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| Issue date: | 04/27/1979 |
| From: | Landwehr J, Matalas N, Wallis J IBM CORP., INTERIOR, DEPT. OF, GEOLOGICAL SURVEY |
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Text
RC 7626 ( 4 3.a v 0 4 ) 4/27/79 Environ:nental Sc:.ences 3; pages Researc, Repor:
QUANTILE ESTIMATION WITH NIORE OR LESS FLOOD-LIKE DISTR!-
B UTIONS
- 1. \\ tac:unas Landwent N. C. Statalas U.S. Geolog: cal Survey Nat:onal Center Reston. Vir;:n:a 20092 J. R. Watlis IBN! Thorras J. Watson Researen Center Yornown Hergnts. New Yoric 10598 LIMITED DISTRIBUTION NOTICE This report has been submitted for publication outside of IBM and weil probably be copyrighted if accepted for publication. It has been issued as a Research Re po rt for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer Communications and SDecific requests. After outside publicatiOM. requests should be filled only by reprints Or legally obtained CGQses 3f the artiCie to.g.,
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RC 7626 (#33004) 4/27/79 Environmental Sciences 31 pages QUANTILE ESTIN1ATION WITH NIORE OR LESS FLOOD-LIKE DISTRI-BUTIONS J. Maciunas Landwehr N. C. Matalas U.S. G
, :al Survey National tenter Reston, Virginia 22092 J. R. Wallis IBM Thomas J. Watson Research Center Yorktown Heights, New York 10598 Abstract:
The desirable prope*ies of an estimator relative to an hypothetical population may be irrelevent in practice unless the population at issue more or less resembles the hypothetical population. Evidence that floods are distributed with long, stretched upper tails suggests that use of the more common distributions r'esults in a rather precise underestimation of the extreme quantiles and thereby in the underdesign of flood protection measures.
g> r/ g l.,c
I Introduction Various distribution functions have been proposed for flood frequency analysis. The most recent distribution, introduced by Houghton [1977,1978], is Thomas' Wakeby distribu-
- tion, x = m + a(1-(1 - F) ) - c(1-(1 - F) }
(1) where F = F(x) = PlXsx]. The Wakeby distribution is defined by fise parameters and so a reasonably good fit to a sample might be expected. On the other hand, the more familar Gumoel distribution, x = m-a in (- In F)
(2) is defined by only two parameters. And in contrast to Wakeby and other ' flood distributions',
the Gumbel distribution has a unique value of skewness, y 1.14, and of kurtosis A 5.4.
=
=
Whatever other appeal the Gumbel and Wakeby distributions may have, they serve as paradigms of simplicity and complexity, respectively, of flood distributions. There are several distributions of intermediate complexity; one in particular is the much-used Log-Normal distribuuon, 1
-1 ~ In (x - a) - m 2
f(x )
=
exp (3) 6b(x - a) 2 b
dF(x )
where f(x)
For the Log +ormal distribution, values of y 2 0 and A 2 3 are
=
dx admissibic, but such that y and A are related by virtue of both being functions of a single parameter, narr. elf b (see e.g., Wallis et al.: 1974).
Empirical evidence, in relation to the condition of separation (see Ntataias er al.: 1975),
suggests that the distriba: ions of floods are more nearly Wakeby-like with b > 1 and d > 0 t i.e.
long stretened upper tails) than like any of the other more commomiy suggested flood
,qr
- .3 7 0)D juJ
2' distributions (see Houghton: 1977 and Landwehr er al.: 1978). That the Wakeby distribution can satisfy the condition of separation does not imply that indeed floods are distnbuted as Wakeby. However, the Wakeby distribution provides a plausible description of flood se-quences, and it also provides a means for representing the seemingly long, stretched upper tail structures of flood distributions, as well as the tail structures of the distnbutions of other hydrologic phenomena. Thus the Wakeby distribution provides a convenient analytical and a reasonable hydrologic basis for assessing the relative performances of alternative techniques of estimating the unknown quantiles of the distribution of hydrologic phenomena.
For the specific Wakeby populations considered by Landwehr er al. [1979a i;], the Wakeby, Leg-normal, and Gumbel distributions, with alternative methods of fitting, were used to determine the biases and mean square errors of the estimates of the upper quantiles for each of the populations. Also, the expected underdesign losses associated with the estimates were assessed under the assumptions of linear and quadratic loss functions. The results provide an assessment of i) the relative performance of alternative techniques (i.e., choice of distribution and method of fitting) for estimating the quantiles of Wakeby distributions, and ii) the relative performance of the more common distributions (i.e., Log-Normal, and Gumbel) in estimating the unknown quantiles of flood distnbutions in hydrologic environments that are Wakeby-like.
Experimental Design Landwehr er al. [1979a,b] considered six specific Wakeby distributions, each with lower
- 0. Values of the parameters (a, b, c, d) and of the statistical characteristics, bound m
=
mean, standard deviation, and coefficients of variation, skewness, and kurtesis, (g, o, C,, y, A) for each of :he distributions are given in Table 1.
The distributions are depicted in Fig.1, emphasizing differences in the left tails, and in Fig. 2, emphasizing differences in the right tails.
O f*
- t, A
- 'T i/
3 Figure y ~~ Cumulat5 distribution functions IV 4. g.
IVg.6 1.0
~~-~.~.
,W j
//A 0.8
~
,/./ / '
/
./7/
E p)?'i
'9 !
"- 0.6
~
I/
El
/}
l' 0.4
/.j i'
/
//
9 i
$'/?
/
0.2 fo i
,/
i I
0.0 '
0 1
2 3
A 5
g I
8 X-
- d. C :
e O c~
hG)
~1
.i Figure 2 -- Cumulatise distribution functions : 5 5 '.4 - 1, 15'.4 - 6 10.
,i 9 '_
-4 l
8H
]
~
5 l
~
7r
/
1 D/
i 4'/
l 6-nj
~
Ni
/
{
/
5-x 1
S e
hv 4-
,i
~'
/
~/
2 3_
' +9..
l V
-> ls
-lQ '
1 2-
.._.' J,, C
'~,
\\
l
,.-pq'3 s '*
d 1-y s'
0'
.0001.001
.01.02.05.1
.2
.5
.95.98.99.995.999.9999 F
A0A
- l ;i r.G
>/a l
6
+
the Log-Normal conditioned quantile estimates by solution (numerical integration) of j
A A
r exp <
in (y - a ) - m, -
^
2 g
i i
y F=
dr (5)
A A
7 A
/2sb
- * (y - a )
L b
4 q
4 and the Gumbel conditioned quantile estimates St, - a, in (- In F)
(6) were determined for specific values of f in the range (0.5,0.999).
For F specified, the quantile value x is, in practisc, the design flood; i.e., the flood magnitude upon which protective measures (structural or nonstructural) are sized. Thus the measure is underdesigned if (S - x) < 0 and is overdesigned if ($ - x) > 0. The expected over-and underdesign losses, L+ and L, are defined as
<} a+k+El $ - x l', $ - x > 0 L+
i (7)
=
,$-xs0 k El x - x l'. $ - x > 0
<{j a
L=
(8)
,$-xs0 where. a+ and a-1 - a+ denote the probabilities of over-and underdesign, k+ 2 0 and
=
k 2 0, weighting factors reflecting the scale of over-and underdesign costs and r 2 0, a factor defining the analytical shape of the loss functions (see Slack et al. 1975). Hence the expected design loss is given by L=
L+ + L-(9)
If k+
k-k (i.e. the loss functions are symmetric), the bias, e, in the estimates
=
=
. O ',
1.
7 I
j,/ ;
{
L' I
5 Table 1 -- Wakeby distributions Distribution Parameters Statistical Characteristics C,
y A
m a
b c
d a
IVA ~ l 0
1 16 0 4
0.20 1.94 1.34 0.69 4.14 63.74 IVA - 2 0
1 7.5 5
0.12 1.56 0.90 0.58 2.01 14.08 It'.i - 3 0
1 1.0 5
0.12 1.18 1.03 0.87 1.91 10.73 IVA - 4 0
1 16.0 10 0.04 1.36 0.51 0.38 1.10 7.69 IVA - 5 0
1 1.0 10 0.04 0.92 0.70 0.76 1.11 4.73 IV.i - 6 0
1 2.5 to 0.02 0.92 0.46 0.50 0.00 2.65 Relatise to y = 1.14 for the Gumbel distribution, three of the six distnbutions may be regarded as having high skews (IVA-1, EVA-2, IVA-3), two moderate skews (IVA-4, IVA-5), and one low skew (IVA-6). The distributions IPA-5 and IVA-6 are less kurtotic and the other Wakeby distributions are more kurtotic than the Gumbel distribution for which A =
5.4.
In contrast to the distributions IVA-3, IVA-5, IVA-6, the distnbutions IVA-1, IVA-2, and IVA-4 are more kurtotic than the Log-Normal distribution for comparable values of y and satisfy the condition of separation.
With respect to each Wakeby distribution, y sequences, each of length n = 31 were generated in the manner described by Landwehr et af. [1978]. For the q-th sequence conditioned on a particular Wakeby distribution the Wakeby, Log-Normal, and Gumbel parameter estimates, ( A, S $, 2, d ), ( A, 3, b ), and (m, S ).
respectively, were q q 7 7 q q 7 q
q q obtained and hence the Wakeby ecnditioned quantile estimates q
q y,1 - (1 - F)\\
0; _1 - t 1 - F) "\\
.E A
=
+a (4)
O k,-
/ 5.T
.)
7 ef x is given by (L+ - L-)
0 (10)
=
k where r 1 (i.e., linear loss functions), and the mean square error, 4, of the estirr.ates is
=
given by 4aL (L* + L-)
(11)
=
k k
where r 2 (i.e., quadratic loss functions). Thus the statistical measures of goodness of
=
estimation, O and 'D, are directly related to the economic measures of goodness of design, L+
and L, if the loss functions are symmetric. The use of <> as a criterion upon which to choose among alternative estimates of x implies that i) the economic loss functions are symmetric and quadratic, and ii) one is indifferent to an over-or an underdesign loss.
Given the n sequences, where n was at least 20,000, conditioned upon a particular Wakeby distribution, the estimates of x, for specific values in the range (0.5, 0.999), were identified as being greater or less than x The probabilities of over-and underdesign were approximated by
$* = h (12)
~-
y-a
= -
(13) rl where r;+ denotes the number of estimates that were greater than x, and 3, the number less than x. Given the set of estimates $ > x, the value E l$ - x l for r 1,2 was approximat-
=
ed by
=
~
r r
El$ - xi
= 2 l x, - x l J n (14) 4-1
(: '~) D
\\ e > 'l r
r
8 Similarly, the value El 2 - xl' was approximated gisen the set of estimates E s v.
The expected oser-and underdesign losses were taken to be L+
'+k+ E l $ - x l ' - x-x>0 (15)
=
~
~
~ ~ k E l x - x l,. x, - x > 0 (16) 4 L-
=a whereby
~
~
~
L+
+ L-(17)
L
=
For k
- 1 (i.e., symmetric, unsealed loss functions), the bias. 8. and the mean k-
=
=
square error. 4, where approximated by
~
~
~
L* - L-(18) 8
=
~
~
& =L (19) 4 7'
bI i <
4 r
9 The Log Normal, ( LN), conditioned quantile estimates were obtained by the method of (conventional) moments,.E 31, tsee e.g., Johnson and Kotz: 1970). The method of maximum likelihood VxL, a method which in principle is more efficient than Afo3f when the population is distributed as Log-Normal was not considered. On the basis of some explorato y work, it was noted that for a non-neglible percentage cf sequences, the iterative solutions in the course of estimating m showed little ii any tendency towards convergence; a drawback to the use of AfrL as noted in the statistical literature (see Aitcheson and Brown: 1957 and Johnson and Kotz: 1970).
Three methods for obtaining the Gumbel conditioned qt.antile estimates were considered:
i) 3fo3f, ii) AfxL, and iii) probability weighted moments, PW31. These methods were consid-ered previously by Landwehr et al. [1979c] in the case of sampling from Gumbel populations.
The probability we.ghted moment, Afg,, is defined as k
Afa,= E[X(1 - f) ]
(.J) where k is a real number (see Greenwood et al.: 1979). In the case where k is a non-negative integer, an unbiased estimate of Afu,, here denoted as Af i,, is given by n-k 1 x ("f/)/(n ("{I))
k = 0, 1, (21) 3/ g,
=
j joi where t 5 x, 5 sx,_3 (see Landwehr et al.: 1979c ).
i The Wakeby, ( WA ), conditioned quantile estimates were obtained v,ith the algorithm given by Landwehr et al. [1979b], as 3f[43 witn path order [Y m m = Ol. where Af 4,, a biased estimate of 31,g,, is given by Stu, = p x [(n - j 4
a)<n) fn (22)
+
m j j=l 0.35 and xi s x, s s x,,.
where a
=
3 $
j } T'r
\\i
10 Experimental Results: Skenness For a given population, samples of size n reflect, on the aserage. skewness less than that characterizing the population. Moment estimates cf y, denoted as y, are algebraically bounded (see Kirby: 1974).nd biased downward (see Wallis er al.: 1974 and Landwehr er al.:
1978). Based on the n sequences (i.e., samples) of length n = 31, the means '(y), and standard deviations,'(y), of the moment estimates of y were determined and are gisen in Table 2.
Table ! -- Values of g'($) and o($) conditioned on IVA
~
~3 Distribution y
g( )
a(y)
IVA - 1 4.14 1.69 0.93 li'A - 2 2.01 1.06 0.82 IVA-3 1.91 1.24 0.66 IV.1 - 4 1.10 0.66 0.82 IVA-5 1.11 0.87 0.48 IVA-6 0.00
- 0.06 0.37 On the average, the samples reflect populations with moderate to low skews. The reflection is particularly prcnounced in the case of IVA-1. In addition, had n been smaller, the downward bias in the estimates of y would have been even more pronounced. The results for
$ when A is large are even more striking than for y, for instance 5 = 7.02 for 30.1-1 with n =
A 31 compared to the true value of 63.74 It would appear almost impossible to obtain reasona-ble moment estimates of y or A, at least from samples of normal hydrologic lencth, and if one n- 0:
' T )
/:.
I !,
il admits the hypothesis that floods may be distributed in a Wakeby-like manner with relatisely high y and A, then y and $ estimated by the method of moments are likely to be severely A
underestima'ad, and fitting procedures that use y or $ are likely to gise results that do not A
distinguish between different values of y or A.
The high skew, high kurtosis hypothesis for flood sequences can be used to explain the apparent lack of success shown by those research studies that hate attempted to deselop either physically based regional maps of y, or regres-sion equaticns for y in real space. Note, this is a new and separate difficulty from those that have hase already been shown to exist for the estimate of skew in log space (see, Landwehr er al. 197 8 ).
Esperimental Results: Quantiles For each of the six Wakeby populations, the estimates of the upper quantiles, for which 0.5 s F s 0.999, as gisen by the five estimating techniques, were assessed according to bias 8, mean square errors, 'b, and expected underdesign loss, L, where k-1,
=
~ ~
~
Tables 3 througth 8 present the approximate values 0,
'b, and L, as well as a, for WA-1 through WA-6.
Table 9 presents an aggregate assessment of the results. From the values gisen, the values of 1* where k +
1 and the values of a* may be determined. Also, the
=
values of 1+, b, and 1 may be determined for any arbitrary values of k + and k-Q
- 7 r.) /O
- 1J
12 Bias -- H for ';te upper quantiles, the IVA-PiV3f estimites generally display the smallest bias.
The LN - Afo31 techniques underestimates the quantiles for all six populations. The G-techniques underestimate the quantiles for the populations with high values of skewness. For populations of moderate skew (i.e., y - 1.14, the skew of the Gumbel distribution) the biases may be positive or negative. For low values of skewness the G-techniques overestimate the quantiles.
Mean square error -- 4 The mean square error associated with the astimates provided by the IVA - PiV3f technique were larger or at least as large as those given by the other tcchniques over all populations. For sampics from low skew distributions, i.e., un-floodlike distributions (e.g.,
IVA-6), the LN - Afo3f estimates had the smallest & salues. However, for IVA-1 through IVA-5, the estimates given by the three G-techniques gave varying values of 4, but in all cases the values were smaller than those given by the other techniques. It is noted that for the high and low skew populations, <D 2
Also it is noted that among the G-the 3fxl estimates tended to yield the larger values of
@ relative to the high and low skew populations. Thus amongst the G-iechniques the Af.v.L may lead to minimal values of + when the population is distributed with near Gumbel skew, but it is not likely to do so if the population is characterized by y substantially different from YG.
Expected underd uign loss - L-As noted above, the mean square error. 4. is proportional to the e:<pected design loss if the loss functions are symmetric. If such loss functions attain, and if one is indifferent to over-or underdesign, then 4 is a convenient and meaningful measure upon whien to base the q'!.}
} I il
13 choice of quantile estimate. However, if underdesign losses are of greater concern, then 1 more meaningful measure is the expected underdesign loss, L-In the case of the high y and high A distributions (i.e., WA-1, WA-2), the WA - P43/
estimates gave the smaller L values, inspite of having larger @ values. For the other less
~
flood-like distributions, the estimates based on one of the G-techniques gave the smaller L-values. To reiterate, the use of only a mean square error criterion could lead to some poten-tially dangerous enderdesign, and should only be advocated if one is indifferent to over-or underdesign losses.
Aggregate assessment If the six Wakeby distributions are considered to attain at different sites in a region, the aggregate (i.e. regional) performance of the quantile estimating techniques may be assessed by 6
3 the cumulative mean square error, I 4-(i). With respect to the techniques from smallest to i-1 largest cumulative mean square errors is: G - P H3f
- s G - 3fo3I 5 G - AfxL
$ LN - 3fo3f s WA - P R3/
It is noted, however, that the magnitudes of the differ-ence.; are not large, except with respect to WA - PW3f The assessment masks bias effects, hence the over-or under-design aspects of flood protection measures, but it does provide some appraisal of the total regional error in estimating the quantiles for all sites in a region.
6 A similar statistic, cumulative squared biases, I O'(i), yields a ranking for the five i-l techniques of WA-1 s LN-Afo3I s G-3fo3/ s G-PW3/ $ G-3fxL, an almost perfect reversal of the results for cumulative mean square error. Overall, the WA biases are far smaller than those for any of the other techniques.
Anticipating caveats It might be said that if you fit Wakeby's to Wakeby's she results should be good so let us add that this was net th-Se purpose of the study. Rather the interest was upon the hypothe-
14 sis that annual flood sequences are distributed with high skew and high kurtosis and what effect, if any, such an hypothesis would have upon the quantile estimates of int,erest in flood frequency analysis. However, fitting the Wakeby distribution with the algorithm identified by Landwehr et al. [1979b] as,5f and path [Y m; m = 0], performs fairly creditably with other high skew but more conventionally kurtotic worlds. For instance, it is possible to generate Log Normal " floods" (see eq. 3), that have p, o, and y identical to those of WA-1 " floods" (see Table 1), and to repeat the analysis. The results, given in Table 10, are quite similar to those given in Table 3. Thus whether the distribution is WA-1 or LV -1, the WA-PW#* techni-que yields estimates that are statistically similar.
There are many other distributions and fitting procedures that might have been selected for inclusion in this study. It appears improbable that any of the more conventional choices would have resulted in at-site extreme quantile estimates that were both less biased and more precise. Without some probability of success in finding a method for which L and h would both be minimal the incentive to test other methods becomes somewhat marginal.
However, the possibility exists that if there were sufficient data available in a region to show a separation effect, that there would also be sufficient data to allow for a statistical regionalization, in which the quantile estimates for each site could be made on the basis of the data for the site in question in combination with the data from all other sites in the region. It is believed that such an approach may result in more precise estimates for each site, while showing minimal overall bias (see Landwehr er al.: 1979d).
Conclusions The biases. O, mean square errors, $ and expected underdesign losses, L, were determined for five quantile estimating techniques relative to the upper quantiles, where 0.5 s F s 0 999, for six specific Wakeby distributions. Although the Wakeby populations had widely varying values of skewness, the samples from the populations reflected populations 1
't N
/')
f\\
15 of more moderate to lower skewness. From an assessment of O,4, and L, the following conclusions were drawn.
- 1. Over all populations considered, the smallest bios quantile estimates were obtained with the IVA - PiVM techniques.
- 2. If the population skew is much different from the Gumbel skew of about 1.14, either larger or smaller, then the Gumbel techniques give precise (small variability) but inaccurate 2
estimates of the quan:ile values as shown by 4 a 0 conditioned upon IVA-1, IVA-2, IVA-3, IVA-4, and IVA-6.
3.
No one of the three Gumbel techniques performed consistently better than the other with respect to 8,4, and L for all populations. Thus if the population is not known to be Gumbel, one can be rather indifferent to the choice of Gumbel technique, and it does not follow that MxL is preferred.
4.
The Gumbel and Log-Normal techniques consistently underestimate the quantiles of high skew populations. To the extent that floods are highly skewed, contrary to the view provided by estimated values of skewness, the use of these techniques lead, on the average, to the underdesign of flood protection measures.
5.
If indeed flood distnbutions are highly skewed, then among the techniques investigated for estimating the upper quantiles, the expected underdesign losses are the smaller when the IVA - PIVM technique is used, even though the mean square errors tend to be larger.
6.
When considered on an aggregate basis, the Gumbel techniques lead to the smaller values of cumulated mean square errors. However, the value masks the biases which differ considerably among the five techniques.
3 /f
- f[
t
16 Table 3 Bias, h, and mean square error, b. probability of underdesign, a, expected linear under
~
~
design loss, Li. expected quadratic underdesign loss, Lf, of quantile estimates gisen by five fitting procedures conditioned on distribution IV.4-1.
F
.500
.900
.950
.980
.990
.995
.999 Quantile 1.59 3.34 4.28 5.75 7.05 8.54 12.92 Method of Estimation
~
O IV.4 - PIV.tf 0.02
- 0.05
- 0.11
- 0.16
- 0.14
- 0.00 1.26 G - PIV3/
0.16 0.07
- 0.24
- 0.S9
-1.58
- 2.46
- 5.48 G - Afx L 0.14
-0.12
-0.50
-1.23
-1.99
- 2.93
- 6.05 G - Afo.tf 0.14 0.21
- 0.50
- 0.62
- 1.25
- 2.08
- 4.91 LN - 3/o.tf 0.09 0.12
- 0.07
- 0.48
- 0.93
-1.53
- 3.60
~
'D IV.4 - PIV3f 0.03 0.31 0.76 2.43 5.90 14.4 110.7 G - PIV3f' O.07 0.38 0.67 1.82 3.88 7.9 32.5 G - 3fxL 0.05 0.22 0.56 2.02 4.62 9.5 38.1 G - 3fo3f 0.06 0.74 1.23 2.53 4.54 8.3 31.0 LN - 3fo3I 0.04 0.53 1.21 3.17 6.10 11.0 36.7 Ykb
'?
l to
17 Table 3, Continued
~
il li'.4 - Pil'Af 0.47 0.58 0.60 0.61 0.61 0.60 0.59 G - PIV.t/
0.21 0.51 0.68 0.84 0.91 0.95 0.99 G - 3fxL 0.22 0.65 0.83 0.95 0.98 0.99 1.00 G - 3/o31 0.23 0.47 0.61 0.76 0.83 0.89 0.96 LN - 3fo3f 0.33 0.50 0.61 0.71 0.76 0.79 0.85
~
L{
II'.4 - Pil'31 0.05 0.25 0.40 0.69 0.99 1.38 2.69 G - Pil'3t 0.02 0.20 0.45 1.02 1.66 2.52 5.45 G - 3fxL 0.02 0.25 0.58 1.26 2.00 2.91 6 05 G - Afo3/
0.02 0.19 0.42 0.93 1.52 2.30 5.05 LN - Afo3f 0.04 0.20 0.43 0.91 1.43 2.10 4.35
~
L:
15'.4 - PlV3f 0.01 0.28 0.38 1.09 2.19 4.23 15.9 G - Pii31 0.00 0.11 0.41
- 1.,
3.70 7.74 32.4 G - AfxL 0.00 0.14 0.52 1.20 4.62 9.35 38.1 G - Afo3/
0.00
- 0. I 1 0.39 1.49 3.43 7.11 29.8 LN - 3/03f 0.01 0.12 0.43 1.56 3.21 6.86 26.7 s.
o indicates Af (k) used to estimate PiV3I co indicates $f (k) used to estimate PiV3I
+ j ^/
i,
' '/ r
18 Table 4 Bias. O, and mean square error, b. probability of underdesign,', expected linear under
~
~
design loss, L[, expected quadratic underdesign loes, Li, of quantile estimates gisen by fise fitting procedures conditioned on distribution IVA-2.
F
.500
.900
.950
.980
.990
.995
.999 Quantile 1.43 2.59 3.16 4.00 4.69 5.44 7.45 Niethod of Estimation
~
8 frA - Pli'M 0.00
- 0.03
- 0.06
- 0.07
- 0.01
- 0. I 3 1.04 G - PiVM' O.00 0.07
- 0.03
-0.25
- 0.48
- 0.78
-1.73 G - MxL 0.00 0.08
-0.02
-0.24
-0.48
- 0.77
-1.73 G - mom 0.00 0.08
- 0.01
- 0.22
- 0.45
- 0.74
- 1.68 LX - mom 0.01 0.06
- 0.05
- 0.27
- 0.50
-0.78
-1.64
~
li'A - PII'M 0.02 0.13 0.28' O.77 1.69 3.S0 25.0 G - PiVM '
O.02 0.13 0.19 0.38 0.67 1.17 39 G - MxL 0.02 0.08 0.12 0.25 0.48 0.92 3.5 G - mom 0.02
- 0. I 8 0.29 0.55 0.88 1.46 4.4 LN - mom 0.02 0.15 0.32 0.83 1.56 2.76 S.2 I
j
(
e
19 Table 4, Continued
~_
a li'.4 - PlVM" O.51 0.57 0.59 0.59 0.57 0.56 0.53 G - PlVM 0.52 0.46 0.57 0.71 0.79 0.86 0.95 G - MxL 0.51 0.42 0.55 0.73 0.84 0.91 0.98 G - mom 0.52 0.47 0.57 0.69 0.77 0.83 0.91 L N - mom 0.49 0.48 0.59 0.69 0.74 0.77 0.82
~
L{
IV.4 - PlVM" O.05 0.16 0.24 0.38 0.50 0.65 1.13 G - PIVM' O.06 0.10 0.19 0.38 0.58 0.86 1.77 G - MxL 0.06 0.08 0.15 0.33 0.53 0.80 1.73 G - mom 0.06 0.11 0.21 0.40 0.61 0.88 1.77 L N - mom 0.05 0.12 0.24 0.50 0.76 1.08 2.03
~
L,_
IV.1 - PIVM 0.01 0.07 0.15-0.35 0.62 1.06 3.20 G - P lVM '
O.01 0.03 0.09 0.28 0.57 1.10 3.91 G - MxL 0.01 0.02 0.06 0.20 0.45 0.90 3.50 G - mom 0.01 0.04 0.10 0.32 0.65 1.20 4 12 LN - mom 0.01 0.04 0.14 0.49 1.03 1.91 6.16 4
- indicates M (k) used to estimate PIVM on indicates $f (k) used to estimate PIVM s
.'l.
iI!
P '; /
20 Table 5 Bias, b, and mean square error. D probability of underdesign 2, expected linear under
~
~
design loss. LJ, expected quadrati: underdesign loss, Li, of quantile estimates given by five fitting procedures conditioned on distribution li'A-3.
F
.500
.900
.950
.980
.990
.995 999 Quantile 0.93 2.49 3.11 3.98 4.68 5.44 7.45 Method of Estimation
~
O II'A - Pit'3f
- 0.00
- 0.01
- 0.02
- 0.07
- 0.12
- 0.14 0.10 G - Pit'3/
0.09
- 0.03
- 0.10
- 0.26
- 0.43
- 0.65
-1.44 G - AfxL 0.07
- 0.21
-0.34
- 0.58
- 0.81
-1.09
- 2.02 G - Afo3/
0.09
- 0.03
-O.10
-0.24
- 0.41
-0.64
-1.42 LN - Afo3/
0.08
- 0.05
- 0. I 1
- 0.24
- 0.38
- 0.56
- 1. I 6
~
li'A - Fif'3/
0.03 0.18 0.35 0.90 1.88 3.88 21.4 G - P it'Af 0.03 0.17 0.27 0.48 0.72 1.12 3.2 G - 3fxl 0.03 0.17 0.30 0.62 1.02 1.66 4 80 G - 3fo3f 0.03 0.21 0.35 0.62 0.94 1.42 17 LN - 3fo3f 0.04 0.18 0.37 0.63 1.44 2.37 6.6 i e i j ' '. I
21 Table 5. Continued
~_
a 15'.4 - PI53f "
O.54 0.53 0.55 0.59 0.60 0.62 0.63 G - PI531 0.31 0.56 0.61 0.68 0.74 0.80 0.91 G - 3fxl 0.34 0.74 0.80 0.86 0.90 0.94 0.98 G - Afo3I 0.31 0.57 0.61 0.68 0.73 0.78 0.88 LN - 3fo3f 0.34 0.58 0.63 0.67 0.70 0.73 0.78
~
L' 55'A - P W.i!'*
0.08 0.17 0.24 0.41 0.59 0.82 1.55 G - Pii3f' O.03 0.17 0.26 0.40 0.57 0.77 1.51 G - AIxL 0.03 0.27 0.40 0.62 0 84 1.12 2.02 G - Afo31 0.03 0.19 0.28 0.44 0.60 0.81 1.55 LN - Afo3f 0.03 0.19 0.29 0.48 0.66 0.89 1.63
~
L.'
15'.4 - Pif 3/
0.02 0.08 0.16 0.41 0.81 1.46 4.93 G - PII3/
0.00 0.08 0.16 0.34 0.59 1.00 3.14 G
.tixl.
0.01 0.14 0.27 0.65 1.00 1.64 4.S0 G.tfo3f 0.00 0.09 0.18 0.39 0.67 1.12 3.40 LN.ifo31 0.01 0.09 0.19 0.46 0.84 1.44 4.30
^.
o indicates 3f (k) used to estimate Pii3f
- k) used to estimate Pil'.if oc indicates.tf t
9 lt
} !b E
22 Table 6 Bias, 9. and mean square error, h. probability of underdesign. ', expected linear under
~
design loss, L. expected quadratic underdesign loss, L,. of quantile estimates given by five t
fitting procedures conditioned on distribution 15'.4-4.
F
.500
.900
.950
.980
.990
.995
.999 Quantile 1.28 1.96 2.27 2.69 3.02 3.36 4.18 Method of Estimation
~
O li'A - PlVM 0.01
-0.02
- 0.01 0.03 0.10 0 21 0.75 G - P 50/
0.00 0.03 0.00
- 0.07
- O.13
-- 0. 20
- 0.41 G - MxL 0.00 0.15 0.17 0.16 0.15 0.12 0.02 G - mom 0.00 0.03 0.00
- 0.06
- O. I 2
-O.t9
-0.40 LN - mom 0.03 0.02
-0.05
- 0.17
-0.28
- 0.10
- 0.71
~
IV.4 - P il'M 0.00 0.04 0.07 0.40 0.40 0.86 4.97 G - P!VM 0.01 0.04 0.05 0.10 0.14 0.20 0.44 G - MxL 0.01 0.06 0.10 0.15 0.20 0.26 0.44 G - mom 0.01 0.04 0.06 0.12
- 0. I 7 0.23 0.49 LN - mom 0.01 0.04 0.08 0.20 0.37 0.62 1.61 I
El
~> ;) !,
23 Table 6, Continued
~_
a 11".4 - Pii3f 0.48 0.57 0.55 0.52 0.49 0.47 0.44 G - Pii3f' O.53 0.46 0.53 0.62 0.67 0.71 0.78 G - MxL 0.52 0.21 0.26 0.32 0.36 0.40 0.480 G
.ifo M 0.53 0.47 0.54 0.62 0.67 0.71 0.78 LN - mom 0.37 0.48 0.61 0.70 0.74 0.77 0.80
~
L{
15'.4 - Pit'M "
O.02 0.08 0.12 0.16 0.19 0.24 0.38 G - P15'.tI 0.04 0.06 0.09 0.15 0.21 0.29 0.48 G
.tIxL 0.04 0.02 0.04 0.07 0.10 0.14 0.25 G - mom 0.04 0.06 0.10 0.16 0.23 0.30 0.49 LN - mom 0.02 0.07 0.14 0.27 0.39 0.53 0.90
~
L.:
It'.4 - Pii3f 0.00 0.02 0.04 0.07 0.11 0.18 0.45 G - Pll3I 0.00 0.01 0.02 0.06 0.10 0.I6 0.34 G
.tixL 0 00 0.00 0.01 0.02 0.04 0.07 0.19 G - mom 0.00 0.01 0.02 0.06
- 0. I 1 0.I7 0.42 LN - mom 0.00 0.01 0.05 0.14 0.27 0.48 1.26 3,
o indicates M (k) used to estimate Pli3f co indicates I'*(k) used to estimate Pii3I
,. U,
l i. i
~ ) E, J/J s
24 Table 7 Bias,9, and mean square error,4, probability of underdesign, ', expected linear under design loss, L, expected quadratic underdesign loss, Ly of quantile estimates given by five i
fi ing procedures conditioned on distribution IVA-5.
tt F
.500
.900
.950
.980
.990
.995
.999 Quantile 0.78 1.86 2.22 2.67 3.01 3.36 4.18
.\\le thod of Estimation
~
O IVA - PIVM
- 0.01 0.00 0.00
- 0.01
- 0.02
- 0.01 0.17 G - P Il M 0.02
-0.03 0.00 0.06 0.10 0.14 0.20 G - Mx L 0.02
- 0. I 1
-O.10
- 0.08
-0.07
- 0.05
-0.06 G - mom 0.02
- 0.07
- 0.05
- 0.01 0.02 0.05 0.07 LN - mom 0.05
- 0.07
- 0.08
- 0.09
-0.10
-0.13
-0.20
~
IV.4 - PlVM 0.02 0.07 0.12 0.24 0.45 0.85 4.16 G - PlVM 0.01 0.06 0.08 0.14 0.19 0.25 0.41 G - MxL 0.01 0.07 0.09 0.12 0.16 0.19 0.26 G - mom 0.01 0.06 0.09 0.14 0.19 0.25 0.41 LN - mom 0.02 0.06 0.10 0.19 0.30 0.45 1.02 o
j
25 Table 7 Continued
~_
a It'A - PlV M "
O.55 0.52 0.52 0.55 0.56 0.57 0.59 G - PiVM' O.45 0.57 0.51 0.45 0.41 0.40 0.38 G - MxL 0.46 0.69 0.65 0.60 0.57 0.56 0.55 G - Mo M 0.43 0.63 0.58 0.53 0.50 0.49 0.48 LN - mom 0.37 0.62 0.62 0.62 0.63 0.63 0.66 L[
IVA - PiVM 0.06 0.10 0.14 0.20 0.27 0.36 0.64 G - PIVM' O.04 0.11 0.12 0.12 0.12 0.13 0.15 G - MxL 0.04 0.16 0.17 0.18 0.19 0.21 0.26 G - mom 0.03 0.13 0.15 0.15 0.16 0.17 0.22 LN - mom 0.03 0.13 0.17 0.22 0.27 0.32 0.49
~
L:
IVA - PiVM 0.01 0.03 0.05 0.11 0.19 0.32 0.92 G - PIVM 0.00 0.03 0.04 0.05 0.05 0.06 0.10 G - MxL 0.00 0.05 0.07 0.08 0.09
- 0. I 1 0.17 G - M o.ti 0.00 0.04 0.07 0.08 0.09 0.09 0.15 LN - mom 0.00 0.04 0.07
- 0. I 1
- 0. I 6 0.23 0.51
^.
o indicates M (k) used to estimate PlVM co indicates $f (O used to estimate PlVM I '!
- ! E
- u. l
26 Table 8 Bias U, and mean square errer, h, probability of underdesign, a', expected linear under design loss. [i, expected quadratic underdesign loss, Lp of quantile estimates given by five fitting procedures conditioned on distri'oution 15'.4-6.
F
.500
.900
.950
.980
.990
.995
.999 Quantile 0.96 1.47 1.62 1.81 1.96 2.12 2.48 Method of Estimation
~
H 85'.4 - P153/
-0.03 0.01 0.01 0.00 0.00 0.03 0.3 I G - Pil3f'
- 0.12 0.08 0.21 0.36 0.47 0.58
- 0. S 3 G
.tfxl
- 0.11 0.18 0.34 0.54 0.69 0.83 1.15 G - mom
- 0.12 0.04 0.14 J.2 i 0.37 0.46 0.66 LN - mom
-0.04
- 0.02 0.03 0.02
- 0.01
- 0.05
- O. I 8
~
li'.1 - PI53/
0.01 0.01 0.02 0.05 0.10 0.19 1.32 G - Pil3/
0.02 0.02 0.06 0.16 0.26 0.38 0.77 G - MxL 0.02 0 05 0.14 0.32 0.52 0.74 1.42 G - mom 0.02 0.02 0.04 0.10 0.17 0.25 0.50 LN - mom 0.01 0.02 0.02 0.03 0.05 0.07 0.I6
'n j
8
27 Table 8, Continued a-IV.4 - PlV3f 0.58 0.46 0.49 0.54 0.54 0.53 0.50 G - PlV3/
0.93 0.22 0.05 0.01 0.00 0.00 0.00 G - AfxL 0.91 0.08 0.01 0.00 0.00 0.00 0.00 G - 31o.iI 0.92 0.37 0.13 0.03 0.02 0.01 0.00 LN - Afo3f 0.67 0.44 0.43 0.51 0.58 0.65 0.76 L{
IV.4 - PlV31 0.06 0.04 0.05 0.09 0.12 1 15 0.26 G - PIV3f 0.13 0.01 0.00 0.00 0.00 0.00 0.00 G - 3fxL 0.12 0.00 0.00 0.00 0.00 0.00 0.00 G - Afo31
- 0. I 2 0.03 0.01 0.00 0.00 0.00 0.00 LN - Afo3f 0.06 0.03 0.03 0.06 0.09 0.13 0.25
~
L.~
IV.4 - PlV31 0.01 0.00 0.01 0.02 0.04 0.06 0.18 G - PlV3/
0.02 0.00 0.00 0.00 0.00 0.00 0.00 G - AfxL 0.02 0.00 0.00 0.00 0.00 0.00 0.00 G - 3fo3/
0.02 0.01 0.00 0.00 0.00 0.00 0.00 LN - 3fo3f 0.01 0.00 0.00 0.01 0.02 0.04
- 0. I 1
.s,
o indicates 3f (k) used to estimate PlV3f co indicates f (k) used to estimate PlV3f 5I.f f) l ') '.,)
s.
28 Tabis 9 Aggregate outcome: cumulative squared biases, and cumulative mean square errors for the median and right tail quaatiles, considering the six populations to occur in one r;gion.
F
.500
.900
.950
.980
.990
.995
.999 Method of Estimation 2
8 i= l IVA - PIV3/
0.00 0.00 0.01 0.04 0.04 0.1 3.4 G - PIV3/
0.05 0.02
- 0. I 2 1.06 3.I7 7.5 36.0 G
.\\fxL 0.04
- 0. I 3 0.52 2.22 5.34 11.1 45.0 G - Sto3I 0.03 0.06 0 26 0.49 1.96 5.3 29.2 LN - Afo31 0.02 0.03 0.03 0.40 1.35 3.5 17.6
?0 IVA - PlV3f 0.11 0.74 1.59 4.80 10.43 23.9 167.
G - P W3/
0.17 0.79 1.32 3.06 5.86 11.0 41.
G - 3fxL 0.14 0.65 1.30 3.50 7.02 13.3 49.
G - Afo3/
0.15 1.25 2.07 4.04 6.92 11.9 41.
LN - 3/o3f 0.14 0.98 2.10 5.24 9.80 17.3 54
^.
o indicates 3I (k) used to estimate PW3I oc indicates $f'*(k) used to estimate PW3/
))
29 Table le
~
~
~
Bias, 9 and mean square error,4, probability of underdesign, a, expected linear under design loss, L, expected quadratic underdesign loss, 'L,, of quantile estimates gisen by five i
fitting procedures conditioned on distribution LN - 1.
F
.500
.900
.950
.980
.990
.995
.999 Quantile 1.55 3.34 4.32 5.35 7.21 S.75 13.18 Method of Estimation
~
8 WA-PlVM" 0.02
- 0.06
- 0.15
- 0.28
- 0.32
- 0.23 1.08 G-PWM 0.21 0.01
- 0.35
-1.09
-1.85
- 2.81
- 5.87 G - MxL 0.18
- 0.31
- 0.78
- 1.66
- 2.53
- 3.60
- 6.91 G - mom 0.19 0.18
- 0.11
- 0.76
- 1.45
- 2.34
-5.24 LN - mom 0.12 0.08
-0.13
- 0.56
-1.03
-1.63
- 3.56
~
'D WA-PWM" 0.03 0.40 0.90 2.69 6.3 14.8 110.
G-PWM 0.09 0.45 0.86 2.37 5.1 10.0 38.
G - MxL 0.07 0.37 1.06 3.46 7.3 14.1 50.
G - Mo M 0.08 0.79
' 35 2.86 5.3 9.7 35.
LN - mom 0.05 0.58 1.27 3.261
- 6. I 7 11.0 36.
) 'N b l
l
30 Table 10, Continued
~_
a IVA - PWM 0.52 0.58 0.60 0.63 0.63 0.62 0.60 G-PWM 0.15 0.54 0.70 0.85 0.92 0.96 0.99 G - MxL 0.18 0.74 0.88 0.96 0.99 1.00 1.00 G - mom 0.17 0.47 0.62 0.77 0.84 0.89 0.96 LN - mom 0.31 0.51 0.61 0.72 0.76 0.80 0.84
~
L{
WA-PWM 0.06 0.28 0.45 0.79 1.13 1.57 2.96 G-PWM 0.01 0.26 0.55 1.20 1.93 2.85 5.89 G - MxL 0.01 0.41 0.84 1.69 2.55 3.60 6.91 G - mom 0.01 0.23 0.49 1.07 1.70 2.55 5.36 LN - mom 0.03 0.25 0.61 0.99 1.52 2.18 4.34 L'
WA-PWM 0.01 0.20 0.48 1.38 2.77 5.3 18.7 G-PWM 0.00 0.18 0.60 2.18 4.89 9.9 37.9 G - MxL 0.00 0.30 1.01 3.45 7.35 14.I 49.6 G - mom 0.00 0.16 0.54 1.95 4.32 8.8 33.8 LN - mom 0.00 0.17 0.55 1.82 3.91 7.60 27.0
^.
- indicates M (k) used to estimate PWM
^..
- indicates M (k) used to estimate PWM
,p-
<r7 nyo
- n
31 References Aitcheson, J. and J. A. C. Brown. The Lognormal Distribution Cambridge University Press, Cambridge, Great Britain,1957.
Greenwood, J. Arthur, J. Ntaciunas Landwehr N. C. N1atalas, and J. R. Wallis, Probabr/itr neighted moments: Jefinition and application Water Resources Research in press, 1979.
Houghton, Jahr. C., Robust estimations of the frequency of extreme events in a flood frequency context, PhD dissertation, Harvard University, Cambridge, N1 ass.,1977.
Birth of a parent: The Wakeby distributwn for modeling flood flows, Water Resources Research, Vol.14:6, Dec.1978.
Johnson, Norman L. and Samuel Kotz. Continuoia Univariate Distribution - 1. Distribution in Statisacs., John Wiley and Sons, New York,1970.
Kirby, W.
4 /gebraic boundedness in Sample Statistics, Water Resources Research, 10:2, 220-222, 1974 Landwehr J. N1aciunas, N. C. N1atalas and J. R. Wallis, Some comparisorts of170oJ statistics in real and log spare, Water Resources Research,14:5. 902-920, 1978.
. Esnmation of Par.2 meters and Quantiles of Wakeby distributions: Part 1: Known Lower Bounds, Water Resot es Research in press,1979a.
Estimation of Parameters and Quantiles of Wakeby distributions: Part 2:
Unknown lower Bounds Water Resources Research in press,1979b.
, Probability seighted moments compared with some traditional technique techni-ques in estimating Gumbel parameters and Qu zntiles., Water Resources Research in press,1979c.
Regional analpis of flood-like distributions., Water Resources Research in press,1979d.
N1atatas, N. C., J. R. Slack and J. R. Wallis. Regional skew m search of a parent. Water Resources Research,11:6 S15-826,1975.
Slack J. R., J. R. Wallis, and N. C. Ntatalas. On the Value of Information to Flood Frequency
.4nalvsis, Water Resources Research, 11:5, 629-647, 1975.
Wallis. J. R., N. C. Nf atalas and J. R. Slack. Just a Moment! Water Resources Research,10:2, 211-219, 1974.
J
.