Estimation of the GEV distribution from censored samples by method of partial probability weighted moments

Journal of Hydrology, 120 (1990) 103-114 Elsevier Science Publishers B.V., A m s t e r d a m

103 --

Printed in The Netherlands

[4]

E S T I M A T I O N OF THE G E V D I S T R I B U T I O N FROM C E N S O R E D S A M P L E S BY METHOD OF P A R T I A L PROBABILITY W E I G H T E D MOMENTS

Q.J. WANG Department of Engineering Hydrology, University College Galway (Ireland) (Received November 17, 1989; accepted for publication February 1, 1990)

ABSTRACT Wang, Q.J., 1990. Estimation of the GEV distribution from censored samples by method of partial probability weighted moments. J. Hydrol., 120: 103-114. The concept of partial probability weighted moments (PPWM), which can be used to estimate a distribution from censored samples, is introduced. Unbiased estimators of PPWM are derived. An application is made to estimating parameters and quantiles of the generalized extreme value (GEV) distribution from censored samples. Censored samples yield high quantile estimates which are almost as efficient as those obtained from uncensored samples. This could be a very useful technique for dealing with the undesirable effects of low outliers which occur in semiarid and arid zones.

INTRODUCTION An a n n u a l m a x i m u m series of floods o v e r a period is often t r e a t e d as a r a n d o m s a m p l e f r o m a p r i o r specified h o m o g e n e o u s p r o b a b i l i t y d i s t r i b u t i o n s u c h as t h e g e n e r a l i z e d e x t r e m e v a l u e (GEV) d i s t r i b u t i o n . T h e p u r p o s e of a n a l y z i n g a n n u a l m a x i m u m series of floods is, in m o s t cases, to p r e d i c t m a g n i t u d e of flood of r e l a t i v e l y l a r g e r e t u r n p e r i o d s u c h as 100 years. As s m a l l floods a r e of little r e l e v a n c e to t h e l a r g e r ones, i n c l u s i o n of d a t a on small floods in e s t i m a t i n g h i g h r e t u r n p e r i o d floods c a n s o m e t i m e s be o n l y of n u i s a n c e value. T h i s is e s p e c i a l l y t r u e w h e n a n a l y z i n g floods of a r i d or s e m i a r i d r e g i o n s w h e r e m a n y v e r y low (or zero) floods occur. C u n n a n e (1987, pp. 60-61) s u g g e s t e d t h a t in s u c h c a s e s a c e n s o r e d s a m p l e s h o u l d be used a n d t h e a n a l y s i s be b a s e d on o n l y t h o s e floods w h o s e m a g n i t u d e s h a v e e x c e e d e d a c e r t a i n t h r e s h o l d x0. H e p o i n t e d o u t t h a t " C e n s o r i n g f r o m b e l o w as a m a t t e r of c o u r s e m i g h t be a d v a n t a g e o u s . I t m a y be t h a t s m a l l e r s a m p l e v a l u e s h a v e o n l y a n u i s a n c e v a l u e in t h e c o n t e x t of u p p e r q u a n t i l e e s t i m a t i o n a n d also in m o d e l f o r m t e s t i n g a n d v e r i f i c a t i o n . " M e t h o d s of m a x i m u m l i k e l i h o o d (ML) e s t i m a t i o n of t h e G E V d i s t r i b u t i o n f r o m c e n s o r e d s a m p l e s h a v e b e e n p u b l i s h e d ( P r e s c o t t a n d Walden, 1983; P h i e n a n d F a n g , 1989). T h e y are, h o w e v e r , n o t c o m p l e t e l y s a t i s f a c t o r y . T h e y

0022-1694/90/$03.50

© 1990 Elsevier Science Publishers B.V.

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Q.J. WANG

sometimes b r e a k down and fail to c o n v e r g e even w h e n the samples h a v e come from a true GEV distribution. The m e t h o d of probability weighted m o m e n t (PWM) e s t i m a t i o n of the GEV d i s t r i b u t i o n from a complete sample is found to possess s m a l l e r bias and b e t t e r efficiency t h a n M L e s t i m a t i o n w h e n sample size is small (Hosking et al., 1985). T h e m e t h o d of P W M is also c o m p u t a t i o n a l l y m u c h simpler and more r o b u s t t h a n t h a t o f ML. In this paper, application of the c o n c e p t of P W M to estimating the GEV distribution from censored samples is examined. PROBABILITY WEIGHTED MOMENTS, PARTIAL PROBABILITY WEIGHTED MOMENTS AND THEIR UNBIASED ESTIMATORS P r o b a b i l i t y weighted m o m e n t s are a g e n e r a l i z a t i o n of the usual m o m e n t s of a probability d i s t r i b u t i o n ( G r e e n w o o d et al., 1979). T h e probability weighted m o m e n t s of a r a n d o m v a r i a b l e X with d i s t r i b u t i o n f u n c t i o n F ( x ) = P ( X <. x) are the q u a n t i t i e s 1

Ms

=

| [x(F)]PF~(1 - F ) ' d F

(1)

,2

0

w h e r e p , r and s are real numbers. W h e n p = i and s = 0, the m o m e n t s become l

[~r

= Ml.r. o = [ x ( F ) F r d F d 0

(2)

Given a r a n d o m sample of size n from the distribution F, e s t i m a t i o n of fir is most c o n v e n i e n t l y based on the o r d e r e d sample, x(,)~< x(2)~< ... < x(n). The statistic br

1

(i - 1)(i - 2)...(i - r) 2, (n 1)(n -~ ~.. -(n - - r) x(') ni=l

(3)

is a n unbiased e s t i m a t o r of fir ( L a n d w e h r et al., 1979). P W M so defined c a n only be used for a complete sample. The c o n c e p t of PWM, however, can be easily e x t e n d e d so as to be applied to a c e n s o r e d sample. I n s t e a d of i n t e g r a t i n g F from zero to one as shown in eqn. (1), we i n t e g r a t e F from a lower b o u n d F0 to one as 1

M~ .... = ~o Ix(F)]p Fr(1 - F)" d F

(4)

w h e r e F 0 = F(x0), x0 being the c e n s o r i n g threshold. We call M'p.rz p a r t i a l probability weighted m o m e n t s (PPWM). Similar q u a n t i t i e s can be defined for u p p e r b o u n d c e n s o r i n g and for double b o u n d censoring. W h e n p = 1 and s = 0, the P P W M become 1

fl'~ = M'~,r,o = [ x(F)F~ d F

(5)

105

ESTIMATION OF GEV DISTRIBUTION

Given a complete sample x(,) ~< x(~) ~< ... ~< x(.), the following statistic is an unbiased estimator of fl'~ :

1 b'~-

(i-

n i ~ZL l (n

1)(i - 2)...(i - r) . 1)(n 2-)Y.('n --- r) x (')

(6)

where 0 x(i ) <~ x o X(i) X(i) > XO

X~i ) =

(for proof, see Appendix). W h e n F0 = 0, fir and b'r become fir and br respectively. PPWM ESTIMATION OF THE GEV DISTRIBUTION FROM CENSORED SAMPLES The generalized extreme value distribution combines into a single form the three possible types of limiting distribution for extreme values. The distribution f u n c t i o n is 1

k # = exp

{ [1 - exp

- ~ ( x - ~)

]}

0

k = 0

(7)

with x bounded by ~ + (a/k) from above if k > 0 and from below if k < 0. In flood frequency analysis, k is u s u a l l y in the r a n g e of ( - 0.4, 0.4). The inverse distribution function is x(F)

=

a

~ + ~ [ 1 - ( - l o g / O k]

= ~ - alog(-

k ~

logF)

0

k = 0

(8)

The P P W M of the GEV distribution for k # 0 are given by = ( ~k +)

1~

(1 - F~ + ,)

~a F(1 . +k + (k ) r ~-)iTkY(l + k, - (r + 1) log Fo) (9)

where P(., .) is an Incomplete G a m m a function, - (r + 1) l°gF0

P(1 + k, - (r + 1) logF0)

=

f0

0d 0 - k e - ° - -

r(1 + k)

(10)

(for proof, see Appendix). F r o m eqn. (9), we have fl~) = @ + k ) ( 1 -

Fo)-

kF(1 + k)P(1 + k, - log F0)

(11)

106

Q.J. WANG

2fl; 1 - F02

fl~ _ 1 - Fo

~ [ P ( 1 + k , - 2 log Fo) P ( I + k , - log F0).] k 2h(1 - F~) 1 2_ Fo (12)

and 2fl; 1 -

fl~

F~

1 -

F0

=

fl~

3fl~

1-F3

P(1 + k, - 2 log F0) 2k(1 - F 02)

P(1 + k, - log 1%) 1 - F0

P(1

P(1

1-Fo

+

k,

-

log F0)

3

3'(1-F3)

+

k,

-

(13)

log F0)

1-Fo

W h e n F0 is known, we can replace fl: by b'~ and estimate p a r a m e t e r s ~, ~ and k as the solutions of (11), (12) and (13). T h e exact solution of (13) requires i t e r a t i v e m e t h o d s which are cumbersome. T h e following simple m e t h o d c a n be used instead. Let z equal the r i g h t - h a n d side of (13), t h a t is

z =

P(1 + k, - 2 log Fo) 2k(1 - F~)

P(1 + k, - l o g

P(1 + k, - 3 log Fo) 3h(1 - F3)

P(1 + k, - log F0)

1 -

]

Fo)

F o

(14)

Fo

W h e n z is plotted vs. k for a fixed Fo, the c u r v e is v e r y smooth, as shown in Fig. 1.

2.4

2.2 Fo = 0.5

Fo

=

0.2

1,4

1.2

i

-0.4

-0'.2

I (~

I

0.2

I

0.4

Fig. 1. Plot of z vs. k under different thresholds Fn.

107

ESTIMATION OF GEV DISTRIBUTION

The exact location of the curve changes with F0 value. The curve can be a c c u r a t e l y approximated by a q u a d r a t i c function of form k

=

ao + a l z

(15)

+ a2 z2

For fixed F0, three z values can be calculated by eqn. (14) corresponding to three chosen k values, e.g. k = - 0.4, - 0.01 and + 0.4, avoiding use of k = 0 as the limiting form of eqn. (14) would be required. S u b s t i t u t i n g them into eqn. (15), we have a set of linear equations. We can t h e n find the solutions for a0, al and a2 corresponding to t h a t F 0. W h e n z is replaced by its sample estimate,

=

2b'1

b~

1 -F02 3b~

1-F 0 b~

l-F0

l-F0

(16)

and substituted into eqn. (15), we can find the estimate for k. The other two parameters can t h e n be estimated successively from eqns. (11) and (12) as 2b] =

k 1 - F02 F(1 + /~)P(1 + ~, - 2 log F0) 2~(1 - F02) 1-Fob~ + ~

IF(l+ k')P(I+

b~ 1 - Fo P(1 + ~, - log F0) 1 - F0

1-F0~'-log F0) - 1]

(17)

(18)

For a specified x 0, we do n o t k n o w the exact value o f f 0 from a sample. Instead we can use an empirical frequency w h e n e v a l u a t i n g eqns. (14)-(18) f0

= no n

(19)

where no is the n u m b e r of occurrences of values which do n o t exceed x0 in the sample. As shown in Fig. 1, the slope of the curve reduces rapidly w h e n the censoring threshold increases. W h e n F 0 = 0.5, the curve has very small slope. This is n o t surprising because as we exclude larger parts of the GEV distribution, the shape becomes less defined. Thus, the censoring threshold should not be u n d u l y h i g h for p a r a m e t e r estimation. A more general form of P P W M w i t h double bound censoring, of which lower bound censoring and upper bound censoring are special cases, and their unbiased estimators are outlined in the Appendix. The a n a l y t i c a l expression of P P W M w i t h double bound censoring for the GEV distribution is also given there.

108

QJ WANG

TABLE 1 Bias, standard error (SE) and root mean square error (RMSE) of PPWM estimates of parameters and quantiles of the GEV distribution from censored samples

Fo 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE

~

~

k

~oo/x~,

~/x2oo

0.013 0.213 0.214 0.007 0.212 0.212 - 0.008 0.228 0.228 0.064 0.308 0.315 - 0.274 1.714 1.736 1.502 32.493 32.528 - 1496 104432 104443

0.000 0.189 0.189 0.003 0.251 0.251 0.039 0.328 0.330 0.143 0.473 0.494 0.495 6.186 6.206 4.002 184.495 184.539 12014 856582 856666

0.031 0.163 0.166 0.029 0.190 0.192 0.043 0.216 0.220 0.080 0.253 0.265 0.161 0.319 0.358 0.327 0.455 0.560 0.674 0.749 1.008

0.008 0.433 0.433 0.012 0.436 0.436 0.004 0.437 0.437 - 0.015 0.439 0.439 0.042 0.445 0.447 0.077 0.454 0.460 0.113 0.465 0.479

0.031 0.530 0.531 0.042 0.543 0.544 0.033 0.548 0.549 0.007 0.551 0.551 0.037 0.555 0.556 0.096 0.558 0.566 0.164 0.558 0.582

Parent distribution parameter value: ~ = 0.0, ~ = 1.0 and k = -0.2. Sample size: n = 30. Censoring level: F 0.

PROPERTIES OF PPWM ESTIMATION OF THE GEV DISTRIBUTION FROM CENSORED SAMPLES Monte Carlo experiments have been performed to investigate the properties o f P P W M e s t i m a t i o n o f t h e G E V d i s t r i b u t i o n f r o m c e n s o r e d s a m p l e s . As a n a n n u a l m a x i m u m f l o o d s e r i e s is u s u a l l y s h o r t , size n - 30 is u s e d i n t h e e x p e r i m e n t s . T h r e e s h a p e p a r a m e t e r v a l u e s , k = - 0 . 2 , 0.0 a n d 0.2, a r e considered. Location and scale parameters ~- 0 and a = 1 are used throughout. Different levels of censoring threshold are considered, namely, F0 = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5 a n d 0.6. W h e n F0 - 0, P P W M b e c o m e t h e o r d i n a r y P W M . T h e n u m b e r o f r e p l i c a t i o n s u s e d i n t h e s i m u l a t i o n f o r e a c h c a s e is 10 000. The properties investigated are bias, standard error and root mean square e r r o r o f t h e e s t i m a t e s o f t h r e e p a r a m e t e r s a n d t w o q u a n t i l e s o f 100- a n d 200year return periods. T a b l e s 1-3 s u m m a r i z e t h e r e s u l t s o f t h e e x p e r i m e n t s . T w o o b s e r v a t i o n s c a n b e m a d e f r o m t h e m . (1) A s c e n s o r i n g t h r e s h o l d i n c r e a s e s , e s t i m a t e s o f p a r a m e t e r s d e t e r i o r a t e v e r y r a p i d l y . (2) D e s p i t e t h e d e t e r i o r a t i o n i n p a r a m e t e r

109

ESTIMATION OF G E V DISTRIBUTION

TABLE 2

Bias, standard error (SE) and root mean square error (RMSE) of PPWM estimates of parameters and quantiles of the GEV distribution from censored samples

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE

0.011 0.208 0.208 0.004 0.208 0.208 - 0.007 0.222 0.222 - 0.063 0.303 0.309 - 0.291 2.194 2.213 - 2.228 63.686 63.725 - 185 8767 8769

- 0.003 0.161 0.161 - 0.006 0.223 0.223 0.025 0.301 0.302 0.135 0.476 0.494 0.573 8.902 8.921 7.837 400.383 400.459 1226 64211 64223

0.017 0.146 0.147 0.009 0.177 0.177 0.021 0.206 0.207 0.064 0.250 0.258 0.168 0.324 0.365 0.395 0.482 0.623 0.884 0.818 1.205

0.012 0.305 0.305 0.026 0.315 0.316 0.020 0.318 0.319 0.000 0.320 0.320 - 0.038 0.322 0.324 - 0.090 0.319 0.332 - 0.147 0.307 0.341

0.026 0.357 0.358 0.049 0.378 0.381 0.044 0.387 0.390 0.016 0.391 0.392 - 0.040 0.389 0.391 - 0.120 0.374 0.393 - 0.208 0.340 0.398

Parent distribution parameter value: ~ = 0.0, ~ = 1.0 and k = 0.0. Sample size: n = 30. Censoring level: F 0. e s t i m a t e s a s c e n s o r i n g t h r e s h o l d b e c o m e s h i g h , t h e t w o q u a n t i l e e s t i m a t e s , xl00 and ~, have only slight increments in standard error and root mean square e r r o r c o m p a r e d w i t h e s t i m a t e s f r o m u n c e n s o r e d s a m p l e s . T h e b i a s is a l s o s m a l l within a large range of censoring threshold. T h e s e c o n d o b s e r v a t i o n shows t h a t m o d e r a t e l y h i g h t h r e s h o l d s c a n be used i f o u r i n t e r e s t is m a i n l y i n e s t i m a t i o n o f h i g h q u a n t i l e s . T h i s is e s p e c i a l l y u s e f u l if we suspect t h a t the l o w e r a n n u a l m a x i m u m v a l u e s do not c o n f o r m to the s a m e d i s t r i b u t i o n as t h e m e d i u m a n d u p p e r v a l u e s , as f o r i n s t a n c e w h e n l o w o u t l i e r s a p p e a r in the sample, w h i c h can h a p p e n f r e q u e n t l y in arid zones. SUMMARY Partial probability weighted moments, which are extensions of the usual p r o b a b i l i t y w e i g h t e d m o m e n t s , h a v e b e e n i n t r o d u c e d . P P W M c a n be used to estimate a distribution from censored samples. Unbiased estimators of PPWM have been formulated. Estimators of parameters and quantiles of the GEV distribution from censored samples have been derived using the method of PPWM.

110

QJ. WANG

TABLE 3 Bias, standard error (SE) and root mean square error (RMSE) of PPWM estimates of parameters and quantiles of the GEV distribution from censored samples Fo

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE Bias SE RMSE

2

~

0.005 0.206 0.206 - 0.003 0.208 0.208 0.013 0.222 0.222 - 0.066 0.304 0.311 - 0.328 2.990 3.008 4.133 144.290 144.349 -6.81 x 104 5.44 × 106 5.44 × 106

- 0.005 0.151 0.151 - 0.013 0.210 0.210 0.016 0.291 0.291 0.140 0.477 0.497 0.730 13.513 13.533 19.286 996.348 996.535 7.23 × 10~ 5.87 x 107 5.87 × 107

xl~/xlo~,

k 0.007 0.142 0.142 0.010 0.174 0.175 0.002 0.206 0.206 0.059 0.251 0.258 0.202 0.338 0.393 0.514 0.523 0.733 1.185 0.913 1.496

0.016 0.219 0.220 0.038 0.229 0.232 0.032 0.231 0.234 0.004 0.228 0.228 ().046 0.222 0.226 0.108 0.208 ().234 0.162 (].184 0.245

:~Ix~

0.026 0.251 0.253 0.057 0.271 0.277 0.053 0.278 0.283 0.015 0.274 0.274 0.053 0.259 0.264 0.140 0.230 0.269 0.214 0.188 0.285

Parent distribution parameter value: ~ = 0.0, a = 1.0 and k = 0.2. Sample size: n = 30. Censoring level: F(, Monte Carlo experiments have been conducted to examine the properties of estimation of the GEV distribution from censored samples by the method of PPWM. The results show that moderately high lower bound censoring threshold can be used with only slight increments in standard error and mean square error for high quantile estimation.

ACKNOWLEDGEMENT T h e a u t h o r t h a n k s D r . C. C u n n a n e f o r h i s s u p e r v i s i o n , a n d P r o f . J , E . N a s h , D r . K . M . O ' C o n n o r a n d D r . R. K a c h r o o f o r t h e i r g e n e r a l s u p p o r t . T h e a u t h o r also gratefully acknowledges the financial assistance of the Irish Government during

his stay in Ireland.

REFERENCES Cunnane, C., 1987. Review of statistical models for flood frequency estimation. In: V.P. Singh (Editor), Hydrological Frequency Modelling. D. Reidel, Dordrecht, pp. 49-95. Greenwood, J.A., Landwehr, J.M., Matalas, N.C. and Wallis, J;R. 1979. Probability weighted

111

ESTIMATION OF G E V DISTRIBUTION

moments: definition and relation to parameters of distribution expressible in inverse form. Water Resour. Res., 15(5): 1049-1054. Hosking, J.R.M., Wallis, J.R. and Wood, E.F. 1985. Estimation of the Generalised Extreme Value distribution by the method of probability weighted moments. Technometrics, 27(3): 251-261. Landwehr, J.M., Matalas, N.C. and Wallis, J.R. 1979. Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resour. Res., 15(5): 1055-1064. Phien, H.N. and Fang, T.S.E. 1989. Maximum likelihood estimation of the parameters and quantiles of the general extreme-value distribution from censored samples. J. Hydrol., 105: 139-155. Prescott, P. and Walden, A,T. 1983. Maximum likelihood estimation of the parameters of the three-parameter generalized extreme-value distribution from censored samples. J. Statist. Comput. Simul., 16: 241-250.

APPENDIX

Unbiased estimators of the partial probability weighted moments Partial probability weighted moments have been defined as 1

M~ .... =

~0 [ x ( F ) ] P F r ( 1 -

F)'dF

(A1)

w h e r e F 0 = F(x0), x0 b e i n g t h e c e n s o r i n g t h r e s h o l d . W h e n p = 1 a n d s = 0, PPWM become 1

fl'r = M'l,r,o = ~o x(F)Fr d F

(A2)

W e n o w p r o v e t h a t t h e f o l l o w i n g s t a t i s t i c is a n u n b i a s e d e s t i m a t o r o f fl~:

b:

n1i =~l

=

(i - 1)(i - 2 ) . . . ( i - r) , (n 1)(n 5 ~.. ('n - - r ) x (i)

(A3)

where 0 X~i )

~

x(i) <~ x o

X(i) X(i) > X 0

and xo) ~< x(2) ~< ... ~

X(n)

X(i), X*(i) a n d b : a r e s a m p l e v a l u e s . W e d e n o t e t h e i r c o r r e s p o n d i n g v a r i a t e s a s X(i), X*(i ) a n d B' r r e s p e c t i v e l y . T h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f X(i) i s g(x(o)

=

i(?~[F(x(i))] ' - 1 1 1 -

F(x(i))] "-i fix(i))

(A4)

112

Q.J. WANG

The expected

v a l u e o f X~i) i s t h e n

E(X*
x0 1

= i(?)Ix(F)F'-l(1-F)"-'dF

(A5)

As the first r terms in the summation

of eqn. (A3) equal zero, b: can also be

written

as 1

b'~ -

~

(i

-

1)(i

ni=r+,(n

1 ~

[(

n i=r+, The expected

E(B'~)

-

2)...(i

1)(n

ii -

1 )j( r-

-

2)...(n

1

n-

r)

.

- r) x(i)

n - 1 )] r1 x~i)

(A6)

v a l u e o f B : is t h e n

ni=~+l

i -

r-

1

n -

r-

1

n i=r+l

i -- r -- 1

n -

r -

1

1

(I

_x(F) t0

i=r+l

-

F)"-~dF

i -

r -

1

n

-

r -

1

1

F0

=

X

i= 1

r

F i-~

1(1 -

-

'dF

ESTIMATION OF G E V DISTRIBUTION

substituting j

=

i -

113

r -

1

1 = ~ox(F) [ " ] ~ = o l ( n -jr - 1 ) F i ( 1 - F ) . . . . 1-,]FrdF 1 = I x(F)[F +

(1

F)]" -r -1F

-

rdF

1 I x(F)F rdF t h a t is

E(B'~) =

(A7)

Partial probability weighted moments for the GEV distribution The inverse function of the GEV distribution is x(F)

=

~ + ~[1 - (-

=

~ - alog(-

ThePPWMfork

B'r=

logF)'] logF)

#

k

4= 0

k

=

0

0isthen

Pl,r,O

1

= fo x(F)F~dF 1 =

{~ + ~ [ 1 - ( - l o g ~ q } F ' d F

substituting u - I~F

=

=

-

log F

0

f

[~ + ~ (1 -

u~)]e-('+l)~du

0

-

=

('"~)

log

F0

-

~ e-(r+l'Udu-~ o

substituting 0

--

log F 0

f o

u k e - (" + l)u d u

(r + 1)u - (r + 1) l o g F o

(~ +

r-V~ (I - F;

+ 1)

a F(1 + k ) k (r + 1) 1 + k

f 0

~e-0d0 F(1 + k )

(A8)

114

Q.J. WANG

t h a t is

(A9)

P P W M with double bound censoring Partial probability weighted moments with censoring from both below and a b o v e a r e defined as | [x(F)]pFr(1 - F)SdF F01

=

(A10)

w h e r e Fo, = F(xol ) a n d F02 = F(xo2), x0, a n d x02 a r e l o w e r b o u n d t h r e s h o l d a n d u p p e r b o u n d t h r e s h o l d r e s p e c t i v e l y . W h e n p = 1 a n d s = 0, t h e P P W M b e c o m e fo~

fl"~ = M"l'r'° =

i x(F)F~dF

(All)

,J

G i v e n a c o m p l e t e s a m p l e , x(1) <~ x(2) ~< ... ~< x(n), we c a n p r o v e t h a t t h e foll o w i n g s t a t i s t i c is a n u n b i a s e d e s t i m a t o r of fl"~:

!

(i - 1)(i - 2)...(i Z~ (n 1)(n 2)...(nhi=,

b':

r)

,

(A12)

r) x (i)

where now x*(~ =

xoOcix(~ ~< x01 ) Xo~ (x(~) ~< Xo2 X(i) > X02

T h e P P W M w i t h d o u b l e b o u n d c e n s o r i n g for t h e G E V d i s t r i b u t i o n c a n be f o u n d as fir

=

( ~k +)

~ 1

(F~2. . ' .- . F~, ~)

a r ( 1 + k) k (r + 1) 1 + k

[P(1 + k, - (r + 1 ) l o g F0,) - P(1 + k, - (r + 1 ) l o g F02)]

(A13)

The solution method used for lower bound censoring can then be similarly a p p l i e d to find p a r a m e t e r e s t i m a t e s .