The generalized method of moments as applied to the ...

Stochastic Hydrology S t o c h a s t i c H y d r o l . H y d r a u l . 2 (1988) 161-174

a n d Hydraulics © Springer-Verlag 1988

The generalized method of moments as applied to the generalized gamma distribution F. Ashkar, B. Bobée, D . Leroux a n d D. Morisette Institut National de la Researche Scientifique I N R S - E a u , C . P . 7500, Sainte-Foy, Québec G l V 4C7, Canada

Abstract: The generalized gamma ( G G ) distribution has a density function that can take on many possible forms commonly encountered in hydrologie applications. This fact has led many authors to study the properties of the distribution and to propose various estimation techniques (method of moments, mixed moments, maximum likelihood etc.). We discuss some of the most important properties of this flexible distribution and présent a flexible method of parameter estimation, called the "generalized method of moments" ( G M M ) which combines any three moments of the G G distribution. The main advantage of this gênerai method is that it has many of the previousiy proposed methods of estimation as spécial cases. We aiso give a gênerai formula for the variance of the T-year event Xj obtained by the G M M along with a gênerai formula for the variance of the parameter estimâtes and also for the covariances and corrélation coefficients between any pair of such estimâtes. By applying the G M M and carefully choosing the order of the moments that are used in the estimation one can significantly reduce the variance of F-year events for the range of retum periods that are of interest. Key words: Floods, estimation, quantiles, generalized gamma, generalized moments, standard error.

1

Introduction

Flood frequency analysis is an important c l é m e n t in the design of hydraulic structures and in the planning and management of water resources projects. I t is often concerned w i t h the search of a statistical distribution which could be used to f i t adequately a given sample of m a x i m u m annual flood flows. From the chosen dist r i b u t i o n , one can make extrapolations in order to estimate an e x t r ê m e flood event corresponding to a high return period i.e., to a low probability of exceedance (100year flood, for example). This estimation can then be used to design flood control structures that are able to withstand this estimated e x t r ê m e flood. Associated w i t h the choice of a statistical distribution is a certain amount of uncertainty called "model uncertainty". Consequently, one would want to choose a model for which this amount of uncertainty would be m i n i m u m . The question of which distribution should be used to reach this objective has been discussed by several authors. On that issue, the U.S. Water Resources Council (Benson 1968) recommended the use of the log Pearson type I I I ( L P ) distribution for representing flood flows. Following that recommendation, several methods have been proposed for fitting this distribution. T a k i n g a close look at the L P distribution, one realizes that i f the skew

162

Figure 1. Régions in plane for various distributions, where C} and

coefficient in log space is négative, then the distribution has a finite upper bound. Some investigators, w i t h or without valid justification, have considered this property to be unsatisfactory, and have looked for other distributions for fitting flood data. One of t h è s e , is the generalized gamma ( G G ) distribution which differs from the L P in that: (1) i t is always unbounded above, and (2) i t is always, bell-shaped ( w i t h a skew) which is one other property that is useful in practice. The 3-parameter G G distribution which will be the subject of our investigation has two shape parameters and one scale parameter, but no location parameter. I t has a fixed lower bound equal to zéro (unlike other three-parameter distributions such as the Pearson type I I I , log Pearson type I I I and WeibuU) but its lower tail can be flexible enough to be able to give a good f i t to the lower values of flood flows even i f the lower bound of thèse flows is différent from zéro. The fourparameter G G distribution can be obtained from its three-parameter counterpart by adding a location parameter m , i.e., i f 2 £ ' ~ G G ( 3 ) then {X + m)~GG{A). In what foUows, the abbreviation G G w i l l be used to designate the generalized gamma distribution G G ( 3 ) . The G G distribution is very flexible in shape and can take on many possible forms commonly encountered in hydrologie applications. This fact can be verified by taking a look at the représentation of this distribution i n a moment ratio diagram ( M R D ) (Leroux et al. 1987) which shows the relationships between the coefficient of skewness ( Q ) and the coefficient of kurtosis ( Q ) of the distribution. I n such a diagram ( F i g . 1), G G occupies one of the largest régions, thus showing its large shape flexibility compared to other commonly used distributions. I t is also interesting to note that the G G ( 4 ) distribution has many useful distribution as spécial cases like the Pearson type I I I (3-parameter), WeibuU (3- and 2-parameter), gamma and exponential (2- and 1-pararneter) as shown in Fig. 2. Obviously i t also

163

ER(ai)

EX(a)

EV(U6) f=-l..,6:l/5

.=0 I=f.Slll». Intfjff

LKaLil

^1 Pe(a,()

EXPldi)

Figure 2. Relations between the GG(4) distribution and other well known distributions

ER EV EX EX2 Gl G2 GG4 HN LG

with V degrees of freedom Erlang Extrême value type 1 or Gumbel Exponential 1 - parameter Exponential 2 - parameter Gamma 1 - parameter Gamma 2 - parameter Generalized gamma 4 - parameter Half-normal Log-gamma

LN LP N(0,1) P PA RA SHN WE WE3

Log-normal Log-Pearson type 3 Standard normal Pearson type 3 Pareto type 1 (Pearson type 4) Rayleigh Standard half-normal WeibuU 2 - parameter WeibuU 3 - parameter

has G G ( 3 ) and G G ( 2 ) (for which the scale parameter is set cqual to unity) as spécial cases. Morcovcr, Fig. 2 shows that the 3-parametcr log N o r m a l ( L N ) and the N o r m a l ( N ) distributions arc l i m i t i n g distributions of G G ( 4 ) (note that the 2-paramctcr L N is a l i m i t i n g distribution of G G ( 3 ) ) . Also, the Gumbcl distribution bcing a spécial case of L N ( 3 ) (Sangal and Biswas 1970) is also a l i m i t i n g dist r i b u t i o n of G G ( 4 ) . Finally, studics have shown (Paradis and B o b é e 1983) that the G G d i s t r i b u t i o n can bc a powerful tool for studying samples of various degrees of skewness. N o t e that the G G distribution has first been used succcssfuUy in modcUing pollution concentration (Nicholson 1975) and also in the study of lifc distributions (Parr and Webster 1967). Morcovcr, the G G is the most widely used distribution in the Soviet Union for flood frequency analysis ( K r i t s k y and M c n k c l 1969). I n hydrology, some extensive research (Hoshi and Yamaoka 1980, Paradis and Bobée 1983) has been donc i n order to dérive mathematical and statistical properties of the G G distribution, to develop methods for estimating its parameters and to obtain the associated variance of the estimated F-year flood event (Xj-). Investigations concerning estimation o f - t h e parameters of the G G distribution led to severaj methods:

164

indirect method of moments (Stacy and M i h r a m 1965) which uses the first three moments in log space, direct method of moments (Hoshi and Yamaoka 1980), which uses the first three moments in real space, method of m a x i m u m likelihood (Hager and Bain 1970), and four other methods ( M M l , M M 2 , M M 3 , M M 4 ) referred to as methods of " m i x e d moments" (Phien et al. 1987). They are ail based upon the first two moments of X and the first two moments of F = I n Y : they correspond to the four ways of choosing three moments among t h è s e four. We shall propose for the G G distribution a g ê n e r a i method of estimation which we shall call "generalized method of moments" ( G M M ) which combines any three moments of the distribution and we shall show how many of the previously proposed methods of moments are spécial cases of this more g ê n e r a i method. We shall p r é s e n t a spécial case of the G M M which we shall call "sundry averages m e t h o d " ( S A M ) which uses the harmonie, g é o m é t r i e and arithmetic means of the distribution. O u r main aim will therefore be to focus on flexibility by considering: (1)

a very flexible distribution (the G G ) which has the large majority of commonly used distributions as spécial cases; and

(2)

a very flexible method of parameter estimation ( G M M ) which has many of the previously used methods of estimation as spécial cases. I n hydraulic design, one is concerned not only w i t h the estimation of quantiles X J (specified annual flood discharge corresponding to a specified return period of 7-years) but also w i t h the construction of confidence intervais for t h è s e quantiles. N o t e that it is always possible to dérive asymptotic confidence intervais for Xj using the assumption that the estimation Xj is normally distributed w i t h mean Xj and variance o

(1)

where (3 is a scale parameter and both X and 5 are shape parameters. Both P and X must be positive but 5 can be either positive or n é g a t i v e . Non-central moments of the generalized gamma distribution are given by (Stacy 1962):

165

T h è s e moments are defined only i f {X + r/s) > 0, i.e., they exist for ail /• i f J > 0 and only for r > —sX iïs < 0. I n the remainder of the présent study, whenever we shall deai w i t h p.;.(x:), we shall assume that [i-^M exists, i.e., {X T r/s) > 0. Using /• = 1 in Eq. (2), the mean PiCx) of the G G distribution is obtained: riw

=

J 5 a ^ .

Central moments

(3) can be expressed in terms of non central moments p^ using the

following relationship ( K e n d a l l et al. 1987): p,u) =

i:c/p;_/-pi)À

(4)

7=0

Using r = 2 in the obtained: -^^[nX)nX+2/s)

above

-

équation,

the

variance

r\X+\/s)].

of the G G distribution is

(5)

3 Description of the generalized method of moments and its application to the generalized gamma distribution The generalized method of moments ( G M M ) applied to the G G distribution can be described as follows: (1)

For a given sample x ,

drawn from a G G population, define the non

central moment of order r as follows '«3-'w = ^ t ^ r (2)

(6)

Choose any three distinct real numbers t, u and v différent from zéro (the case where one of t h è s e numbers tends zéro w i l l be explicitly considered i n the following section since i t is an important spécial case of this method) and write the following three é q u a t i o n s :

m',{x) =

p,'(x),

m^{x)

p^(x),

=

wj(x)

=

p^(x),

(7)

where terms of the left-hand side of E q . (7) are moments of the sample (Eq. (6)) and those of the right-hand side are the corresponding moments of the population ( E q . (2)); (3)

Solve the above é q u a t i o n s for the parameters P, X and j to give the desired e s t i m â t e s P, X, s. This is done by isoiating the parameter p in the first équation and substituting its value in the last two é q u a t i o n s so that the resulting System of é q u a t i o n s becomes:

gi{X,\/s)

= rnX+u/s)

-

ml{x)nX)

= 0

g2{X,l/s)

= ^TiX+v/s)

-

mXx)r{X)

= 0

(8>

166

P=

nx

+ tis)

•

This later System is then solved for X and J using the Newton-Raphson method where partial derivatives needed for applying this method are:

ôg,/a>. = p"ra-t-«/j)fv(?.+M/j)+(u/o[v(/W-v(>v+//U]}-'7i;(x)raM>.) (9) dgjd(\/s)

= u^''Y{X+u/s){^i/(X+u/s)

dg^ldX = ^'Y{X+v/s){^{X+v/s) dg2/d{\/s)

-

M/{X+t/s)]

(10)

+ {vIt)[V(/L)-Mf{X+tls)]}-ml(x)Y{X)Mf{X)

= vp''r(>t+v/5){Hf(A+v/j) -

( 11 )

v(^+r/5)}

(12)

where \)/( ) is the digamma function defined as r'( )/r( ) where r'( ) is the derivative of r(-). The generalized method of moments which uses the System of Eq. (7) w i l l be denoted by G M M {t,u,v).

4 A spécial case of the G M M : the sundry averages method ( S A M ) I t is obvious that putting r = 0 in Eqs. (2) and (6) give Po(^) ' m^ix) = 1, respectively (independently of the parameters p, À and s) since \Iq corresponds to the area under the density curve. Therefore, the moment of order zéro is not interesting because i t cannot be used for estimating the parameters of a distribution. I t is interesting to note, however, that when r tends to zéro the é q u a tion: mXx)

= p;(x)

(13)

becomes équivalent to (sec Ashkar and Bobée 1986): gx = G, where. g^ is the g é o m é t r i e mean of the sample and

(14) is the g é o m é t r i e mean of the

population. L e t y be the arithmetic mean of the sample of logarithmic values y,- = Inx; (which is the log of the géométrie mean of the observed values x , ) and PiCy) be the population mean of the variable Y = InA". T a k i n g natural logs on both sides of Eq. (14) i t can be shown that this é q u a t i o n is é q u i v a l e n t to: y =

niU)

(15)

where \ii{y) csn be easily calculated using i n t é g r a t i o n of E q . ( 1 ) (sec also Stacy and M i h r a m 1965) to give: Pi'Cy) = Inp + ^v{X)/s. W e shall therefore refer to this case of r tending to zéro as the case of moment of order "quasi z é r o " . I n the remainder of the paper we shall use the simplified notation " r = 0" to mean " r 0" and shall define W o ( x ) and Po(^) follows: WQ) = y = H}r\Xi)/N

(16)

167

P o U ) = PY) = Inp + v W / j .

(17)

I t is seen from Eq. (17) that Po(x) does not have the same functional form as \i.rix) in E q . (2). This remark w i l l later be used for caiculaling the variance of Xj

for

certain interesting spécial cases of the G M M . Two of thèse interesting spécial cases are: the method M M l introduced by Rao (1980) for the L P distribution and used by Phien et al. (1987) for the G G distribution. This method can be denoted by G M M ( 0 , 1 , 2 ) because it uses the moment of order 0 ( g é o m é t r i e mean), 1 (arithmetic mean) and 2 (variance) of the distribution. I t can be easily shown that, when the moment of order 1 is part of the System of équations, there is é q u i v a l e n c e between using the non-central moment of order 2 or the cent r a l moment of order 2 (variance). Therefore in both cases it is correct to d é n o t e the method M M l by G M M ( 0 , 1 , 2 ) ; a method w h i c h we shall call "sundry averages method" ( S A M ) which can be denoted by G M M ( - 1 , 0 , 1 ) because i t uses moments of order -1 (harmonie mean), 0 ( g é o m é t r i e mean) and 1 (arithmetic mean) of the distribution (and thus the name "Sundry Averages M e t h o d " ) . 5

Estimation of Xj

and calculation of its variance

The estimation Xj- of the flood magnitude w i t h return period T under a G G distribution is obtained from: Wj

=

i.e. Xr

where

Wj

-

^W^'

(18)

is the estimator of the event

Wj

of return period T under a one-

parameter gamma distribution w i t h p.d.f.: =

v 1r n(A)

'

^ > 0' ^ > 0-

(19)

I n practice, Wj is calculated from: Wj. = KjVl

where Kj

+X

= 2 ^

+ -4t

(since C, =

2/Vl)

(20)

is a frequency factor (standardized Pearson type 3 variate) which has

been extcnsively tabulated as a function of the estimated coefficient of skewness Q of the random variable W (Harter 1969). I n s u b s é q u e n t calculations we shall drop the symbol " — " , for simplicity. I f X follows a G G distribution, then obviously Xj is a function of the three parameters of the distribution, and for large sample sizes we have: ÔXT

VtsrXj

ÔXT

ÔXT

= (^)'VarP + (^)'VarÀ + dXj

dXj

dXj

(^)'Var^ dXj

168 (21) Using X J =

Ç

the partial derivatives in Eq. (21) are obtained as follows; (22)

dP

^ (23)

(24) and by E q . (20),

is given by: 1 (25)

1 + where

is obtainable from the polynomial

given by Bobée and

Boucher

(1981) or f r o m the formula given by Hoshi and Burges (1981). I t is obvious that the parameter variances and covariances involved in Eq. (21) are functions of the variances and covariances of the three moments p , ( x ) , p û ( ^ ) and Pv(^) I h ^ t

employed in the estimation (Eq. (7)).

Using matrix notation,

t h è s e variances and covariances of the parameters are given by: Vp =

K - ' - Vn,

(26)

I.e., -1 Varp

^16"

"VarMi

Vari^

^^21

.

.

^-26

VarMj

Varj

1^31

•

•

^^36

VarMj

Cov(p,3,)

1^41

•

•

^-46

C o v ( A / i , A/2)

1^51

•

Cov

Cov(A/,, A/3) V66

.^61 where A / j =

(27)

Cov(A/2 A/3)

p , ( x ) ; A/2 = p „ ( x ) ; A/3

liv(^)-

The m a t r i x is given by:

2,^11,413

24,3

Ah

Ah2

Ah

2,4,,^12

Ah

Ah

Ah

2A2\A22/121.423

2422423

Ah

Ah

Ah

24j,432

2432433

4 4 4 (4, + 4, (4, + 4 (4,+4, 4,

4 4 (4, (4,

,432 + where

(28)

169

(29)

1,2,3 and = p, 6 3 =

The terms are therefore obtained from Eq. (2) by partial differentiation and are given in the following matrix:

(31) ^33.

.^31 where ,-ir{x 41,1

nx) '13

^nx) r

nx)

(32)

^31

vp

^^^^

,432

^^^^^^[V(^+v/.)-V(X)]

^33 = - = ^ [ r ( ^ + v / 5 ) v ) r ( X + v / i ) ] . ^T(X) The entries of the vector i n E q . (26) are easily obtained i f we recall from Kendall et al. (1987) that for arbitrary and and arbitrary sample size we have: V a r ( p ; ) = (p2, -

p;2)/7V

Cov(p;, p,) = ( p , + , -

p;p,)/7V.

(33) (34)

Since the c l é m e n t s of the vector (Eq. (26)) and those of the matrix ( E q . (28)) are now well determined, the vector can be easily deduced using p

m

Hence by knowing the variances and covariances of the parameters, and (i.e., the vector and the partial derivatives of w i t h respect to each of t h è s e parameters (Eqs. (22), (23), and (24)), the calculation of Var becomes straightforward via Eq. (21) and can be easily programmed on a computer. One can also be interested i n Computing corrélation coefficients between moments or between parameters. This can be done by using the following relationship: cov(x,>')/(varxvar>')'7^ and letting, for example, x = M , and >> = A f , in the moments case or, x = p and the parameter case.

170

6

Spécial case of moment of order quasi zéro

In the p r é s e n t a t i o n of the sundry averages method ( S A M ) it was mentioned that the moment of order quasi zéro,Q{X), Prix)-

has a functional form différent from that of

This means that certain terms involved i n the calculation of V a r A j would

have to be modified whenever the moment Po(x) is used in the estimation. Such is the case not only when the method S A M ( G M M ( 0 , - 1 , 1 ) ) is used, but also whenever the

moment

of order

"quasi-zero"

is involved

(like i n the method

M M l or

G M M ( 0 , 1 , 2 ) for example). Terms to be modified are a i l those involving t = 0, or A / | = Pr(jc) = Po(^)- T h è s e precisely are: the terms V a r M , , C o v ( M , , M j ) and C o v ( M , , M3) of the vector in E q . (26); ail terms i n the first row of the matrix in E q . (31). The new terms w i l l be (see appendix):

VarAf, =

(36)

where \|;'( ) is the trigamma function Cov(M„

P"^^^^^"^'^

Cov(A/„

A/3)

(37)

= P'^^^^^^^^^

v/s) -

1/P 41.2 = 41.3 =

(40) - ¥ U ) A '

A comparison of WarXj

(41)

calculated by différent versions o f the G M M has been

undertaken and the results are presented in the following section. 7

Comparison of WarX-p obtained by various versions of the G M M

The G M M is a method of estimation which is very broad i n the sensé that i t allûws one to combine orders of moments in such a way that the chosen combination performs better than others with respect to reducing \arXj-. N o t e however that one should not expect to find one method of estimation to be best for a i l return periods. T h e example provided here consists of the comparison between the following methods: G M M ( - 1 , 0, 1) or S A M , G M M (0, 1, 2) or M M l and G M M ( 1, 2, 3) T h e three différent methods were applied to a set o f observations made o f annual m a x i m u m daily discharge recorded from 1916 to 1974 at the station located on St-Mary's river (at Stillwater) in Nova Scotia (Canada). This set o f data was also used by K i t e (1977). Results are presented i n Fig. 3 where the standard error o f Xj is given for différent values of the return period T. Each curve is associated w i t h one of the three methods of estimation. For this set of observations. Figure 3 indicates that lowest values for standard error o f Xj are obtained by the sundry averages method or G M M ( - 1 , 0, 1) indicating t h a t this method should probably be preferred over the two others in this particular case. Here, i t appears that making use of moments of lower order ( - 1 ,

171 800

0.0001

0 001

0.01

0 05

0 2

P R O B A B I L I T Y

Figure 3. Standard error for ties of exceedance

0 5 O F

0 8

0 95

0 99

0.999 0.9999

E X C E E D A N C E

obtained by différent versions of the G M M for various probabili-

quasi-zero and 1 as compared to 1, 2 and 3 for example) helps in reducing the standard error of Xj for ail return periods considered. Further investigations would be needed i n order to confirm this preliminary resuit and to examine i n more détail (by M o n t e Carlo simulation for example) how the différent versions of the method of moments would compare to each other when applied to the G G distribution. 8

Summary and conclusion

The m a i n purpose of the p r é s e n t study was to apply a very flexible method of parameter estimation ( G M M ) to a very flexible distribution ( G G ) . This was done by: (1)

pointing ont the interest of the generalized gamma distribution as a flexible distribution for hydrological applications and showing in a schematical diagram how many of the widely used distributions are either spécial cases or l i m i t i n g cases of the G G distribution;

(2)

showing how the generalized method of moments ( G M M ) , can be considered as a generalization of the various methods of moments;

(3)

deriving the g ê n e r a i é q u a t i o n s needed to apply the G M M to the G G distribution;

(4)

presenting an interesting spécial case of the G M M which was called the sundry averages method ( S A M ) . This method makes use of the arithmetic, g é o m é t r i e and harmonie means of the sample, which are respectively moments . of order 1, quasi zéro and - 1 . Reducing the order of the moments in this manner can lead to a lower value of VaxiXj) for certain values of T\

172

(5)

determining the g ê n e r a i formula for VarXj

for the G M M applied to the G G

distribution inciuding the spécial case of moment of order quasi zéro; (6)

providing an exampie of the use of the G M M for annual flood séries and pointing eut the need for further comparison between the différent versions of the G M M .

Acknowiedgements This study was made possible by grants from tbc Natural Sciences and Engineering Research Council of Canada (operating Grants A-8399 and A-5797 and Stratégie Grant G-1617).

Références Ashkar, F . ; Bobée, B. 1986; Comment on: Variance of the T-year event in the Log-Pearson type 3 distribution. J . of Hydrology 84, 181-187 Ashkar, F . ; Bobée, B. 1988: Confidence intervais for extrême flood events under a Pearson type 3 or log Pearson type 3 distribution. Water Resour. Bulletin (accepted for publication) Benson, M.A. 1968: Uniform flood frequency estimating methods for fédéral agencies. Water Resour. Res. 4, 891-908 Bobée, B.; Boucher, P. 1981: Calcul de la variance d'un événement de période de retour T: cas des lois log-Pearson type 3 et log-gamma ajustés par la méthode des moments sur la série des valeurs observées. I N R S - E a u , Rapp. Sci. 135, 17 pp Hager, H.W.; Bain, L . J . 1970: Inferential procédures for the generalized gamma distribution. J . Amer. Statist. Assoc. 65, 1601-1609 Harter, H . L . 1969: A new table of percentage points of the Pearson type 3 distribution. Technometrics 11, 177-187 Hoshi, K . ; Burges, S.J. 1981: Approximate estimation of the derivative of a standard gamma quantile for use in confidence interval estimâtes. J . of Hydrology 53, 317-325 Hoshi, K . ; 'Yamaoka, I. 1980: The generalized gamma probability distribution and its application in hydrology. Hydraulics papers, 7. The research laboratory of civil and environmental engineering, Hokkaido University, Japan, 256 pp Kendall, M.J.; Stuart, A.; Ord, J . K . 1987: Kendall's advanced thcory of statistics. Vol. 1. 5th éd., Oxford University Press, New York, 604 pp Kite, G.W. 1977: Frequency and risk analysis in hydrology. Water Resour. Publications, Colorado: Fort CoUins, 224 pp Kritsky, S.N.; Menkel, M . F . 1969: On principles of estimation methods of maximum discharge. Floods and their computation. A.I.H.S. 84 Leroux, D; Bobée, B.; Ashkar, F. 1987: Diagrammes des relations entre certaines fonctions de moments pour quelques lois fréquemment utilisées en hydrologie. I N R S - E a u , Rapp. Sci. 237 Nicholson, D . L . 1975: Application of parameter estimation and hypothesis test for a generalized gamma distribution. Proc. I E E E conf. decis. control incl. symp. adapt. processes. Houston, Tex., Dec. 10-12, 321-326 Paradis, M.; Bobée, B. 1983: L a distribution Gamma Généralisée et son application en hydrologie. I N R S - E a u , Rapp. Sci. 156, 52 pp Parr, V.B.; Webster, J . T . 1965: A method for discriminating between failure density functions used in reliability prédictions. Technometrics 7, 1-10 Phien, H.N.; Nguyen, V.T.V.; Jung-Hua. K. 1987: Estimating the parameters of the generalized gamma distribution by mixed moments. In: V.P. Singh (éd.). Hydrologie Frequency Modeiing. D. Reidel Publ. Co., Dordrecht, HoUand Rao, D.V. 1980: Log Pearson type 3 distribution: Method of mixed moments. Soc. Civ. Eng. 106, 999-1019

J . Hydraul. Div. Am.

Sangal, B.P.; Biswas, A . K . 1970: The 3-parameter log normal distribution and its applications in hydrology. Water Resour. Res. 2. 505-515

173 Stacy, E . W . 1962; A generalization of the gamma distribution. Ann. of Math. Stat. 33, 1187-1 192 Stacy, E.W.; Mihram, G . A . 1965: Parameter estimation for a generalized gamma distribution. Technometrics 7, 349-358

Appendix A

Dérivation of équations (36) through (41)

In Eqs. (36) through (41) we have: A/, = F estimate of p-oG) = PiCv) = '"P + V(X)/s A/j = m„(x) estimate of p „ ( x ) = E{X") = P"r(X + u / s ) / r ( X ) A mlix) estimate of p^x) = E^X") = ^T{X + v/s{X) From the expression for A/,, Eqs. (39), (40) and (41) are easily dcrived; in fact we have: Au = âA/,/ôP = ô(lnp + v(X)/s

1/p,

4,2 = aA/,/ôX = a(lnp + \\i(X)/s)/dX

Y(X)/s,

=

Au = dM^/ds = ô(lnP + \\t(,X)/s)/ds

Y(X)/s.

Equation (36) is obtained as follows: Since A ~ G G (P, X ,s), let F = InA and Z = l n ( A / P ) ' i.e., InA = (Z/s) Then:

+ Inp.

VarA/, = V a r F = ( l / A ) V a r F where V a r F = Var(lnA) = V a r [ ( Z A ) + Inp] = ( l / i 2 ) V a r Z = (l/j^)!£(Z^) -

[£(Z)]^1.

From Stacy and Mihram (1965),

where U^^X) is the Xth derivative of r(X) with respect to the parameter X. Then after some manipulations we obtain: VarF = ( i A

-

[^Ç^l^l =

(lAVW.

Finally, VarA/, =

Y(X)/Ns\

Equation (37) is derived using A/, and A as follows: Cov(A/|, A/j) = E[r

mi(x)]

-

E{r)

E{mUx))

= £ [ ( 1 / A ) 5 : F ) ((1/A)2:A/')] =

(1/A2)[££(F,A,'') +

=

(1/A^)[A£(FA'') +

(A^

=

(1/A)£(FA") +

-

= (1/A)[£(FA") -

(1

(A'

-

£ ( F ) £(A") A ) £ ( F ) £(A'")] -

A)£(F)-£(A")]

1/A)£(F) £(A") -

£ ( F )

£(A'')]

-

£ ( F )

£(A")

£ ( F )£(A")

E(Y)

E(X")

(Al)

where: £ 0 0 = PiCx)

(A2)

£ ( A " ) = ti:(x)

(A3)

and

174 £ ( F A " ) = £ ( l n A A") = J x N n x / G ) dx

• (A4)

where / ( x ) is the probability density function of the generalized gamma distribution (Eq. (1)) and the intégration is over the whole domain of variation of this distribution. Making the appropriate change of variable and after intégration and some mathematical manipulations E q . (A4) can be shown to be equal to: E{X"Y)

=

^"^^r(X)"^^^^'"'^ ^ "^^^

Substituting Eqs. (A5), (A3) and (A2) into Eq. ( A l ) gives the desired E q . (37). Equation (38) is derived in an exactiy similar manner.

Accepted May 20, 1988

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