Favorable Estimators for Fitting Pareto Models: A Study Using
A comparative evaluation of the estimators of the three-parameter generalized pareto distribution
17
This article was downloaded by: [University of Bath] On: 02 November 2014, At: 11:12 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Statistical Computation and Simulation Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gscs20 A comparative evaluation of the estimators of the three-parameter generalized pareto distribution V. P. Singh & Maqsood Ahmad a Department of Civil and Environmental Engineering , Louisiana State University , Baton Rouge, LA, 70803-6405, USA Published online: 13 May 2010. To cite this article: V. P. Singh & Maqsood Ahmad (2004) A comparative evaluation of the estimators of the three-parameter generalized pareto distribution , Journal of Statistical Computation and Simulation, 74:2, 91-106, DOI: 10.1080/0094965031000110579 To link to this article: http://dx.doi.org/10.1080/0094965031000110579 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms- and-conditions
Transcript of A comparative evaluation of the estimators of the three-parameter generalized pareto distribution
This article was downloaded by: [University of Bath]On: 02 November 2014, At: 11:12Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Journal of Statistical Computation andSimulationPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gscs20
A comparative evaluation of theestimators of the three-parametergeneralized pareto distributionV. P. Singh & Maqsood Ahmada Department of Civil and Environmental Engineering , LouisianaState University , Baton Rouge, LA, 70803-6405, USAPublished online: 13 May 2010.
To cite this article: V. P. Singh & Maqsood Ahmad (2004) A comparative evaluation of the estimatorsof the three-parameter generalized pareto distribution , Journal of Statistical Computation andSimulation, 74:2, 91-106, DOI: 10.1080/0094965031000110579
To link to this article: http://dx.doi.org/10.1080/0094965031000110579
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Journal of Statistical Computation & Simulation
Vol. 74, No. 2, February 2004, pp. 91–106
A COMPARATIVE EVALUATION OF THE
ESTIMATORS OF THE THREE-PARAMETERGENERALIZED PARETO DISTRIBUTION
V. P. SINGH* and MAQSOOD AHMAD
Department of Civil and Environmental Engineering, Louisiana State University,Baton Rouge, LA 70803-6405, USA
(Received 22 December 1999; In final form 7 March 2003)
The parameters and quantiles of the three-parameter generalized Pareto distribution (GPD3) were estimated using sixmethods for Monte Carlo generated samples. The parameter estimators were the moment estimator and its twovariants, probability-weighted moment estimator, maximum likelihood estimator, and entropy estimator.Parameters were investigated using a factorial experiment. The performance of these estimators was statisticallycompared, with the objective of identifying the most robust estimator from amongst them.
Keywords: Monte Carlo simulation; Method of moments; Maximum likelihood; Probability weighted moments
1 INTRODUCTION
The generalized Pareto (GP) distribution was introduced by Pickands (1975), and has since
been applied to a number of areas, encompassing socio-economic processes, physical and
biological phenomena (Saksena and Johnson, 1984), reliability analysis, and analyses of
environmental extremes. Davison and Smith (1984) pointed out that the GP distribution
might form the basis of a broad modeling approach to high-level exceedances.
DuMouchel (1983) applied it to estimate the tail thickness, whereas Davison (1984a,b)
modeled contamination due to long-range atmospheric transport of radionuclides. Ochoa
et al. (1980) found the Pareto distributions to be more suitable for annual peak flow
data than the exponential-tailed distributions common to hydrologic frequency analysis.
Van Montfort and Witter (1985; 1991) applied the GP distribution to model the peaks-
over-threshold (POT) streamflows and rainfall series, and Smith (1984; 1987) applied it
to analyze flood frequencies. Similarly, Joe (1987) employed it to estimate quantiles of
the maximum of N observations. Wang (1991) applied it to develop a POT model for
flood peaks with Poisson arrival time, whereas Rosbjerg et al. (1992) compared the use
of the 2-parameter GP and exponential distributions as distribution models for exceedances
with the parent distribution being a generalized GP distribution. In an extreme value
* Corresponding author. E-mail: [email protected]
ISSN 0094-9655 print; ISSN 1563-5163 online # 2004 Taylor & Francis LtdDOI: 10.1080=0094965031000110579
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analysis of the flow of Burbage Brook, Barrett (1992) used the GP distribution to model
the POT flood series with Poisson interarrival times. Davison and Smith (1990) presented a
comprehensive analysis of the extremes of data by use of the GP distribution for modeling
the sizes and occurrences of exceedances over high thresholds. These studies show that the
3-parameter generalized Pareto distribution is more flexible and is, therefore, better suited
for modeling those environmental and hydrologic processes that have heavy-tailed
distributions.
Quandt (1966) used the method of moments (MOM), and Baxter (1980), and Cook and
Mumme (1981) used the method of maximum likelihood estimation (MLE). Methods for
estimating parameters of the 2-parameter GP distribution were reviewed by Hosking and
Wallis (1987). Along with the aforementioned estimators, they also included probability
weighted moments (PWM) in the review. Van Montfort and Witter (1986) used MLE to fit
the GP distribution to represent the Dutch POT rainfall series, and used an empirical correc-
tion formula to reduce the bias of the scale and shape parameter estimates. Davison and
Smith (1990) used MLE, PWM, a graphical method, and the least squares method to estimate
the GP distribution parameters. Singh and Guo (1995a,b; 1997) presented the entropy (ENT)-
based parameter estimation method for 2-parameter, 2-parameter generalized Pareto and
3-parameter generalized Pareto distributions. Singh (1998) summarized entropy-based para-
meter estimation for the Pareto family. Moharram et al. (1993) compared MLE, MOM, PWM
and least squares method, using Monte Carlo simulation. They found the least squares
method to be superior to other methods for shape parameter greater than zero but PWM
for shape parameter less than zero.
The objective of this paper is to comparatively evaluate six methods of parameter estima-
tion for the three-parameter generalized Pareto distribution: MOM and its variants, MLE,
PWM and ENT using Monte Carlo simulated data, and statistically evaluate the performance
of these estimators. These methods were evaluated based on their relative performance in
terms of robustness and variability and bias of large quantile estimates from small
sample sizes.
2 GENERALIZED PARETO DISTRIBUTION
The 3-parameter generalized distribution (GPD3) has the cumulative distribution
function (CDF):
F(x) ¼1 � 1 � a
x� c
b
� �1=a
, a 6¼ 0
1 � exp �x� c
b
� �, a ¼ 0
8><>: (1)
and the probability density function (PDF):
f (x) ¼
1
b1 � a
x� c
b
� �(1=a)�1
, a 6¼ 0
1
bexp
x� c
b
� �, a 6¼ 0
8>><>>: (2)
The range of x is c � x � 1 for a � 0 and c � x � b=ðaþ cÞ for a � 0. Figures 1–3 graph
the PDF for different values of parameters a, b, and c, respectively.
92 V. P. SINGH AND M. AHMAD
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3 PARAMETER ESTIMATORS
In environmental and water resources engineering, the commonly used methods of parameter
estimation are the methods of moments, maximum likelihood and probability-weighted
moments. The entropy-based method is also becoming popular these days. Therefore,
these methods and their variants were considered in this study. Specifically, six methods
of parameter estimation were compared for the three-parameter generalized Pareto distribu-
tion: 3 methods of moments, and probability-weighted moments, maximum likelihood
FIGURE 1 GPD3 probability density function for various shape parameter, a.
FIGURE 2 GPD3 probability density function for various scale parameter, b.
THREE-PARAMETER GENERALIZED DISTRIBUTION 93
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estimation, and entropy methods. The parameter estimators resulting from these methods are
briefly outlined.
3.1 Regular Method of Moment Estimators (RME)
The regular moment estimators of the generalized Pareto distribution were derived by
Hosking and Wallis (1987). It is important to note that E[1 � a(x� c)=b]r ¼ 1=(1 þ ar) if
[1 þ ra] > 0. Thus, the rth moment of X exists only if a � [1=r]. Provided that these
moments exist, the moment estimators are
�xx ¼ cþb
1 þ a(3)
S2 ¼b2
(1 þ a)2(1 þ 2a)(4)
G ¼2(1 � a)(1 þ 2a)1=2
1 þ 3a(5)
where �xx, S2 and G are, respectively, the mean, variance and skewness coefficient. Equations
(3) to (5) constitute a system of equations which are solved for obtaining parameters a, b, and
c, given the value of �xx, S2 and G. The regular method of moments is applicable only when
parameter a > �1=3.
3.2 Modified Moment Estimators
The regular method of moments is valid only when a > �1=3, which limits its practical
application. Secondly, its estimation of shape parameter (a) is dependent on the sample skew-
ness alone, which in reality could be grossly different from the population skewness.
Following Quandt (1966), two modified versions of this method have been suggested,
which involve restructuring Eq. (5) in terms of some known property of the data. The two
modified methods of moments are briefly outlined below.
FIGURE 3 GPD3 probability density function for various location parameter, c.
94 V. P. SINGH AND M. AHMAD
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3.2.1 Modified Moment Estimators (MM1)
In the first modification, Eq. (1) is replaced by E[F(x1)] ¼ F(x1), which yields
1 � ax1 � c
b¼
n
nþ 1
� �a
(6)
Eliminating b and c by substitution of Eqs. (4) and (5) into Eq. (6), one gets an expression for
estimation of a:
�xx� x1 ¼1
1 þ a�
1
aþ
1
a
n
nþ 1
� �a� �S(1 þ a)(1 þ 2a)1=2, a 6¼ 0 (7)
With the value of a obtained from the solution of Eq. (7), estimates of b and c are obtained
from the solution of Eqs. (4) and (5).
3.2.2 Modified Moment Estimators (MM2)
In the second modification, Eq. (4) is replaced by E(x1) ¼ x1. Using Eq. (1), the CDF of x1 is
given by
Fx1(x) ¼ 1 � [1 � F(x)]n ¼ 1 � 1 � ax� c
b
h in=a(8)
and its PDF is
fx1(x) ¼n
b1 � a
x� c
b
� �(n=a)�1
(9)
Thus,
E(x1) ¼ x1 ¼b
nþ aþ c (10)
Substitution of Eqs. (4) and (5) in Eq. (10) yields the expression for estimation of a:
�xx� x1 ¼s(nþ 1)(1 þ 2a)1=2
nþ a(11)
With the value of a obtained from the solution of Eq. (11), estimates of b and c are obtained
from the solution of Eqs. (4) and (5).
3.3 Probability Weighted Moment (PWM) Estimators
The probability-weighted moments (PWM) estimators for GPD3 are given (Hosking and
Wallis, 1987) as
a ¼W0 � 8W1 þ 9W2
�W0 þ 4W1 � 3W2
(12)
b ¼ �(W0 � 2W1)(W0 � 3W2)( � 4W1 þ 6W2)
( �W0 þ 4W1 � 3W2)2(13)
c ¼2W0W1 � 6W0W2 þ 6W1W2
�W0 þ 4W1 � 3W2
(14)
THREE-PARAMETER GENERALIZED DISTRIBUTION 95
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where the rth probability-weighted moment, Wr, is given by
Wr ¼1
r þ 1cþ
b
a
� ��b
a
1
(aþ r þ 1), r ¼ 0, 1, 2, . . . (15)
Thus, Eqs. (12)–(14) yield parameter estimates in terms of PWMs. For a finite sample size
consistent moment estimates ( �WWr) can be computed as
�WWr ¼1
n
Xni¼1
(1 � Fi;n)rxi;n (16)
where xi;n � x2;n � � � � � xn;n is an ordered sample and Fi;n ¼ (iþ a)=(nþ b). a and b are
constants. Gringorten (1963), Cunnane (1978) and Adamowski (1981) have computed values
of these constants. Landwehr et al. (1979) recommended a ¼ �0:35 and b ¼ 0:0 for the
Wakeby distribution. As the generalized Pareto distribution is a special case of it, these
same values turn out be the best choice.
3.4 Maximum Likelihood Estimators (MLE)
The maximum likelihood (ML) equations of Eq. (2) are given (Moharram et al., 1993) as
XNi¼1
(xi � c)=b
1 � a(x1 � c)=b¼
n
1 � a(17)
Xni¼1
ln [1 � a(xi � c)=b] ¼ �na (18)
A maximum likelihood estimator cannot be obtained for c, since the maximum likelihood
function (L) is unbounded with respect to c. However, since c is the lower bound of the random
variable X , one may maximize L subject to the constraint c � x1, the lowest sample value.
Clearly, L is maximum with respect to c when c ¼ x1. Thus, the regular maximum likelihood
estimators (RMLE) are given as c ¼ x1 and the values of a and b given by Eqs. (17) and (18).
This causes a large scale failure of the algorithm, particularly for sample sizes <20. By
ignoring the smallest observation x1, however, the pseudo-MLE (PMLE) can be obtained
as follows: First assume an initial value of (c < x1). Then estimate a and b. Thereafter,
re-estimate c, and then repeat this procedure until the parameters no longer change.
3.5 Entropy (ENT) Estimator
Following Jaynes (1968), Levine and Tribus (1979), Tribus (1969), and Singh and Rajagopal
(1986), the entropy-based parameter estimation equations for GPD3 given by Eq. (2) are
given as
E ln 1 � ax� c
b
� �h i¼ �a (19)
E(x� c)=b
1 � a(x� c)=b
� �¼
1
1 � a(20)
Var ln 1 � ax� c
b
� �h i¼ a2 (21)
where Var[X ] is the variance of the bracketed quantity. For finite samples of observed data,
Eqs. (19) and (20), based on entropy, become equivalent to Eqs. (17) and (18), respectively,
96 V. P. SINGH AND M. AHMAD
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of the MLE method. As the results of MLE and entropy based parameter estimator give the
identical results, therefore, only MLE results are presented in the latter discussions.
4 EXPERIMENTAL DESIGN
To compare the aforementioned parameter estimation methods, error-free data were generated
by Monte Carlo simulations using an experimental design commonly employed in environ-
mental and water resources engineering a brief discussion of which is given in what follows.
4.1 Monte Carlo Simulation
The inverse of Eq. (1) is
x(F) ¼ cþb
a[1 � (1 � F)a], a 6¼ 0 (22)
x ¼ c� b ln (1 � F), a ¼ 0 (23)
where X (P) denotes the quantile of the cumulative probability P or nonexceedance probabil-
ity 1 � P. To assess the performance of the parameter estimation methods outlined above,
Monte Carlo sampling experiments were conducted using Eqs. (22) and (23).
In engineering practice, observed samples may be frequently available, for which the first
three moments (mean, variance and skewness) are computed. In dimensionless terms, the
coefficient of variation and the coefficient of skewness are obtained. Thus, parameter esti-
mates and quantiles for the commonly encountered peak characteristics data are frequently
characterized using the coefficients of variation and skewness. Since this information is read-
ily derivable from a given data set, a potential candidate for estimating the parameters and
quantiles of GPD3 will be the one which performs best in the expected observed ranges
of the coefficient of variation and skewness. Thus, the selection of the population parameter
ranges are very important for any simulation study. To that end, the following considerations
were, therefore, made in this regard:
1. For any given set of data, the distribution parameters are not known in advance; rather
what is usually known is the sample coefficients of variation and skewness, if they exist.
Therefore, if one were somehow able to classify the best estimators with reference to these
readily knowable data characteristics, that would be preferable from a practical standpoint
as compared to classifying the best estimators in terms of an a priori unknown parameter.
2. Not all the estimators perform well in all population ranges, so a range where all the
estimators can be applied has to be selected. Fortunately, in real life most commonly
encountered data lie within the range considered in this study.
3. As the sample skewness in generalized Pareto distribution may correspond to more than
one variances, therefore parameter estimation based on skewness alone is misleading. To
avoid this folly evaluation of the estimators is based on the parameters.
Keeping the above considerations in mind, parameters were investigated using a factorial
experiment within a space spanned by {ai, bj, ck}, where, (ai) ¼ ( � 0:1, � 0:05, 0:0,
0:05, 0:1) , (bj) ¼ (0:25, 0:50), (ck ) ¼ (0:5, 1:0), twenty GPD3 population cases, listed in
Table I, were considered. The ranges of data characteristics were also computed to so that
the results of this study could be related to the commonly used data statistics. For each popu-
lation case, 20,000 samples of size 10, 20, 50, 100, 200, and 500 were generated, and then
parameters and quantiles were estimated.
THREE-PARAMETER GENERALIZED DISTRIBUTION 97
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4.2 Performance Indices
Although standardized forms of bias, standard error and root mean square error are com-
monly used in engineering practice (Kuczera, 1982a,b), the performance of parameter esti-
mators was evaluated by using their well-defined statistical meaning. These performance
indices were defined as:
Standardized bias, BIAS ¼ E(yy) � y (23)
Standard Error, SE ¼ S(yy) (24)
Root Mean Square Error, RMSE ¼ E[(yy� y)2]1=2 (25)
where yy is an estimate of y (parameter or quantile), E[�] denotes the statistical expectation,
and S[�] denotes the standard deviation of the respective random variable. If N is the total
number of random samples then the mean and standard deviation were calculated as
E[yy] ¼1
N
XNi¼1
yyi (26)
S[yy] ¼1
N � 1
XNi¼1
[yyi � E(yy)]2
( )1=2
(27)
The RMSE can also be expressed as
RMSE ¼N � 1
NSE þ BIAS2
� �1=2
(28)
TABLE I GPD3 Population Cases Considered in the Sampling Experiment.
Population parameters Population statistics
a b c Mean Coefficient of variation Skewness
�0.10 0.25 0.5 0.78 0.40 2.811.0 1.28 0.24
0.50 0.5 1.06 0.591.0 1.56 0.40
�0.05 0.25 0.5 0.76 0.36 2.341.0 1.26 0.22
0.50 0.5 1.03 0.541.0 1.53 0.36
0.00 0.25 0.5 0.75 0.33 2.001.0 1.25 0.20
0.50 0.5 1.00 0.501.0 1.50 0.33
0.05 0.25 0.5 0.74 0.31 1.731.0 1.24 0.18
0.50 0.5 0.98 0.471.0 1.48 0.31
0.10 0.25 0.5 0.73 0.29 1.521.0 1.23 0.17
0.50 0.5 0.95 0.431.0 1.45 0.29
98 V. P. SINGH AND M. AHMAD
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These indices were used to measure the variability of parameter and quantile estimates for
each simulation. Although they were used to determine the overall ‘‘best’’ parameter
estimation method, our particular interest lies in the bias and variability of estimates of quan-
tiles in the extreme tails of the distribution (noexceedance probability, P ¼ 0:9, 0:99, 0:999)
when the estimates are based on small samples (N � 50). Due to the limited number of ran-
dom number of samples (20,000 here) used, the results are not expected to reproduce the true
values of BIAS, SE, RMSE, and E[yy], but they do provide a means of comparing the perfor-
mance of estimation methods used. The computed values of bias and RMSE in quantiles are
tabulated as ratios XX (F)=X (F) rather than for the estimator XX (F) itself.
4.3 Robustness
One objective of this study is to identify a robust estimator for GPD3. Kuczera (1982a,b)
defined a robust estimator as the one that is resistant and efficient over a wide range of popu-
lation fluctuations. If an estimator performs steadily without undue deterioration in RMSE
and BIAS, then it can be expected to perform better than other competing estimators
under population conditions different from those on which conclusions were based. Two cri-
teria for identifying a resistant estimator are mini–max and minimum average RMSE
(Kuczera, 1982b). Based on this mini–max criterion, the preferred estimator is the one
whose maximum RMSE for all population cases is minimum. The minimum average criter-
ion is to select the estimator whose RMSE average over the test cases is minimum.
5 RESULTS AND DISCUSSION
The performance of a parameter estimator depends on (1) sample size, N , (2) population para-
meters, (3) distribution parameter, and (4) the probability of exceedence or quantile. Out of 20
population cases considered, results for the five cases are summarized in Tables II–V.
5.1 BIAS in Parameter Estimates
Table II displays the results of bias in parameters. For shape parameter (a), for small sample
sizes (N � 50) MM2 outperformed the other methods, although it tended to underestimate a
for increasing population values (i.e., decreasing skewness). RME and MM1 also exhibited
lesser bias, particularly for increasing population values of a, for smaller sample sizes
(N � 20). All three moment-based methods tended to estimate a without responding to
changes in b and c. MLE did not do well for small sample sizes but improved consistently
while its bias increased with increasing population values of a. PWM exhibited less bias
for a > 0. From the bias results of all 20 population cases, it was concluded that MM2
would be preferred for all sample sizes. However, for increasing sample sizes (N � 50)
PWM and MLE would be acceptable.
For scale parameter (b), RME, MM1 and MM2 responded linearly to the change in popu-
lation values of b. MM1 and MM2 performed better for all sample sizes and population
cases. For larger sample sizes (N � 100), all the estimators, except RME which had a higher
bias for a < 0, showed comparable absolute bias. As far as the location parameter (c) is con-
cerned, all the methods, except MLE, showed a negative bias. The reason for the positive bias
of MLE is rooted in the solution procedure adopted for MLE, discussed earlier. Another
important finding in this regard is that the bias in c was dictated only by the population values
of a and b.
THREE-PARAMETER GENERALIZED DISTRIBUTION 99
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TA
BL
EII
Bia
sin
the
Par
amet
ers
of
GD
P3
for
Sp
ace
Sp
ann
edbya¼
(�0
.1,�
0.0
5,
0.0
,0
.05,
0.1
),b¼
0.5
0,c¼
1.0
.
BIASparametersofGPD3
Size
Method
BIAS(a)
BIAS(b)
BIAS(c)
10
RM
E0
.324
0.2
96
0.2
63
0.2
32
0.1
99
0.2
76
0.2
36
0.2
01
0.1
70
0.1
41
�0
.08
5�
0.0
70
�0
.05
5�
0.0
46
�0
.03
7M
M1
0.2
66
0.2
52
0.2
37
0.2
20
0.2
09
0.1
61
0.1
48
0.1
41
0.1
32
0.1
24
�0
.01
2�
0.0
10
�0
.00
9�
0.0
09
�0
.00
9M
M2
0.0
24
�0
.005
�0
.032
�0
.062
�0
.089
�0
.082
�0
.094
�0
.103
�0
.112
�0
.120
0.1
00
0.0
96
0.0
93
0.0
89
0.0
85
PW
M0
.188
0.1
75
0.1
59
0.1
40
0.1
25
0.1
40
0.1
31
0.1
24
0.1
14
0.1
06
�0
.04
1�
0.0
40
�0
.03
7�
0.0
36
�0
.03
6M
LE
0.2
59
0.2
67
0.2
75
0.2
75
0.2
89
0.1
68
0.1
80
0.1
94
0.1
89
0.2
04
0.0
51
0.0
50
0.0
51
0.0
50
0.0
49
20
RM
E0
.230
0.2
06
0.1
81
0.1
58
0.1
40
0.1
99
0.1
67
0.1
42
0.1
15
0.0
97
�0
.06
9�
0.0
54
�0
.04
3�
0.0
32
�0
.02
4M
M1
0.1
39
0.1
27
0.1
13
0.1
02
0.0
97
0.0
76
0.0
67
0.0
62
0.0
54
0.0
51
�0
.00
3�
0.0
03
�0
.00
2�
0.0
02
�0
.00
2M
M2
0.0
37
0.0
17
�0
.004
�0
.022
�0
.037
�0
.032
�0
.041
�0
.047
�0
.055
�0
.060
0.0
52
0.0
50
0.0
48
0.0
46
0.0
44
PW
M0
.091
0.0
83
0.0
71
0.0
62
0.0
54
0.0
65
0.0
60
0.0
57
0.0
51
0.0
46
�0
.01
8�
0.0
18
�0
.01
7�
0.0
16
�0
.01
5M
LE
0.1
34
0.1
38
0.1
42
0.1
42
0.1
50
0.0
90
0.0
97
0.1
05
0.1
02
0.1
10
0.0
25
0.0
25
0.0
25
0.0
25
0.0
25
50
RM
E0
.136
0.1
12
0.0
94
0.0
77
0.0
64
0.1
24
0.0
96
0.0
77
0.0
59
0.0
46
�0
.05
0�
0.0
36
�0
.02
6�
0.0
18
�0
.01
3M
M1
0.0
65
0.0
52
0.0
46
0.0
40
0.0
36
0.0
34
0.0
26
0.0
25
0.0
21
0.0
18
�0
.00
10
.00
00
.00
00
.00
00
.00
0M
M2
0.0
28
0.0
13
0.0
04
�0
.006
�0
.014
�0
.007
�0
.014
�0
.017
�0
.022
�0
.025
0.0
22
0.0
21
0.0
20
0.0
19
0.0
18
PW
M0
.038
0.0
30
0.0
27
0.0
22
0.0
18
0.0
26
0.0
21
0.0
22
0.0
19
0.0
16
�0
.00
7�
0.0
07
�0
.00
6�
0.0
06
�0
.00
5M
LE
0.0
56
0.0
58
0.0
59
0.0
59
0.0
62
0.0
31
0.0
33
0.0
36
0.0
35
0.0
37
0.0
10
0.0
10
0.0
10
0.0
10
0.0
10
10
0R
ME
0.0
90
0.0
70
0.0
56
0.0
43
0.0
35
0.0
86
0.0
63
0.0
46
0.0
34
0.0
26
�0
.03
8�
0.0
25
�0
.01
7�
0.0
11
�0
.00
7M
M1
0.0
35
0.0
28
0.0
23
0.0
20
0.0
18
0.0
18
0.0
14
0.0
12
0.0
10
0.0
09
0.0
00
0.0
00
0.0
00
0.0
00
0.0
00
MM
20
.018
0.0
09
0.0
03
�0
.003
�0
.007
�0
.001
�0
.006
�0
.009
�0
.011
�0
.012
0.0
11
0.0
10
0.0
10
0.0
09
0.0
09
PW
M0
.018
0.0
15
0.0
13
0.0
10
0.0
09
0.0
12
0.0
11
0.0
10
0.0
09
0.0
08
�0
.00
3�
0.0
03
�0
.00
3�
0.0
03
�0
.00
3M
LE
0.0
27
0.0
27
0.0
28
0.0
28
0.0
30
0.0
13
0.0
13
0.0
14
0.0
14
0.0
15
0.0
05
0.0
05
0.0
05
0.0
05
0.0
05
20
0R
ME
0.0
57
0.0
44
0.0
32
0.0
25
0.0
17
0.0
57
0.0
40
0.0
27
0.0
20
0.0
13
�0
.02
6�
0.0
16
�0
.01
0�
0.0
06
�0
.00
4M
M1
0.0
19
0.0
15
0.0
12
0.0
10
0.0
08
0.0
10
0.0
08
0.0
06
0.0
05
0.0
04
0.0
00
0.0
00
0.0
00
0.0
00
0.0
00
MM
20
.010
0.0
06
0.0
02
�0
.001
�0
.004
0.0
00
�0
.002
�0
.004
�0
.005
�0
.006
0.0
05
0.0
05
0.0
05
0.0
05
0.0
04
PW
M0
.009
0.0
08
0.0
07
0.0
06
0.0
03
0.0
06
0.0
06
0.0
05
0.0
05
0.0
04
�0
.00
2�
0.0
02
�0
.00
1�
0.0
01
�0
.00
1M
LE
0.0
12
0.0
12
0.0
13
0.0
13
0.0
13
0.0
07
0.0
07
0.0
08
0.0
07
0.0
08
0.0
02
0.0
02
0.0
02
0.0
02
0.0
02
50
0R
ME
0.0
32
0.0
23
0.0
15
0.0
10
0.0
08
0.0
33
0.0
21
0.0
14
0.0
08
0.0
06
�0
.01
6�
0.0
09
�0
.00
5�
0.0
03
�0
.00
2M
M1
0.0
08
0.0
07
0.0
04
0.0
04
0.0
04
0.0
04
0.0
04
0.0
02
0.0
02
0.0
02
0.0
00
0.0
00
0.0
00
0.0
00
0.0
00
MM
20
.005
0.0
03
0.0
00
0.0
00
�0
.001
0.0
01
0.0
00
�0
.002
�0
.002
�0
.002
0.0
02
0.0
02
0.0
02
0.0
02
0.0
02
PW
M0
.003
0.0
04
0.0
02
0.0
02
0.0
02
0.0
02
0.0
03
0.0
02
0.0
02
0.0
02
�0
.00
1�
0.0
01
�0
.00
1�
0.0
01
0.0
00
ML
E0
.004
0.0
04
0.0
04
0.0
04
0.0
05
0.0
02
0.0
02
0.0
03
0.0
03
0.0
03
0.0
01
0.0
01
0.0
01
0.0
01
0.0
01
100 V. P. SINGH AND M. AHMAD
Dow
nloa
ded
by [
Uni
vers
ity o
f B
ath]
at 1
1:12
02
Nov
embe
r 20
14
TA
BL
EII
IR
MS
Ein
the
Par
amet
ers
of
GD
P3
for
Spac
eS
pan
ned
bya¼
(�0
.1,�
0.0
5,
0.0
,0
.05,
0.1
),b¼
0.5
0,c¼
1.0
.
RMSEin
parametersofGPD3
Size
Method
RMSE(a)
RMSE(b)
RMSE(c)
10
RM
E0
.408
0.3
89
0.3
64
0.3
44
0.3
24
0.4
35
0.3
93
0.3
53
0.3
23
0.2
97
0.1
51
0.1
28
0.1
12
0.0
99
0.0
90
MM
10
.489
0.4
95
0.4
96
0.5
02
0.5
13
0.3
91
0.3
88
0.3
85
0.3
88
0.3
88
0.0
61
0.0
59
0.0
61
0.0
60
0.0
59
MM
20
.244
0.2
52
0.2
64
0.2
75
0.2
93
0.2
05
0.2
09
0.2
13
0.2
14
0.2
20
0.1
15
0.1
10
0.1
08
0.1
03
0.0
99
PW
M0
.386
0.3
81
0.3
72
0.3
61
0.3
59
0.3
22
0.3
12
0.3
03
0.2
91
0.2
84
0.0
78
0.0
75
0.0
74
0.0
71
0.0
70
ML
E0
.456
0.4
10
0.3
90
0.3
98
0.3
86
0.3
79
0.3
22
0.3
06
0.3
13
0.3
03
0.0
72
0.0
71
0.0
72
0.0
70
0.0
69
20
RM
E0
.300
0.2
85
0.2
69
0.2
57
0.2
49
0.2
98
0.2
65
0.2
44
0.2
22
0.2
08
0.1
13
0.0
93
0.0
79
0.0
68
0.0
62
MM
10
.270
0.2
71
0.2
71
0.2
69
0.2
79
0.2
04
0.1
98
0.1
97
0.1
94
0.1
98
0.0
27
0.0
27
0.0
26
0.0
26
0.0
27
MM
20
.189
0.1
91
0.1
95
0.1
99
0.2
10
0.1
49
0.1
47
0.1
49
0.1
51
0.1
54
0.0
59
0.0
56
0.0
54
0.0
52
0.0
51
PW
M0
.263
0.2
58
0.2
52
0.2
46
0.2
43
0.2
08
0.1
99
0.1
96
0.1
88
0.1
85
0.0
49
0.0
47
0.0
46
0.0
45
0.0
44
ML
E0
.305
0.2
74
0.2
61
0.2
66
0.2
58
0.2
47
0.2
10
0.2
00
0.2
04
0.1
98
0.0
36
0.0
35
0.0
35
0.0
35
0.0
35
50
RM
E0
.187
0.1
72
0.1
62
0.1
53
0.1
49
0.1
82
0.1
58
0.1
43
0.1
31
0.1
23
0.0
75
0.0
60
0.0
50
0.0
44
0.0
40
MM
10
.154
0.1
50
0.1
48
0.1
47
0.1
50
0.1
12
0.1
09
0.1
07
0.1
05
0.1
06
0.0
10
0.0
10
0.0
10
0.0
10
0.0
10
MM
20
.132
0.1
30
0.1
30
0.1
30
0.1
35
0.0
97
0.0
97
0.0
95
0.0
95
0.0
97
0.0
24
0.0
23
0.0
22
0.0
21
0.0
21
PW
M0
.164
0.1
60
0.1
56
0.1
53
0.1
52
0.1
23
0.1
19
0.1
16
0.1
13
0.1
12
0.0
29
0.0
27
0.0
27
0.0
26
0.0
26
ML
E0
.182
0.1
64
0.1
56
0.1
59
0.1
54
0.1
39
0.1
18
0.1
13
0.1
15
0.1
11
0.0
14
0.0
14
0.0
14
0.0
14
0.0
14
10
0R
ME
0.1
33
0.1
22
0.1
14
0.1
08
0.1
03
0.1
27
0.1
12
0.0
99
0.0
91
0.0
86
0.0
56
0.0
44
0.0
36
0.0
32
0.0
29
MM
10
.106
0.1
03
0.1
01
0.1
00
0.1
01
0.0
75
0.0
74
0.0
72
0.0
72
0.0
72
0.0
05
0.0
05
0.0
05
0.0
05
0.0
05
MM
20
.098
0.0
95
0.0
94
0.0
94
0.0
96
0.0
69
0.0
69
0.0
68
0.0
68
0.0
69
0.0
12
0.0
12
0.0
11
0.0
11
0.0
10
PW
M0
.115
0.1
12
0.1
10
0.1
08
0.1
07
0.0
83
0.0
83
0.0
81
0.0
79
0.0
78
0.0
20
0.0
19
0.0
19
0.0
18
0.0
18
ML
E0
.113
0.1
02
0.0
97
0.0
99
0.0
96
0.0
83
0.0
71
0.0
67
0.0
69
0.0
67
0.0
07
0.0
07
0.0
07
0.0
07
0.0
07
20
0R
ME
0.0
99
0.0
91
0.0
83
0.0
77
0.0
72
0.0
92
0.0
81
0.0
71
0.0
65
0.0
61
0.0
40
0.0
32
0.0
27
0.0
24
0.0
21
MM
10
.078
0.0
74
0.0
71
0.0
69
0.0
70
0.0
54
0.0
52
0.0
51
0.0
50
0.0
50
0.0
02
0.0
02
0.0
02
0.0
02
0.0
02
MM
20
.074
0.0
71
0.0
69
0.0
67
0.0
68
0.0
51
0.0
50
0.0
49
0.0
49
0.0
49
0.0
06
0.0
06
0.0
06
0.0
05
0.0
05
PW
M0
.083
0.0
80
0.0
78
0.0
76
0.0
76
0.0
60
0.0
58
0.0
56
0.0
56
0.0
55
0.0
14
0.0
13
0.0
13
0.0
13
0.0
12
ML
E0
.079
0.0
71
0.0
67
0.0
69
0.0
67
0.0
58
0.0
49
0.0
46
0.0
47
0.0
46
0.0
04
0.0
04
0.0
04
0.0
04
0.0
04
50
0R
ME
0.0
68
0.0
62
0.0
55
0.0
50
0.0
47
0.0
63
0.0
55
0.0
47
0.0
43
0.0
39
0.0
28
0.0
22
0.0
18
0.0
16
0.0
14
MM
10
.050
0.0
47
0.0
44
0.0
43
0.0
43
0.0
34
0.0
33
0.0
32
0.0
31
0.0
31
0.0
01
0.0
01
0.0
01
0.0
01
0.0
01
MM
20
.049
0.0
46
0.0
44
0.0
43
0.0
43
0.0
33
0.0
32
0.0
31
0.0
31
0.0
31
0.0
02
0.0
02
0.0
02
0.0
02
0.0
02
PW
M0
.052
0.0
50
0.0
49
0.0
48
0.0
48
0.0
37
0.0
36
0.0
36
0.0
35
0.0
35
0.0
09
0.0
08
0.0
08
0.0
08
0.0
08
ML
E0
.047
0.0
42
0.0
40
0.0
41
0.0
40
0.0
34
0.0
29
0.0
27
0.0
28
0.0
27
0.0
01
0.0
01
0.0
01
0.0
01
0.0
01
THREE-PARAMETER GENERALIZED DISTRIBUTION 101
Dow
nloa
ded
by [
Uni
vers
ity o
f B
ath]
at 1
1:12
02
Nov
embe
r 20
14
TA
BL
EIV
Bia
sin
the
Qu
anti
les
of
GD
P3
for
Sp
ace
Sp
ann
edbya¼
(�0
.1,�
0.0
5,
0.0
,0
.05,
0.1
),b¼
0.5
0,c¼
1.0
.
BIASin
quantilesofGPD3
F(x)¼0.900
F(x)¼0.950
F(x)¼0.999
Size
Method
(X(F)¼1.795,1.720,1.651,1.587,1.528)
(X(F)¼2.246,2.116,1.998,1.891,1.794)
(X(F)¼5.476,4.625,3.953,3.421,2.994)
10
RM
E�
0.0
18
�0
.034
�0
.04
5�
0.0
53
�0
.071
�0
.073
�0
.081
�0
.08
4�
0.0
86
�0
.097
�0
.342
�0
.30
2�
0.2
60
�0
.219
�0
.189
MM
1�
0.0
35
�0
.038
�0
.03
6�
0.0
37
�0
.039
�0
.077
�0
.075
�0
.06
7�
0.0
63
�0
.061
�0
.201
�0
.16
1�
0.1
24
�0
.086
�0
.058
MM
2�
0.0
85
�0
.085
�0
.07
8�
0.0
74
�0
.071
�0
.104
�0
.096
�0
.08
3�
0.0
72
�0
.064
�0
.030
0.0
32
0.0
92
0.1
51
0.1
96
PW
M�
0.0
07
�0
.008
�0
.00
3�
0.0
01
0.0
00
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102 V. P. SINGH AND M. AHMAD
Dow
nloa
ded
by [
Uni
vers
ity o
f B
ath]
at 1
1:12
02
Nov
embe
r 20
14
TA
BL
EV
RM
SE
inth
eQ
uan
tile
so
fG
DP
3fo
rS
pac
eS
pan
ned
bya¼
(�0
.1,�
0.0
5,
0.0
,0
.05,
0.1
),b¼
0.5
0,c¼
1.0
.
RMSEin
quantilesofGPD3
F(x)¼0.900
F(x)¼0.950
F(x)¼0.999
Size
Method
(X(F)¼1.795,1.720,1.651,1.587,1.528)
(X(F)¼2.246,2.116,1.998,1.891,1.794)
(X(F)¼5.476,4.625,3.953,3.421,2.994)
10
RM
E0
.28
90
.26
20
.23
60
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70
.204
0.3
16
0.2
87
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58
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36
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94
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10
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40
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90
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00
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40
.190
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87
0.2
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0.2
28
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0.5
32
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20
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27
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50
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.13
20
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80
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MM
10
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90
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20
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MM
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70
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10
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10
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MM
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0.2
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0.0
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MM
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0.0
58
THREE-PARAMETER GENERALIZED DISTRIBUTION 103
Dow
nloa
ded
by [
Uni
vers
ity o
f B
ath]
at 1
1:12
02
Nov
embe
r 20
14
5.2 RMSE in Parameter Estimates
The root mean square error in the results of parameter estimates is summarized in Table III.
MM2 did particularly well in estimating a for small sample sizes (N � 20) but its improve-
ment was slower with the increasing sample sizes. It did not perform as well in estimating c
in all ranges. MM2 did not perform as good for c as it did in estimating a and b. MM1 per-
formed good only for c but for other parameters, it showed a pattern similar to that of RME.
PWM demonstrated a consistent improvement for all population cases as sample size
increased. MLE did not do well for when N � 20; however, as the sample size increased,
it outperformed other methods. With increasing population a, MLE exhibited a consistent
improvement in RMSE for both a and b. RME, MM1 and MM2 did not respond to the
changes in population c and b while estimating parameter a, as was indicated by the bias
results earlier.
5.3 BIAS in Quantile Estimates
The BIAS results of quantile estimation by GPD3 are summarized in Table IV. In general, for
all nonexceedance probabilities, among the three moment based methods RME performed
better for lower values of P in terms of quantile bias when a > 0 and sample size
N � 20. As the sample size increased (N � 50), MM1 and MM2 performed comparatively
better than did RME in all ranges. Generally, the moment-based methods exhibited a negative
bias but for P ¼ 0:999 when a � 0 and N � 50, MM2 showed a positive bias. RME, MM1
and MM2 tended to further underestimate the quantiles with increasing population c for a
given value of b for P � 0:995. However, for P ¼ 0:999 the trend was reversed. On the
whole, with increasing b the absolute bias of these methods increased for sample sizes
N � 10. PWM and MLE outperformed the other methods in terms of the absolute value
of the bias for all sample sizes and quantile ranges. PWM and MLE responded positively
in terms of bias to both b and c when one was varied keeping the other constant.
5.4 RMSE in Quantile Estimates
The root mean square error in quantile estimates for P¼ 0.9, 0.95, and 0.999 is given in
Table V for selected five population cases. For probability of nonexceedance P � 0:90,
PWM exhibited the least RMSE for small samples (N > 20) when a � 0 for b ¼ 0:25
and c ¼ 0:50. As the population b and c increased, MM1 and MM2 showed better results
in terms of RMSE for all quantiles and sample ranges. The RME performance was better
for small sample sizes when P � 0:90. For the small sample sizes, MLE did poorly but as
the sample size increased, both PWM and MLE exhibited a consistent improvement. MLE
did perform best only when N � 50 with a > 0. RMSE of the quintiles responded positively
to the increase in population b, while the opposite was seen in case of c.
5.5 Robustness Evaluation
The relative robustness of different methods of parameter and quantile estimation can be
judged from Tables II and III. In terms of parameter bias, MM2 and PWM performed in a
superior manner for most of the data ranges, whereas MLE performed better for large sample
sizes. In terms of parameter RMSE, RME did well in small sample sizes. As for the bias in
quintiles was concerned, PWM and MLE performed better. For RMSE in the quantiles, MM1
and MM2 did better in most cases. On the whole, PWM showed the most consistent behavior.
Thus, PWM would be the preferred method.
104 V. P. SINGH AND M. AHMAD
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6 CONCLUDING REMARKS
An evaluation of the relative performance of six methods for estimating parameters and quan-
tiles of the 3-parameter GPD was performed. No one method is preferable to another for all
population cases considered. The weakness in RME stems from the fact that shape para-
meters is solely dependent on the sample skewness which may not be the true representative
of the population skewness. MLE, on the other hand, is handicapped by the fact that for small
sample sizes the maximum likelihood estimator of GPD3 is obtained only through pseudo-
convergence. The modified versions of the method of the moments, namely MM1 and MM2,
do incorporate sample mean and variance in computing a thus having better data inference
into it. However, MM1 may behave erratically as a approaches zero. MM2 has a general ten-
dency to underestimate both shape parameter (a) and scale parameter (b). On the whole,
PWM performs in a consistent fashion. In case, a clear choice of a particular method is in
doubt, PWM can be most reliable and can be the preferred method.
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