P~13(19f~) X) l E' Na,,h and J. Am~roch¢), The accuracy of II'le pred=chon of flood.~ i)l' h=gh relurn period Waler Re.sourL'e.sRe.search, 2 (19hh) N~ 2 9) I E Noah and J. Ammoch~, Leller I¢~ EdHor, Waler Re.s¢~urc'e,~Renearch 3 (19h'7) N~ 2 I0) E'zek=el and Fox. Melh¢~d,~ ol c,~rrelal,~n and rel..,,re~;s=onanalys=.s (J~hn W=ley and S~n~, IOSg) II) .I E Na.~h, R~ver E'nb~n¢¢rm8 and Waler C~m~erval~un W¢~rk,~ Chap. f~. E"d~led by R B 'T'h~rn (Bullerw,~rlh.s, L~nd¢~n, 19~f~) 12) M H (.)uenoullle, F"undam(nlal.s ¢~1Slal=~h~'al Rea.son,nR N,~ ~,~l'Gr=h~n.s Slal=sh~:al M~no~aph~, and Cour.se.s(I 9SI,I) I 1) D O'Br,en, Pr,vale cornmun~cali(m (19h1,1)