A note on the applicability of log-Gumbel and log-logistic ...hydrologie.org/hsj/470/hysj_47_01_0123.pdf · (MLM). These estimation methods are used as methods of approximation of

Hydrological Sciences-Journal~des Sciences Hydrologiques. 47( 1 ) February 2002 123

A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: II. Assumed pdf

STANISLAW WEGLARCZYK Institute of Water Engineering and Water Management, Cracow University of Technology, Warszawska 24, 31-155 Cracow. Poland

[email protected]

WITOLD G. STRUPCZEWSKI Water Resources Department, Institute of Geophysics, Polish Academy of Sciences, ul. Ksieeia Jamisza 64, 01-452 Warsaw, Poland [email protected]

VIJAY P. SINGH Department of Civil and Environmental Engineering, Louisiana Slate University Baton Rouge, Louisiana 70X03-6405. USA

[email protected]

Abstract Applicability of log-Gumbel (LG) and log-logistic (LL) probability distributions in hydrological studies is critically examined under real conditions, where the assumed distribution differs from the true one. The set of alternative distributions consists of five two-parameter distributions with zero lower bound, including LG and LL as well as lognormal (LN), linear diffusion analogy (LD) and gamma (Ga) distributions. The log-Gumbel distribution is considered as both a false and a true distribution. The model error of upper quantiles and of the first two moments is analytically derived for three estimation methods: the method of moments (MOM), the linear moments method (LMM) and the maximum likelihood method (MLM). These estimation methods are used as methods of approximation of one distribution by another distribution. As recommended in the first of this two-part series of papers, MLM turns out to be the worst method, if the assumed LG or LL distribution is not the true one. It produces a huge bias of upper quantiles, which is at least one order higher than that of the other two methods. However, the reverse case, i.e. acceptance of LN, LD or Ga as a hypothetical distribution, while the LG or LL distribution is the true one, gives the MLM bias of reasonable magnitude in upper quantiles. Therefore, one should avoid choosing the LG and LL distributions in flood frequency analysis, especially if MLM is to be applied.

Key words Hood frequency analysis; log-Gumbel distribution; log-logistic distribution; statistical moments; quantiles; model error; asymptotic bias; approximation; method of moments; linear moments; maximum likelihood

Applicabilité des distributions de probabilité log-Gumbel et log-logistique dans les analyses hydrologiques: II. PDF supposées Résumé L'applicabilité des distributions de probabilité log-Gumbel (LG) et log-logistique (LL) aux études hydrologiques est examinée pour un cas réel où la distribution supposée diffère de la véritable distribution. L'ensemble des distributions alternatives est composé de cinq distributions à deux paramètres et dont la borne inférieure est zéro; à savoir les deux distributions LG et LL, ainsi que les distributions lognormale (LN), analogue de diffusion linéaire (DL) et gamma (Ga). La distribution log-Gumbel est considérée d'abord comme une vraie et ensuite comme une fausse distribution. L'erreur du modèle des quantiles supérieurs et des deux premiers moments est déterminée analyttquement par trois méthodes d'estimation: la méthode des moments (MMO), la méthode des moments linéaires (MML) et la méthode de probabilité maximale (MPM). Ces méthodes d'estimation sont utilisées pour

Open for discussion until 1 August 2002

mailto:[email protected]



124 Stanislaw Weglarczyk et al.

l'approximation d'une distribution par une autre. Comme il a été indiqué dans l'article associé, MML s'avère être la pire si la distribution supposée LG ou LL n'est pas vraie. Elle produit une énorme erreur systématique pour les quantiles supérieurs, qui est plus grande d'au moins un ordre que celle qui est obtenue avec les deux autres méthodes. Toutefois, dans le cas inverse, si l'on prend LN, DL ou Ga comme distribution hypothétique et si LG ou LL est vraie, la même méthode donne une erreur systématique raisonnable pour les quantiles supérieurs. Ainsi, on doit éviter le choix de LG ou de LL lors de l'analyse fréquentielle des crues, particulièrement quand on veut utiliser MML.

Mots clefs analyse fréquentielle des crues; distribution log-Gumbel; distribution log-logistique; moments statistiques; quantiles; erreur de modèle; erreur asymptotique; approximation; méthode des moments; moments linéaires; maximum de vraisemblance

INTRODUCTION

The classical problem of parameter estimation for a random sample taken from two-parameter log-Gumbel (LG) and log-logistic (LL) probability distribution functions (pdfs) has been discussed in the first part of this series of papers (Rowinski et ai, 2002). The maximum likelihood method (MLM) proves to be the universal method for these pdfs as it can also be used outside of the parameter range where the first two moments or L-moments are finite. Taking for granted that the first two moments are finite in hydrological applications, the method of moments (MOM) and the linear moments method (LMM) can also be used. However, going one step further and assuming the existence of any order moment brings one to the conclusion that neither of the two distributions can be the true distribution (T) of the hydrological variables.

The possibility of correct identification of a pdf in the case of a normal size of hydrological samples is small even in the ideal case, when the set of alternative pdfs contains the true distribution function (T). Therefore, in reality, one deals with the hypothetical pdf (H), referred to here as the false distribution function (F), which differs, more or less, from the true one. This will result in a model error in any statistical characteristic of the distribution. Its magnitude for a given characteristic depends not only on how close F is to T, but also on the estimation method. The main interest of flood frequency analysis (FFA) is estimation of upper tail quantiles.

The two-parameter LG and LL probability distribution functions differ from all other pdfs used in FFA, with respect to both the moment's existence and the magnitude of skewness coefficient when the first two moments exist. Therefore it seems necessary to examine the problem of parameter estimation from LG and LL assuming the real conditions, where the hypothetical pdf differs from the true one. This study is focused on the model error of upper quantiles and moments. Applicability of the two pdfs is examined critically in the real condition of unknown true distribution and recommendations are given for the estimation method under the assumption that LG and LL are either F or T.

ASYMPTOTIC BIAS OF ESTIMATION METHODS CAUSED BY THE ASSUMPTION OF FALSE PROBABILITY DISTRIBUTION

A study of asymptotic bias can serve as a basis to assess its magnitude and give an idea about the bias for any sample size, and whether the difference in the bias due to various parameter estimation methods can be counterbalanced by their efficiency of

On the applicability of probability distributions in hydrological analyses: II. Assumed pdf 125

estimation. It would also be useful to verify the correctness of Monte Carlo experiments. The theoretical background for the asymptotic bias derivation for various estimation methods when a pdf is falsely assumed is given by Strupczewski et al. (2002). To present a unified treatment for various distributions, the original set of parameters of every distribution was expressed as functions of moments. The estimation methods are used as methods of approximation of one distribution function by another. However, because for LG and LL the original parameters cannot be explicitly expressed by the first two moments, the original parameters are used in this study. Furthermore, its scope is limited to the investigation of the three methods, i.e. MOM, LMM and MLM, while the least squares methods (LSM) of approximation (Strupczewski et al., 2002) are omitted.

The method of moments, used as an approximation method, reduces to the moment matching of F to T distribution:

jxrf(V){x;g)dx = jxrfin{x;h)dx r=l,2,...,R (1) 0 0

where R is the number of parameters of false distribution, i.e. the dimension of the g vector.

Similarly, LMM as an approximation method reduces to the L-moments or the probability weighted moments matching:

Jx(F,,|)[F,,l]'dF(l' =)x(f(TllF ,nJ'dF(n r= 1,2,...,* (2) 0 0

where F " is the cumulative probability distribution. In order to apply MLM as an approximation method of one distribution (T) by

another (F), one has to consider the asymptotic sample of T distribution. Then, the MLM equations have the form:

lim E d\ogLV]{X;g\T) ia io8L'" (x ;*Vm(*;*)dx = 0 (3)

°g

where f<l)(x; h) is the probability density function of the true distribution and h is the vector of its parameters, while g is the vector of parameters of the false distribution /*F )( .Ï; g). The term log//' ' denotes the log-likelihood function for the false distribution.

Having approximated T by F, one can find for any characteristic Z both the value of z of the approximated function, i.e. z(H = T) and the corresponding value of z of the approximating function, i.e. z(H = F| T). The asymptotic bias of any statistical characteristic Z, B(Z), is defined as:

B(Z) = z(H = F\T)-z(H = T) (4)

and the relative asymptotic bias, RB(Z), as:

RB(Z)=-± ;M (5)

where H, F and T stand for hypothetical, false and true distributions, respectively.


The maximum likelihood method, used as an approximation method, is irreversible, i.e. assuming the same dimension of vectors of parameters g and h, if (h = a) => (g = b) then (g = b) =£> (h = a). The asymptotic bias of the MLM approximation of any statistical characteristic Z is not asymmetric as it is for the MOM and LMM, i.e. 5MLM(z(H = <p | T =ff) * -BmM(z(H =f\ T = <p)), where / and q> denote pdfs. Furthermore, to apply MLM as an approximation method, the approximate distribution (H) must have the same range as the true distribution, or its domain must cover that of the true distribution. For MOM and LMM, there are no constraints with respect to the range of both distributions, but the overlapping range enables the fit of the first moments.

SET OF ALTERNATIVE PDFS

The set of alternative pdfs (apdfs) taken for the study consists of the following two-parameter distributions with zero lower bound: log-Gumbel (LG), log-logistic (LL), lognormal (LN), linear diffusion analogy (LD) and gamma (Ga). Although three-parameter distributions are often recommended for flood frequency analysis, two-parameter distributions were chosen in this study for two reasons. First, the constraints in respect to the number of parameters are very rigid for normal hydrological sample sizes. Second, the objective is to show the significance of bias using simpler models. To extend the results to three-parameter lower bounded distributions, it would suffice to start with the matching of the lower bounds.

This study is focused on the LG investigation considering it as both a false and a true distribution. However, all the conclusions remain valid for the case when LL approximates other distributions, or is approximated by them. Therefore, the two following cases are analysed herein: - (H = LG; T = LL, LN, LD, Ga) - (H = LL, LN, LD, Ga; T = LG)

However, for the sake of brevity, only the H = LG; T = LN and H = LN; T = LG cases are presented in full in the following two sections. The derivations for all other pairs (F, T) and the respective graphs are omitted and only the tabulated results are presented. Being aware that a substantial part of the information is missing in such a way, the authors are ready to supply the reader with the full text upon request.

The density functions of the distributions are given below. The formulas for moments, L-moments and quantiles can be easily found in the statistical literature (see, e.g. Hosking & Wallis, 1997; Kaczmarek, 1977), while for LD they are given by Strupczewski et al. (2001):

x ,oc I ( i ,^>0 (6)

a,, ,K,X > 0 (7)

log-Gumbel (LG) distribution:

log-logistic (LL) distribution: i

( x / a , , )K

exp(— fy

/ ( x ; a , j . , K ) = -/ i V

KX 1 + U /a ,


In order to distinguish between the a parameters of LG and LL, the subscripts LG and LL, respectively, are used.

lognormal (LN) distribution:

fix; Lt, 0) = p = exp 0xv2ît 20

T ( l n x - L i ) 2 x , 0 > O (8)

linear diffusion analogy (LD) distribution:

a, /[*;an»Pi.D] = - f = e x p VrocJ

( « L D - P L D * ) 2

X'ai.D'Pl,D > " (9)

gamma (Ga) distribution:

a ; Ax;aGa,X) = —f-xx^ exp(-aG:ix) 1(A)

c,aCia,X> 0 (10)

LOG-GUMBEL AS HYPOTHETICAL DISTRIBUTION AND LN AS TRUE DISTRIBUTION

The asymptotic relative bias of the mean (m), variance (var) and quantile (xp) expressed in terms of parameters of both distributions equal, respectively:

M(W )=^-i=-*aT ( 1-S-i exp1 (LI + 0 2 / 2 )

ftB(var) •• varL0 i = ^ ' " [ r ( l - 2 a ) - r 2 ( l ^ q )j

varIN exp(2jj. + 0 2 | exp (0 2 ) - l j

a i .G < 1

-1 a,,. <

(11)

(12)

RB(Xp): >I.G

>I.N

-1 = •THI-P)

expya + at (13)

The form of relationship of LG parameters (£, ai.o) with LN parameters (u, 0) depends on the approximation method.

MOM approximation

The condition of the equality of first moments takes the form:

^ T ( l - a U i ) = exp f 0 2 ^

a u i <1 (14)


<x_LG

1.2

1.0

0 . 8 -

0.6 CvLG

1.0

0.51

0.2

0.1

0.2

0.0J 0.L

tij_MCT=fLN, Fi

I \y

_/'i'riLri(iT=LG

I ! I I I I y^ >*-""

j - ^ r " I I I I I

"

= LE

;, F = LN)

,~!LI"lr1

non

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 o.o 0.2 0.4 0.6 1.0 1.2 1.4

CvLN

Fig. 1 Relationship between parameters a u i and o for MOM, LMM and MLM. Unlike the first two methods, MLM produces two relationships (i.e. equations (20) and (28)) depending on which of distributions (respectively LN or LG) is true. Additional axes for true coefficients of variations C,. are given for comparison purposes. The axis Crln is valid for MOM and T = LG, while the Crl N axis is valid for all cases except T = LG.

Matching the coefficients of variation, i.e. C,LG = C,-LN, gives:

_, r ( l -2cc 1 ( i ) , / n , Q =~~T7 ^ - l = exp a- - 1

r 2 ( i - a L ( ! ) F l '

(15)

which (through given CvLN) relates the parameters 0 < rj < °° and 0 < OCLG < 0.5. The relevant relationship OCLC;(O>) is plotted in Fig. 1 as the MOM curve,

From equations (13) and (14) one has the relative asymptotic bias of the xp

quantile:

RB (MOM) ,

*,J=-exp(a2/2-atp)(-ln(l-p)Y

r(i-aIXi) (16)

where parameters ate; and a are interrelated through equation (15). Equation (16) is illustrated in Fig. 2. The term RB(M0M)(xp) remains within about a 30-50% variability range for all CvLN under consideration with RBiMOM\xo.\%) increasing with CVI.N up to C,LN = 0.4 and then decreasing with C,-LN- The minimum of RB(mm>(xp) deepens and shifts towards low p.

LMM approximation

The match of the L-coefficients of variation gives:


°% . p > [ \ ] CvLH=0.20 RB ' *p ' . [ \ ] T=ui, F=U6 nori ' , ° 1

CvLN=0,30 R B ' . p ' ' [ \ ] T=LN. F= W v

T w 1 5 ° l 40"

30

20-

10'

CvLM=0.-fO

30 20 10 5 2 1 0.5 0.1 prob. of exceedance, EXJ

50 20 10 5 2 1 0.5 0.1 prob. of exceedance, V/.J

50 20 10 5 2 1 0.5 0.1 prob. of exceedance, [X]

PB' . p t [ \ ]

40

30

20

10 0

-10

pio 20

10

0

-10

- 2 0

, , , , , ,

V ^

, '

50 20 10 prob.

> p t [ \ ]

h i i

50 2 prob.

of

1

'--— n IO

of

T=LN, F=L6 t1

, ^.

--'"'' 5 2 1 0.5 0.1

exceedance, C"<3

CvLH=0.80 T=UI, F=l»6 ,(1

/

' ' ' "' ' '

5 2 1 0.5 0.1 exceedance, [X21

CvLH=0.50 R B ' . p i p J 40-1

30-

20-

10

-10

-20

V

C'vLN = 0.60 PB' .p''['-.] T=LN,' F=L6,:"0r1 4 ° 1

20-10

0

-10

-20

CvLM=0.70 T=LN," F=L6 ' M 0 M

50 20 10 5 2 1 0.5 0.1 prob. of exceedance, ["'0

50 20 10 5 2 1 0.5 0.1 prob. of exceedance, C'̂ 3

CvLH=0.80 RB'vpU'O CvLN = 0.9ÇL„ RB' .pH"'.] T = U I . F=M3,nori t ° ] ; T=WJ, F=t6/T,0M J5] ;

CvLM=1.00

15 10"

-10 -15 -20

151 10

0 - 5

-10 "15 -10

\.. \,, • v 1 < ^ ,

" *l

, , ,

< 1

"~-

' 1 1

J~-

T=UI,

' ' 1 '

v '

/' • , • ' ' '

1 ' 50 20 10 5 2 1 0.5 0.1 prob. of exceedance, I'/.l

50 20 10 5 2 1 0.5 0.1 prob, of exceedance, ['<]

Fig. 2 Asymptotic bias of LG MOM-estimated quantile vs probability of exceedence for some selected values of the true coefficient of variation C,.|,N.

2 " - - 1 = 1 - 2 / 7

where

p{w)= J

a

1 | Z

2 , e x prT 2 ~\

<k

(17)

(18)

The range (0, 1) for OCLG corresponds to (0, +°°) for o. The relationship of these two parameters is displayed in Fig. 1 as the LMM curve.

Using equation (15), one can express a and (XLG by C,LN and C,-LG, respectively. Substituting them into equation (17), one obtains the LM relationship of both coefficients of variation, i.e. CVLG = (PLMM(C,LN)- Hence, the asymptotic relative bias of the variance given by equation (12) is:

RB (I.MM) , var;F=LG, T=LN) = Q,x. C. I N

- 1 (19)

The bias is displayed in Fig. 3 as the function of the true variation coefficient of the LN distribution, C,LN- Because CXLG < 0-5 (see equation (12)), from equation (17) one obtains o < V2 t tl, where tp is the p - N(0, 1) quantile, which gives an upper limit to C-LN of 0.9007. The LMM bias of variance quickly increases with CVLN and exceeds 100%atCvLN = 0.5.


RB 17.1

160T" 1-10

120-

100

80~

6 0 -

40

2 0 i

0

"QT"t"r"ue^ Lb r a l î é

tlLM.'

RB 17.2

160D-I

1-tOO"

1200

1000

800

600-

100- '

200

0

-X

' urTrue'."' nrïaiïB

0.0 0.2 0 .1 0.6 0.8 1.0 CvLN

0.0 0.8 1.0 CvLN

Fig. 3 Asymptotic bias of (a) LG MLM-estimated mean (upper limit: C,.|.N = 1.3108), and (b) LG MLM- and LG LMM-estimated variance vs true coefficient of variation C,,|,N (upper limits: C,,LN = 0.5329, CrLN = 0.9007).

The relative asymptotic bias of the quantile xp for LMM can be expressed by equation (16), where OCLG and o are interrelated through equation (17). The results are shown in Fig. 4. The difference between the RB(LMM)(xp) and RB(M0M)(xp) is small for small values of CVLN (less than 15% for CVI,N

more than 100% forxo.i% for C,-LN = 1-0. 0.2), and increases with Cr, reaching

RB'' SO

40

30

20

10

0

p U \ ] C v L N = 0 . 2 0 RB'

,T=T'\ F = U V m so" -, .M0M BO

.-- -- 40 , , , , , , , . - , _ . , , , 30

^ ^ r > , , , , . »° 50 20 10 5 2 1 0.5 0.1

p r o b . of e x c e e d a n c e , ExJl

R B ' v p H X ] C v L N = 0 . 5 0 RB' 100-

an-

6 0 "

4 0 -

20

0 '

T=LN, ' F = L S . ' L m 1 2 C I

100

80

MOM 60

40

' ' ' ' ' . , > ' > > ' ' ' ' 20

â ^ - ^ ^ i ^ l — - " ' o

50 20 10 5 2 1 0.5 0 .1

p r o b . of e x c e e d a n c e , l"-Q

RB'

120

100 ' 8 0 -

6 0 -

4 f r

2 0 '

p i [ \ ] CvLN = 0 . 8 q m R & r

T * H . F=L6, ' £

100-80

: . : : : : i ] . • : : : I 1 0 H 4 0 .

- ' - ' ' • ' > - " 201

' • a ^ j r ^ j - ^ - " " ' • • o-

50 20 10 5 2 1 0.5 0.1

p r o b . of exceedance. , I'Kl

p i [ \ ] CvLM=r j .30 PB'

] T = U I , F = L 6 . LMM 8 0

j "s 6 0

. , - , .MOM ,_

-f i-~ • y s 20 -

__.- __ . - *J - - -

p • ' [ • ' . ]

1 !

•^

50 20 10 5 2 1 0.5 0.1 50 2

p r o b . of e x c e e d a n c e , ["-;] p r o b .

p i [ \ ] C v L H = 0 . 6 0 R B C X D U X ]

T=LN, F=L6 .LI1M 1-°

, , , , , . , , , ,, 100l 1 1 1 , 1 ' ' ' >J ' 80

' 6 0 -.-" _M0M 40

.s*'* „ , - - ' " ' ' 20

t .—--""_- , -—"" 0

' ' ' ' ' '

5 ^

50 20 10 5 2 1 0.5 0 .1 50 2

p r o b , of e x c e e d a n c e , OQ p r o b .

p ' ' [ \ ] C v L N = 0 . 9 0 PB*

; T=UI, F=U3,LMH J *

, , . ' , 100

..MOM - °

i ^ j j__>--<_J^---r , _20

p > [ \ ]

^ i

B0 20 10 S 2 1 0.5 0.1 50 2

p r o b . of e x c e e d a n c e , C"-:i p r o b .

. non

_r- - r 1 1 1 x - i I j . - " 1 1

^c^-. , , , 3 10 5 2 1 0.6 0,1

of e x c e e d a n c e , Î'-K.1

CVLN=0 .?0

' ' T=LN.- F = L 6 , ' L m

.' ' ,-' ' !

, MOM

.--' _,,---'" --*-"" J-*"""~

0 10 5 2 1 0.5 0.1

of e x c e e d a n c e , t"41

CvLN = 1.00 , , ,T=l r l l J i F=lf6 ,Lt1M

• '

' ' ' , - ' - ' ' ' _J MOM

! _ + - - - * " - + ™ - ~ ™

0 10 5 2 1 0 .5 0.1

of e x c e e d a n c e , I'/.l

Fig. 4 Asymptotic bias of LG MOM- and LG LMM-estimated quantile vs probability of exceedence for some selected values of the true coefficient of variation C,.|.N.


MLM approximation

Application of the MLM method results in the following equations:

= 1 (20) ( o ^

a,

^ = e x p [ - l / 2 + u7a] (21)

The graph of (XLG(<J) corresponding to equation (20) is presented in Fig. 1 as the MLM curve. The bias of mean and variance are given by:

RBlM,M\m) = Eâz^Mil 1 a <! (22) exp(aUi /2 + aj"(i 12)

*fl<"™'(v«r) = r ( 1 - 2 a U i ) - p ( l - « L J _x I ( 2 3 )

exp(afx; + a, (j ) (exp(a[G ) -1 ) 2

The conditions for OCLG in equations (22) and (23) combined with equation (20) give 0 < 1 and G < 0.5, respectively, which gives the corresponding upper limit values of CVLN for the relative bias of the mean and variance: CVLN =1-3108 and 0.5329 (Fig. 3). The MLM relative bias of the variance starting point at CVLN = 0 is not zero as for LMM: the limit of RB{MLM)(var) given by equation (23) (cxLG -» 0) is:

\|/'(1) - 1 = jr/6 - 1 = 0.64493 = 64.5%

where the \|/ is the trigamma function. Then, the bias quickly tends to infinity. The MLM relative bias of the quantile can be calculated using equations (20) and

(21) combined with equation (13):

/îfî(M '-M '(^) = exp[-(0.5 + r / , ) a U i ] ( - l n ( l - p ) ) - a " i - l (24)

The relationship given by equation (24) is presented in Fig. 5 together with the MOM and LMM RB graphs.

LOG-GUMBEL AS TRUE DISTRIBUTION AND LN AS HYPOTHETICAL DISTRIBUTION

The methods MOM and LMM as approximation methods are reversible, i.e. for any statistical characteristic Z, the relative asymptotic bias holds asymmetric property:

B(Z; H = A;T = B) = ^B{Z; H = B;T = A) (25)

Therefore, for the relative asymptotic bias one has:

RB{Z; H = B;T = A)= [RB(Z; H = A;T - B)+1]"1 - 1 (26)

and all algebra presented above for MOM- and LMM-matching and H = LG; T = A applies to the opposite case, i.e. to H = A; T = LG. Thus, proceeding in a similar way to that presented above, one can derive the asymptotic relative bias if log-Gumbel is


RB'' 100-

80-

S0-

40-

20

RB' 400 -350 300 250 200

150 100

5 0 -

RB'-

1000

800 •

600 400

200

p • ' [ • ' . ]

' ' ' l-50 20 10 p r o b . of

p t [ \ ]

, . , ,

50 20 10 p r o b . of

pHX]

, , , ,

50 20 10

p r o b . of

L'vLH=0.20 RB( T = U I , F=U6,'MLH 2 0 0 '

ISO-

j- , HLHM 100-

',^-:' ,^"^"m so

5 2 1 0.5 0.1

e x c e e d a n c e , [ Xl

P'C'-J CvLN = 0.30 RB' T = U I , F=L6 3 0 0 1

, i ( iLn ; 5 0 .

' 200 •

. '- ' ' 150-

_,.LMt1 1 0 0 .

^ ' ^ _ _ ^ ^ - ~ ' , m 50

p i [ \ ] Cvl_H=0.40

E,, , , , T = , " \ F = w v«

__, LMM

, , , '___j^- ' " ' ' ^___i^[___i t10M

SO 2 0 1 0 5 2 1 0 . 5 0 .1 50 2 0 1 0 5 2 1 0 . 5 0.1 p r o b . of e x c e e d a n c e , CK1 p r o b . of e x c e e d a n c e , ["-;]

CvLN=0.50 RB(»p i [ \ ] CvLN = 0.60 R B ' . p t [ \ ] CvLN=0.T0 , T=LN, , F ^ . t t L H " » T=LM, F=U6 | | 1 L f 1 ^ . T=LN, , F=LC% t 1 L t ,

/ 50u " J ,' 600- - !

400- - / 5 0 0 . .

300

-;-^^rrr:---ttoM 10° 5 2 1 0.6 0.1

exceedance. , ["<]

CvLN=0.80 B®' T = U I , F H . S ^ U L M 1 2 0 0

/ 1000

/ 800

é00 1 1 i 1' 1 !

400

' _ , > " " ' ' ' J L M H 200

"T ~~~~ -,m" n 5 2 1 0.5 0.1

e x c e e d a n c e , VA1

. . , • . . 400

300

, L » 1 200

« ' •

--•'"'*' ' E-ELnr1

_ — - - " " ' ——-" """"" - - -nnn

50 20 10 5 2 1 0 . 5 0 .1 50 20 10 5 2 1 0 . 5 0.1 p r o b . of e x c e e d a n c e , L'41 p r o b . of e x c e e d a n c e , ["-;3

p''[' 'J CvLH=0.9Q,.,PB'-T-i >j c i r , i " L r , l S 0 0 , T - | T H ' , F-\6/, 1400

/ 1200 / looo

1 ! 1 , ! ! y |J|J

J 400

-pU'-:] CvLN=1.00 ~ T=LH, F=L6 i riLH

f ' 1 1 1 1 1 i i i J i i

, , , , , . „ - ' " , , '__jUU1

50 20 10 5 2 1 0.5 0 .1 50 20 10 5 2 1 0.5 0.1

p r o b . of e x c e e d a n c e , E"-;] p r o b . of e x c e e d a n c e , V/.1

Fig. 5 Asymptotic bias of LG MOM-, LG LMM- and LG MLM-estimated quantité vs probability of exceedence for some selected values of the true coefficient of variation CvLN-

assumed to be the true distribution. The numerical results are obtained here by the transformation of equation (25) of the results obtained in the previous section. Hence, the problem of this section is the MLM-approximation of the LG distribution by LN (and other lower bounded pdfs), which are used in FFA. One should remember when looking at all the figures that RB always refers to the true value of the coefficient of variation.

The MLM estimates of the lognormal distribution parameters are:

u. = a1 G(C + ln^)

a2 = a2xi V|/'(l)

(27)

(28)

where C = 0.5722... is the Euler constant. Note in Fig. 1 that the relationship given by equation (28) is closer to both the MOM and LMM relationships than that in equation (20) derived for the opposite case, i.e. when LN = T and LG = F.

The relative asymptotic bias of the mean is given by:

RBm.M)(m) = exp(cr/2 -Coc, ( i) {

r(i-aLG)

where (XLG < 1, and from equation (28) one obtains:

(29)

o < ^ ' ( 1 ) = 1.2825 (30)


RB 17.1 0.0- |

- 0 . 5 -

- i . o -

- 1 . 5 -

- 2 . 0 -

- 2 . 5 -

- 3 . 0 -

- 3 . 5 -

~ ~ ^ v "' f T = L 6

\ \....

\

-, !

EX ••T=uC

\

ML M

0 1

- 1 0 -

- 2 0 -

- 3 0 "

- 4 0 -

- 5 0 -

- 6 0 -

- 7 0 "

- R n -

,'.J

t

' • *

-

var X f=L6.':T=LN-

• \

^ - -

inn 1I1LH

0.0 0.2 0.1 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 CvLG CvLG

Fig. 6 Asymptotic bias of (a) LN MLM-estimated mean, and (b) LN LMM- and LN MLM-estimated variance vs true coefficient of variation Crl G.

The relat ive asymptot ic bias of the var iance is given by:

,»„«„> exp(cr + 2 C a , r ) ( e x p ( c r ) - l ) RBim M\var) = — - , 'x' \ / , ' -1

r ( l - 2 a U J ) - r - ( l - a U i ) (see Fig. 6) where ate; < 0.5, and from equation (28) one obtains:

4W) o< 2

= 0.64127

The relative asymptotic bias of quantile is given by:

RB (MI.M)

(*„)=--pi M

",,I.(i

- 1 = -expln + w . exp(q,GC + gf ;J

- im-p)r"=(-Mi-P)r

(31)

(32)

(33)

and is presented in Fig. 7, where it is compared with the MOM and LMM approximations.

DISCUSSION OF RESULTS

A similar analysis to that made above for LN was carried out for other pdfs, i.e. LL, LDA, Ga, and graphs analogous to those in Figs 1-7 were produced for each (F,T) pair. However, for economy of space, all the relevant information is summarized in the following four tables, from which some essential conclusions can be drawn. Tables 1, 2 and 3 illustrate in a concise form the results obtained with respect to the mean and variance for two cases: (a) when LG is the true distribution and other pdfs are false and (b) when the LG pdf is used to approximate other pdfs.

It can be seen from Table 1 that all LG MLM approximations of the pdfs used produce much greater relative asymptotic bias than in the reverse case, when LG is approximated by other pdfs. The use of LG as an approximation of another pdf results even in infinite or indefinite bias of the mean, as for the Ga pdf. The situation is even more dramatic for the MLM relative bias of variance presented in Table 2, from which it follows that the use of LG as an approximation gives very high bias for Cv, rising fast to infinity for Cv = 0.4 and 0.6.


5 ; 10.5 o.i exceedance, ["<]

50 20 10 5 2 1 0.5 0.1

prob. of exceedance, E'<3

CvL6=0.30 RB( . p ' i [ \ ]

T)=L6, Fi=LH 2 0 1

0

"-=:,Lt1t1 .nun

C»U5=0.40

T H . 6 , F *U(

50 20 10 5 2 1 0.5 0.1 p r o b . of e x c e e d a n c e , E''.]

RB( . p t [ \ ] CvLS=0.B0 R B ( . p » [ \ ]

T=L<3, F ÏLN 2 0 1

o- "

20 --ir-iOM

C»L6=0.60 R B ^ . p ' ' [ ' • . ]

Tk.a, F=Lri

"-<-Lnn ""riLn

2U-

0

20

-10-

f^-, , , , , ,

--.__ T=l_a, F=LH

' ' X >- . ' 'X-

LMM t1LH

RB< 40

50 20 10

p r o b . of

p > [ \ 3

5 2 1 0.5 0.1 e x c e e d a n c e , V41

50 20 10 5 2 1 0.5 0.1 p r o b . of e x c e e d a n c e , l'/.l

50 20 10 5 2 1 0.5 0.1 p r o b . of e x c e e d a n c e , Oil

C« 1.6=0.80 RBCptL " ' . ]

T*LG, F+Oi 4 D l

201

I --1H0M - 2 0

— " M L M -60

T^~,

CvL6=0.90 P B ' . p U

T t L 6 , F tLN 4 0 "

SO 20 10

p r o b . of <

5 2 1 0.5 0.1 •xceedance , ["<]

50 20 10 5 2 1 0.5

p r o b . of exceedance. , C'*Q

—"nun

o.i

C»L6=1.00

TSL6, F*LH

50 20 10 5 2 1 0.5 0 .1 p r o b . of e x c e e d a n c e , VK1

Fig. 7 Asymptotic bias of LN MOM-, LN LMM- and LN MLM-estimated quantile vs probability of exceedence for some selected values of the true coefficient of variation Cvi a.

Compared to MLM, LMM (Table 3) extends the range of C, for which asymptotic relative bias of variance exists and lowers the difference between the relative biases of variance for T = LG; H = non-LG and T = non-LG; H = LG cases. As can be expected, the absolute values of bias are lower than the corresponding values of MLM bias shown in Table 2.

The very first glimpse at Table 4 confirms our previous findings with respect to the magnitude of bias of various estimation methods (Strupczewski et al., 2001). The value of KB of the upper quantiles is smallest for MOM and largest for MLM. The bias of LMM occupies an intermediate position. Furthermore, the relative asymptotic bias (RB) of quantiles corresponding to the upper tail is an increasing function of the true value of the coefficient of variation (Cv).

Table 1 Asymptotic relative bias (%) of mean by MLM approximation (NE = not existing)

Pdf

LG

True C,

0.2 0.4 0.6 0.8 1.0

Probability distribution function: LL True

5.2 16.6 35.6 62.6 97.8

False

-1.7 -3.5 -5.1 -6.6 -7.8

LN True False

3.2 -0.1 11.3 -0.7 28.5 -1.5 64.2 -2.5 150.9 -3.4

LD True

3.1 10.6 24.7 49.3 96.4

False

0 0 0 0 0

Ga True

4.3 23.5 152.1 NE NE

False

0 0 0 0 0

On the applicability of probability distributions in hydrological analyses: II Assumed pdf 135

Table 2 Asymptotic relative bias (%) of variance by MLM approximation (NE = not existing).

Pdf

LG

Table 3 availabl

Pdf

LG

True Cr

0.2 0.4 0.6 0.8 1.0

Probability distribution function: LL True 231 863 NE NE NE

False -24.4 -40.2 -52.6 -62.3 -69.8

Asymptotic relative bias (%) of e).

True C,

0.2 0.4 0.6 0.8 1.0

LN True 156 577 NE NE NE

variance

False -21.0 -38.9 -53.4 -64.5 -72.6

by LMM ;

Probability distribution function: LL True 17.5 34.3 59.2 103.1 201.3

False -14.1 -22.9 -31.1 -38.8 -46.1

LN True 26.6 59.3 138.3 541.0 NE

False -19.7 -31.3 ^2 .5 -52.5 -61.0

LD True 153 517 NE NE NE

False -20.3 -36.8 -50.1 -60.7 -68.8

ipproximation (NE =

LD True NA NA NA NA NA

False NA NA NA NA NA

Ga True 215 3,794 NE NE NE

not existing,

Ga True 28.9 73.6 217.9 NE NE

False -16.7 -33.8 -48.6 -60.3 -69.0

NA = not

False -20.6 -34.2 -46.8 -57.5 -66.0

As far as the asymptotic relative bias of the MOM and LMM quantiles forp = 10% is concerned, the T = LG; H = non-LG case can barely be distinguished from the T = non-LG; H = LG case for low Cr values, because Table 4(a) shows almost the same values in the True and False columns. For higher Cv, the LL column still exhibits the pattern for low Cv, while the differences in the bias in the True and False columns for other pdfs increase, especially for LMM. As a rule, the MLM bias for T = non-LG; H = LG is much greater than in the opposite case, i.e. T = LG; H = non-LG.

From among the pdfs used, LL seems to be the best approximation for LG, as the absolute values of LL, MOM, and LMM bias are the lowest. The use of MLM destroys the symmetry exhibited by MOM and LMM: the absolute MLM values for the T = non-LG; H = LG case are about one order or more greater than those for the T = LG; H = non-LG case. The bias of the Ga approximation by LG exhibits the greatest differences, growing fast with true Cr and reaching ca. 550% for C, = 1, compared to 6% for T = Ga; H = LG.

The remarks concerning Table 4(a) refer essentially to the results presented in Table 4(b) and (c). The differences lie generally in the greater absolute values of the corresponding biases, faster increase of bias with the true coefficient of variation Cr, and clearly higher LMM biases compared with MOM.

The 0.1% quantile bias for MOM and LMM does not exceed ca. 60% (in absolute values) while the difference between the MLM bias for T = LG; H = non-LG and T = non-LG; H = LG cases rapidly increases reaching ca. 8 x lO^/o for T = Ga; H = LG.

CONCLUDING REMARKS

It is shown numerically that employing the log-Gumbel distribution in place of some selected non-log-Gumbel true distributions results in an asymptotic bias of quantiles


Table 4 Asymptotic relative bias (%) of quantile with probability of exceedence p (%) got by various approximation methods.

Pdf

(a)p = LG

(b)p-LG

(c)p-LG

True Cr

= 10% 0.2

0.4

0.6

0.8

1.0

= 1.0% 0.2

0.4

0.6

0.8

1.0

= 0.1% 0.2

0.4

0.6

0.8

1.0

Method

MOM LMM MLM

MOM LMM MLM

MOM LMM MLM

MOM LMM MLM

MOM LMM MLM

MOM LMM MLM MOM LMM MLM MOM LMM MLM MOM LMM MLM MOM LMM MLM

MOM LMM MLM MOM LMM MLM MOM LMM MLM MOM LMM MLM MOM LMM MLM

Probability distribution function: LL True

-0.4 1.0

16.3

-2.8 0.1

32.7

-5.1 -1.4 47.2

-6.6 -2.9 59.0

-7.6 -4.1 68.1

6.3 10.4 49.0

5.2 15.1

111.2 1.7

16.2 178.1 -1.5 15.7

240.9 -3.9 14.8

294.9

13.9 20.9 90.9 14.6 33.1

236.3 10.2 38.2

425.4 5.4

39.5 631.1

1.5 39.2

827.9

False

0.4 -1.3 -3.5

2.9 -1.8 -6.0

5.4 -2.5 -7.8

7.1 -3.4 -9.0

8.2 ^1.3 -9.8

-6.0 -9.2

-12.2 -A.9

-13.7 -20.2 -1.7

-16.4 -25.4

1.5 -18.2 -28.8

4.1 -19.5 -31.1

-12.2 -16.7 -20.2 -12.7 -24.7 -32.5 -9.3

-28.9 -40.0 -5.1

-31.4 -44.7 -1.4

-33.0 -47.7

LN True

-2.0 0.0 9.7

-7.0 -2.5 19.8

-12.3 -6.5 29.7

-16.7 -10.8

39.1

-19.9 -14.9 47.7

10.1 16.3 42.1

9.2 26.0 98.0

2.2 29.2

167.4 -6.7 27.9

248.2 -15.0

23.9 337.8

30.0 42.1 92.9 41.5 79.3 258.9 37.2 105.2 529.2 25.5 118.8 930.8 12.0 122.8 1482

False

2.1 -0.2 -0.4

7.5 1.5

-0.7

14.0 3.7

-0.9

20.0 5.8

-1.1

24.9 7.5

-1.2

-9.2 -13.0 -13.3 -8.4

-19.0 -22.0 -2.1

-21.6 -27.7

7.1 -22.5 -31.3

17.7 -22.6 -33.7

-23.1 -27.5 -27.8 -29.3 -40.4 -43.3 -27.2 -46.8 -52.2 -20.3 -50.0 -57.4 -10.7 -51.8 -60.8

LD True

-4.1 NA

7.2

-7.4 NA

17.7

-13.5 NA 23.3

-18.9 NA

25.9

-23.2 NA

25.9

7.9 NA

38.8 9.8

NA 93.7

2.8 NA 150 -6.8

NA 201.5 -16.4 NA 242.6

27.9 NA

88.7 45.5

NA 254.4 45.6

NA 498.5 36.8

NA 808.9 24.2

NA 1151

False

4.3 NA

1.5

8.0 NA

0.5

15.6 NA

1.8

23.4 NA

3.3

30.2 NA

4.7

-7.3 NA -11.7 -8.9

NA -21.3 -2.7

NA -26.0

7.3 NA -28.5

19.6 NA -30.0

-21.8 NA -26.6 -31.2 NA -43.4 -31.4 NA -52.1 -26.9 NA -57.0 -19.4 NA -60.0

Ga True

-2.0 0.2

13.0

-7.8 -2.6 38.1

-15.1 -8.2 91.0

-22.3 -15.7 216.4

-28.5 -24.2 548.2

12.3 19.2 54.6 15.2 36.5

193.5 10.2 48.4

658.5 0.9

52.9 2935 -9.5 49.9

2087

35.6 49.3

122.0 60.1

112.2 586.9

68.0 180.4 3366 62.6

241.8 34714

50.2 285.3

792883

False

2.0 -0.3

0.1

8.4 1.2 1.3

17.8 3.6 2.9

28.7 6.0 4.6

39.9 8.2 6.1

-11.0 -14.6 -13.9 -13.2 -23.3 -23.2 -9.2

-28.2 -29.0 -0.9

-30.8 -32.5

10.5 —32.3 -34.7

-26.3 -30.2 -29.4 -37.5 ^fl-6.6

—46.5 -40.5 -55.5 -56.2 -38.5 -60.6 -61.8 -33.4 -63.6 -65.2


and, depending on the estimation method used, in asymptotic bias of mean and variance. While asymptotic biases of MOM- and LMM-estimated quantités lie relatively close to each other, the MLM-estimated bias of quantités is, in most cases, of at least one order higher, reaching as high a value as ca. 800 000% for x0.i% for T = Ga; H = LG. These findings, illustrated with values in Table 4, essentially diminish the practical value of MLM because its efficiency does not compensate for the (frequently) huge bias produced by the assumption of a false pdf in the region of small exceedence probability quantiles in which the user is often interested. The bias produced by the other methods (MOM and LMM) does not exceed a few dozen percent.

Comparing the results obtained herein with Table 4 allows one to answer the question of how to choose a pdf when each of LG (or LL) and LN, LD and Ga pdfs with MLM estimation is taken into consideration. If one accepts LN, LD, or Ga as a hypothetical distribution, one obtains the MLM bias of reasonable magnitude in upper quantiles of more than one order less than the bias obtained in the case when LG or LL is a hypothetical distribution.

It is hoped that this two-part series of papers provides sufficient rationale for being rather careful with application of LG and LL in hydrology, especially in FFA.

Acknowledgments This work was supported by the Polish State Committee for Scientific Research (Grant KBN no. 6 P 04 D 056 17), entitled "Revision of applicability of the parametric methods for estimation of statistical characteristics of floods".

REFERENCES

Hosking, J. R. M. & Waliis. J. R. (1997) Regional Flood Frequency Analysis: An Approach Based on L-moments. Cambridge University Press, Cambridge, UK.

Kaczmarek, Z. (1977) Statistical Methods In Hydrology and Meteorology. US Department of Commerce, Springfield, Virginia, USA.

Rowinski, P. M , Strupezewski, W. G. & Singh, V. P. (2002) Remarks on the applicability of log-Gumbel and log-iogislie probability distributions in hydrological analyses: I. Known pdf. Hydro!. Sci. J. 47(1), 107-122.

Strupezewski, W. G., Singh, V. P. & Weglarczyk, S. (2001 ) Impulse response of linear diffusion analogy model as a tlood frequency probability density function. Hydrol. Sci. J. 46(5), 761-780.

Strupezewski, W. G., Singh, V. P. & Weglarczyk, S. (2002) Asymptotic bias of estimation methods caused by the assumption of false probability distribution. j . Hydro!. 258(1-4), 122-148 (in press).

Received 20 October 2000; accepted 27 August 2001

Documents

A note on the applicability of log-Gumbel and log-logistic ...hydrologie.org/hsj/470/hysj_47_01_0123.pdf · (MLM). These estimation methods are used as methods of approximation of