Direct estimation of quantile functions using the maximum entropy principle

Structural Safety 22 (2000) 61±79

www.elsevier.nl/locate/strusafe

Direct estimation of quantile functions using the maximum entropy principle M.D. Pandey* Department of Civil Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada

Abstract The paper presents a distribution free method for estimating the quantile function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints speci®ed in terms of the probability-weighted moments estimated from observed data. Traditionally, MaxEnt is used for estimating the probability density function under speci®ed moment constraints. The density function is then integrated to obtain the cumulative distribution function, which needs to be inverted to obtain a quantile corresponding to some speci®ed probability. For correct modelling of the distribution tail, higher order moments must be considered in the analysis. However, the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain. The diculty in obtaining accurate moment estimates from small samples has been the main impediment to the application of the MaxEnt Principle in extreme quantile estimation. The present paper is an attempt to overcome this problem by the use of probability-weighted moments (PWMs), which are essentially the expectations of order statistics. In contrast with ordinary statistical moments, higher order PWMs can be accurately estimated from small samples. By interpreting the PWM as the moment of quantile function, the paper derives an analytical form of quantile function using MaxEnt principle. Monte Carlo simulations are performed to assess the accuracy of MaxEnt quantile estimates computed from small samples. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Probability; Data analysis; Quantile; Information theory; Entropy; Maximum entropy principle; Probability-weighted moment; Order statistics

1. Introduction The problem of estimation of extreme quantiles corresponding to small probabilities of exceedance (POE) is commonly encountered in the risk analysis of engineering systems. Such extreme quantiles often represent the design values of loads and material properties speci®ed by design codes. It is desirable that the quantile estimate be unbiased, i.e. its expected value should be equal * Tel.: +1-519-888-4567; fax: +1-519-888-6197. E-mail address: [email protected] (M.D. Pandey). 0167-4730/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0167-4730(99)00041-7

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M.D. Pandey / Structural Safety 22 (2000) 61±79

to the true value. Furthermore, the unbiased estimate should ideally be ecient, i.e. its variance should be as small as possible. The unbiased and ecient estimation of extreme quantiles is often problematic because of the lack of adequate data. For example, from 20 to 30 samples, the engineer may have to extrapolate a design value corresponding to POE of the order of 10ÿ2±10ÿ4. The ®rst step in quantile estimation involves ®tting an analytical probability distribution for adequate representation of sample observations. To achieve this, the distribution type is judged empirically from the available data, and then distribution parameters are suitably estimated using methods such as the maximum likelihood method, the method of least squares, and the method of moments. However, the bias and eciency of quantile estimates remains sensitive to the type of assumed distribution. An alternative approach to the distribution ®tting comes from the modern information theory in which a mathematically robust measure of probabilistic information, namely, the entropy, has been developed. Jaynes [1] presented the maximum entropy principle (MaxEnt) as a rational approach for choosing the most unbiased probability distribution, amongst all possible distributions, which is consistent with available data and contains minimum spurious information. When only moment constraints are speci®ed, Shore and Johnson [2] proved that the entropy maximization is a uniquely correct method of probabilistic inference that satis®es all consistency axioms. Although the MaxEnt approach, given the moment constraints, leads to the most unbiased distribution, the main problem remains the accurate estimation of moments from limited data. It is well known that the estimates of higher order moments (order >2) from small samples (size less than 50) tend to be highly biased, as shown in Section 4.1. Obviously, the MaxEnt distribution derived from poor moment estimates would lead to inaccurate quantile values. The diculty in obtaining accurate moment estimates from small samples has been the main impediment to the application of the MaxEnt principle in quantile estimation. The present paper is an attempt to overcome this problem by the use of probability-weighted moments (PWMs). The PWMs are essentially the expectations of order statistics, which can be interpreted as moments of the quantile function of any non-negative random variable, as shown in the paper. The quantile function means the inverse of the cumulative distribution function. In contrast with ordinary statistical moments, the main advantage of using PWMs is that their higher order values can be accurately estimated from small samples. Also, PWMs are shown to be fairly insensitive to outliers [3]. The paper applies the maximum entropy principle to derive an unbiased approximation of quantile function based on estimates of the probability-weighted moments obtained from observed data, such as material test data, live loads, and wave heights etc. This approach is conceptually elegant as there is no assumption involved regarding the type of the probability distribution, and it obviates the need for inverting the distribution function. To assess the bias and eciency of MaxEnt quantile estimates, Monte Carlo simulations were performed. 2. Theory of probability-weighted moments 2.1. Expectations of order statistics Consider a sample consisting of n observations, {X1, X2,..., Xn}, randomly drawn from a statistical population. If the sample values are rearranged in an increasing order of magnitude, X1:n

M.D. Pandey / Structural Safety 22 (2000) 61±79

63

< X2:n <...
…1†

kˆr

which can be written in terms of an incomplete Beta function [4]  … F…x† n F…r† …x† ˆ r urÿ1 …1 ÿ u†nÿr du r 0 The probability density function of Xr:n is given by the ®rst derivative of Eq. (2):   n f…r† …x† ˆ r ‰F…x†Šrÿ1 ‰1 ÿ F…x†Šnÿr f…x† r

…2†

…3†

Now the expected value of rth order statistics can be obtained as E‰Xr:n Š ˆ

…1 ÿ1

xf…r† …x†dx

…4†

Substituting from Eq.(3) into Eq.(4) and introducing a transformation, u ˆ F…x† or x ˆ Fÿ1 …u†, 04u41, leads to  … 1 n x…u†urÿ1 …1 ÿ u†nÿr du …5† E‰Xr:n Š ˆ r r 0 Note that x…u† denotes the quantile function of a random variable. The expectation of the maximum and minimum of a sample of size n can be easily obtained from Eq. (5) by setting r ˆ n and r ˆ 1, respectively. …l

E‰Xn:n Š ˆ n x…u†u 0

nÿ1

…1 du; and E‰X1:n Š ˆ n x…u†…1 ÿ u†nÿ1 du 0

…6†

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M.D. Pandey / Structural Safety 22 (2000) 61±79

2.2. Probability-weighted moment The probability-weighted moment (PWM) of a random variable was formally de®ned by Greenwood et al. [5] as Mi;j;k

  ˆ E Xi uj …1 ÿ u†k ˆ

…1 0

x…u†i uj …1 ÿ u†k du

…7†

The following two forms of PWM are particularly simple and useful: Type 1 : k ˆ M1;0;k ˆ

…1 0

x…u†…1 ÿ u†k du

…k ˆ 0; 1; . . . n

…8†

and Type 2 : k ˆ M1;k;0 ˆ

…1 0

x…u†uk du

…k ˆ 0; 1; . . . n†

…9†

Comparing Eq. (6) with Eqs.(8) and (9), it can be seen that k and k , respectively, are related to the expectations of the minimum and maximum in a sample of size k 1 1 k ˆ E‰X1:n Š; k ˆ E‰Xn:n Š …k51† k k

…10†

In essence, PWMs are the normalized expectations of maximum/minimum of k random observations; the normalization is done by the sample size (k) itself. From an ordered random sample of size n, unbiased estimates bk and ak of k and k , respectively, can be obtained as [3] bk ˆ

   n  1X iÿ1 nÿ1 Xi …k ˆ 0; 1; . . . …n ÿ 1†† k k n iˆ1

   n  1X nÿ1 nÿ1 Xi ak ˆ k k n iˆ1

…k ˆ 0; 1; . . . …n ÿ 1††

…11†

…12†

The use of PWMs is fairly common in hydrology for distribution ®tting and parameter estimation [6]. 2.3. PWMs as moments of quantile function For a non-negative random variable, the k can be interpreted as moments of the quantile function. Recall the de®nition of an ordinary statistical moment of kth order (k51)

M.D. Pandey / Structural Safety 22 (2000) 61±79

  E Xk ˆ

…

k

R

x f…x†dx ˆ

…1

f…x†dx ‰x…u†Šk du where du ˆ dF…x† ˆ „ f…x†dx 0

65

…13†

where du ˆ dF…x† is a probability measure, which is a monotonic, continuous and non-negative function with 04F…x†41. Comparing Eqs. (9) and (13), x…u† is analogous to f…x†, whereas the probability u is analogous to the random variable x. The quantile function is known to be a continuous, monotonic function [5], and in the present case it is non-negative, since, 04x…u† <1 for 0 4u41. Now, a probability measure, dT…u†, can be introduced using the following normalizing transformation [7,8]. x…u†du x…u†du dT…u† ˆ „ 1 ˆ 0 0 x…u†du

…14†

where 0 …ˆ 0 † is the area under the quantile function, and is also equal to the average of the random variable. Thus the PWM can be rede®ned using a new measure, dT as …1 …15† k ˆ 0 uk dT…u† …k ˆ 1; 2; . . . n† 0

Comparing Eqs. (13) and (15), ( k = 0 ) can be interpreted as the kth moment of the quantile function, x(u), 00 and pk=1. The self-information of the event Ek is de®ned as [9]:   1 ˆ ÿ log…pk † S…Ek † ˆ log …16† pk The use of a logarithmic measure for information, ®rstly introduced by Hartley [10], is intuitive in the following sense: the information provided by a deterministic event (i.e. pk ˆ 1) is zero, and the rarer an event is (pk <<1), the more information is conveyed by its realization. It is clear from Eq.(16) that the self-information of an event increases as its uncertainty grows, i.e. the probability of occurrence reduces. In this respect, S is regarded as a measure of uncertainty [9].

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3.2. Entropy If the probabilities of various outcomes of the random experiment E are known a priori, can we predict the outcome (Ek) of the experiment in advance? The degree of diculty of this prediction is dependent on the overall uncertainty associated with E. Shannon [11] de®ned a measure of uncertainty, referred to as entropy, similar to that used in thermodynamics and statistical mechanics. Considering the de®nition of self-information from Eq. (16), Shannon's entropy for a random event can be expressed as mathematical expectation of the self-information: n X H…p† ˆ ÿ pk lnpk

…17†

kˆ1

The entropy is a positive, permutationally symmetric quantity which vanishes for completely certain outcome, and is maximum when all outcomes are equi-probable. The axiomatic characterization of entropy and its other mathematical properties can be found in Kapur and Kesavan [12]. For a continuous random variable, x, with the density function f…x†, the entropy is de®ned as … H‰f…x†Š ˆ ÿ f…x†‰lnf…x†Šdx …18† R

3.3. Maximum entropy (MaxEnt) principle Jaynes [1] presented the maximum entropy (MaxEnt) principle as a rational approach for choosing a consistent probability distribution, amongst all possible distributions, that contains minimum spurious information. The principle states that the most unbiased estimate of a probability distribution is that which maximizes the entropy subject to constraints supplied by the available information, e.g. moments of a random variable. The distribution so obtained is referred to as the most unbiased, because its derivation involves a systematic maximization of uncertainty about the unknown information. The consistency is a fundamental requirement in mathematical analysis, i.e. if a quantity can be found in more than one way, the results obtained by di€erent methods must be the same. To ensure this, the inference method must satisfy some basic conditions, referred to as consistency axioms. Shore and Johnson [2] postulated four such axioms. 1. Uniqueness: the result should be unique. 2. Invariance: the result should be independent of the axes of reference. 3. System independence: the result should be independent of the form of information about various independent systems (i.e. consideration of marginals versus a joint density should not matter). 4. Subset independence: the same as (3), but applicable to independent subset of a system. Given the moment constraints only, Shore and Johnson [1] proved that the entropy maximization is a uniquely correct method of probabilistic inference that satis®es all the consistency axioms.

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67

3.4. MaxEnt quantile function The entropy of a quantile function can be written as …1 H…u† ˆ ÿ ‰x…u†lnx…u†Šdu

…19†

0

and the available information is presented in terms of PWMs …1 0

uk x…u†du ˆ bk

…k ˆ 0; 1; . . . N†

…20†

where bk is a sample estimate of population PWM, k , and N is the highest order of PWM considered in the analysis. Note that x…u† in Eq. (19) is not normalized, rather the normalizing condition is included as an external constraint in Eq. (20) corresponding to k ˆ 0. To account for the constraints (20), the entropy function is augmented as … 1  X … 1  N k H ˆ ÿ ‰x…u† ln x…u†Šdu ÿ …l0 ÿ 1† x…u†du ÿ b0 ÿ lk u x…u†du ÿ bk …1 0

0

kˆ1

0

…21†

where lk denotes an unknown Lagrangian multiplier. Note that (l0ÿ1) is used as the ®rst multiplier instead of l0 as a matter of convenience. To derive the quantile function, the entropy is maximized using the usual condition @H ˆ0 @x…u†

…22†

Substitution from Eq. (21) into (22) and subsequent simpli®cation leads to the following solution [12]: " # N X …23† x…u† ˆ exp ÿ lk uk kˆ0

The Lagrangian multipliers are determined by solving a system of nonlinear Eq. (20) along with Eq. (23) by the method of nonlinear least square that utilizes a quasi Newton optimization algorithm [13]. If the Type 1 PWMs, namely, k , were speci®ed as constraints instead of k , the quantile function can be derived as a function of the exceedance probability, q ˆ …1 ÿ u†, similar to Eq. (23). "

N X x…q† ˆ exp ÿ lk qk kˆ0

# …24†

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4. Numerical results 4.1. General To illustrate the biased nature of moment estimates obtained from small samples, a Monte Carlo experiment involving the generalized Pareto distribution (GPD) is considered. The mathematical expressions of ordinary moments and PWMs of GPD are given in Appendix. For preselected values of shape (c) and scale (d) parameters, a sample of size n is simulated and its standard deviation, skewness, and kurtosis are estimated [14]. The population moments are computed from Eq. (A.4) given in the Appendix to assess the normalized bias and root mean square error (RMSE) of sample estimates. Let Dk denote the di€erence between the kth sample estimate of a moment (or quantile) and the exact X value obtained from the parent distribution. Then the bias is de®ned as the average of Dk †=M, M being the number of simulation samples. The root mean square error Dk , i.e. …  q X (RMSE) is de®ned as … D2k †=M. Both bias and RMSE are normalized by the exact quantile value. The normalized bias and RMSE of GPD moments for c ˆ ÿ0:2 and d ˆ 1:0 are plotted in Figs. 1 and 2, respectively, for the sample size ranging from 10 to 50. This exercise was repeated

Fig. 1. The bias of moment estimates obtained from various samples sizes.

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69

Fig. 2. The RMSE of moment estimates obtained from various samples sizes.

to estimate the bias and RMSE of estimates of k . Exact values and sample estimates of k are calculated from Eqs. (A.3) and (12), respectively, and results are shown in Figs. 3 and 4. Fig. 1 shows that the sample size less than 30 results in large underestimation of skewness and kurtosis, whereas the bias of comparable order of k …k ˆ 2 ÿ 4† is almost zero, as shown in Fig. 3. Comparing Figs. 2 and 4, it is concluded that the RMSE of PWM estimates is smaller than that of ordinary moments. This illustration also highlights the limitation of MaxEnt inference based on the small sample estimates of moments. 4.2. MaxEnt approximation of quantile function To examine the accuracy of MaxEnt quantile function, an example involving the generalized Pareto distribution (GPD) is considered. From Eq. (A.3), exact values of four PWMs (k ) are calculated for c ˆ ÿ0:2 and d ˆ 1. Then the Lagrange multipliers associated with the quantile function of the form (24) are computed by entropy maximization. Finally, the estimated function

70

M.D. Pandey / Structural Safety 22 (2000) 61±79

Fig. 3. The bias of PWM estimates obtained from various samples sizes.

is compared with the exact expression (A.4), as shown in Fig. 5. Although the general shape of the Pareto quantile function is well approximated, the attention is focussed on the distribution tail in Fig. 6 in which a semi-log plot of POE versus the quantile value is presented. The MaxEnt approximation is reasonable up to POE of 10ÿ2, and beyond this in the far tail region it underestimates the quantile values. The conclusion is somewhat obvious: with limited information an accurate extrapolation of the distribution tail is not rational. The extrapolation in the upper tail region by ®tting an analytical distribution to the available data assumes continuity of the tail beyond the data, which cannot be defended in a strict mathematical sense. Since the Pareto distribution is known to have a long and heavy tail, it can not be captured from the consideration of a small number of PWMs. To illustrate this, the COV, skewness and kurtosis of GPD are reported in Table 1 for various values of c. In the present case of c ˆ ÿ0:2 and d ˆ 1, the kurtosis is 73.8, and skewness is 4.65. It does, however, appear that quantiles corresponding to POE (or probability) of the order of 10ÿ2 can be reliably estimated from the MaxEnt approximation. Therefore, this approach can be used to estimate quantiles representing nominal values of various mechanical loads (live, dead, and construction loads) and material properties, such as yield strength and fracture toughness. The estimation of higher order PWMs (order >8) from small samples (size <20) is often problematic because certain consistency conditions must be satis®ed by sample estimates [8]. Since this topic requires further investigation, the study did not consider more than 6 PWMs.

M.D. Pandey / Structural Safety 22 (2000) 61±79

71

Fig. 4. The RMSE of PWM estimates obtained from various samples sizes.

4.3. Bias and eciency of quantile estimation A simulation experiment was designed to estimate the bias and RMSE of quantile estimates obtained from MaxEnt quantile function against some benchmark estimates. The steps involved in the simulation experiment, shown in Fig. 7, are brie¯y described as follows. A random sample of size n was simulated from a known distribution, e.g., Pareto and lognormal, with preselected parameters. From the sample, PWMs of order N were estimated and MaxEnt quantile function (QF) was ®tted following the procedure described in Section 3.4. The required quantile value was computed from the MaxEnt QF and benchmark distribution. The simulation was repeated M times to estimate the quantile bias and RMSE. Consider the estimation of a Pareto quantile (POE=10ÿ2) from sample of size n ˆ 10. In the simulation, the generalized Pareto distribution (GPD) is taken as the parent distribution with a ®xed scale parameter (d ˆ 1:0) and varying values of the shape parameter (c), ranging from ÿ0.4 to +0.4. The simulation consisted of M ˆ 10; 000 cycles. Four sample PWMs of order 0±3 (N ˆ 3) were considered in ®tting the MaxEnt QF in the form of Eq.(24). Using the ®rst two PWMs, the GPD parameters were estimated to calculate the benchmark quantile value. Hosking and Wallis [6] have shown that the accuracy of PWM based GPD parameter estimation is far superior to that of maximum likelihood and the method of moments.

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Fig. 5. MaxEnt approximation of the Pareto quantile function (N ˆ 4, C ˆ ÿ0:2).

The variation of normalized bias with the shape parameter is compared in Fig. 8. In general, MaxEnt QF results in slight underestimation, less than 5%, except when c ˆ ÿ0:4. For c > ÿ0.2, the RMSE of MaxEnt estimates is very close to that of the benchmark results (Fig. 9). As expected, the MaxEnt estimates approach benchmark values as the tail heaviness of GPD decreases. The tail heaviness, in the present notation, is inversely proportional to the shape parameter, for example, a GPD with c ˆ ÿ0:4 has much heavier tail than c ˆ ‡0:1 [15]. Large values of skewness and kurtosis for c <0, given in Table 1, also give an idea about long and heavy Pareto tail. In the second example, lognormal distribution is considered as parent distribution in the simulation. The objective is to estimate the lognormal quantile (POE=10ÿ2) from a sample of size 10 using four sample PWMs in the MaxEnt approach. To compute the benchmark quantile estimate, a lognormal distribution was ®tted using the sample mean and variance. The simulation involved 104 cycles, and it was repeated for several COV (coecient of variation) values of the parent lognormal, ranging from 0.1 to 1.0. It is interesting to note from Fig. 10 that the MaxEnt quantile bias is within 5% for the entire range of COV values. However, the RMSE tends to be higher than the benchmark estimates, especially for COV >0.6, (Fig. 11). The nominal values of mechanical design loads (COV <0.6), e.g. live & dead loads, and material properties (COV<0.2) correspond to the POE of order 10ÿ2. Therefore, the proposed MaxEnt approach can provide reliable estimates of such nominal values from a very small sample that may belong to a fairly general distribution.

M.D. Pandey / Structural Safety 22 (2000) 61±79

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Fig. 6. MaxEnt approximation of the Pareto tail region.

Table 1 Moments of the Pareto distribution for scale parameter d=1

Shape Parameter

Moments of the Pareto Distribution

(ÿc)

COV

Skewness

Kurtosis

ÿ0.4 ÿ0.3 ÿ0.2 ÿ0.1 0 0.1 0.2 0.3 0.4

2.24 1.58 1.29 1.12 1.00 0.91 0.85 0.79 0.75

* 16.44 4.65 2.81 2.00 1.52 1.18 0.93 0.73

*a * 73.80 17.83 9.00 5.78 4.20 3.31 2.76

a

*=Does not exist.

74

M.D. Pandey / Structural Safety 22 (2000) 61±79

Fig. 7. Simulation-based assessment of the accuracy of the MaxEnt quantile estimates derived from Pareto samples.

M.D. Pandey / Structural Safety 22 (2000) 61±79

Fig. 8. Variation of the Pareto quantile with shape parameter (POE=10ÿ2, n ˆ 10†.

Fig. 9. Variation of the RMSE Pareto quantile with shape parameter (POE=10ÿ2, n ˆ 10).

75

76

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5. Example Lind [16] and Jaeger and Bakht [17] presented polynomial models for ®tting the cumulative distribution function and estimating the lower fractiles of concrete strength. A data set consisting of test strength of 15 concrete cylinders (Table 2) was considered to estimate a quantile corresponding to 1% probability. Using a lognormal polynomial, the lower fractile was estimated as 3162 psi [16,17].

Fig. 10. Variation of bias of lognormal quantile with COV (POE=10ÿ2, n=10). Table 2 Concrete compressive strength data [16] Specimen

Concrete strength (psi)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2720 2900 3000 3110 3190 3370 3410 3500 3560 3620 3790 3840 3980 4070 4140

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Fig. 11. Variation of bias of lognormal quantile with COV (POE=10ÿ2), n=10).

Using Eq.(11), the ®rst four sample PWMs from concrete data set were estimated as 3480 psi, 1870, 1288, 985. Using this information, the MaxEnt quantile function was derived similar to Eq.(23) as y ˆ exp…7:8614 ‡ 0:9525u ÿ 0:9578u2 ‡ 0:5109u3 †

…25†

For u ˆ 0:01, the required quantile value is estimated from Eq.(25) as 2620 psi. A case study is under preparation to compare this method with several other distribution free approaches for estimating material strength quantile from test data. 6. Concluding remarks The paper presents a distribution free method for unbiased estimation of the quantile function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints speci®ed in terms of the probability-weighted moments (PWMs). The method is conceptually elegant as it provides direct estimates of quantile values avoiding the inversion of the distribution function. Furthermore, the method uses PWMs which can be reliably estimated from small samples, in contrast with ordinary moments. Considering estimates of four PWMs from fairly small sample size (n10), the proposed method appears to be useful for estimating quantiles corresponding to probability of exceedance (or probability) of the order of 10ÿ2. Such quantiles generally represent nominal values of various

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M.D. Pandey / Structural Safety 22 (2000) 61±79

mechanical loads (live, dead, and construction loads) and material properties, such as yield strength and fracture toughness. Monte Carlo simulations were carried out to assess the bias and root mean square error (RMSE) of MaxEnt quantile estimates. It is observed that bias associated with MaxEnt estimates is fairly small, within 5%, for a wide range of parameters of the Pareto and lognormal distributions used in the simulation. Since the extrapolation in the far tail region (POE10ÿ3 to 10ÿ4) of a distribution on the basis of a small data set is dicult, a prior distribution representing some information or beliefs is commonly used by engineers for this purpose. A distribution free approach, similar to that presented in the paper, can be developed using the principle of minimum cross entropy that systematically integrates a prior distribution with the available data. Acknowledgements The author thankfully acknowledges the ®nancial support for this study provided by the Natural Science and Engineering Council of Canada. The author is grateful to Professor J.K. Vrijling and Dr. P. van Gelder of Delft University of Technology, The Netherlands, for introducing him to the theory of probability-weighted moments. The author is also grateful to Professor J.R.M. Hosking for providing programs for computing the probability-weighted moments. Appendix: generalized pareto distribution The cumulative distribution function is de®ned as F…x† ˆ u ˆ 1 ÿ ‰1 ÿ cx=dŠ1=c …for c 6ˆ 0† or F…x† ˆ 1 ÿ exp…ÿx=d†…for c ˆ 0†

…A:1†

where c and d are known as shape and scale parameters of the distribution. The quantile function can be obtained by inverting (A.1) d x…u† ˆ ‰1 ÿ …1 ÿ u†c Š…for c 6ˆ 0† or x…u† ˆ ÿd log…1 ÿ u†…for c ˆ 0† c

…A:2†

The Type 1 PWMs are simply given as k ˆ

d for k ˆ 0; 1; . . . n …k ‡ 1†…k ‡ 1 ‡ c†

…A:3†

which exist provided that c>ÿ1 [18]. An rth order moment of GPD exists only if c > ÿ1/r. In case they exist, the mean (), coecient of variation (COV), skewness (g) and kurtosis (k) are de®ned as 1 2…1 ÿ c† 3…1 ‡ 2c†…3 ÿ c ‡ 2c2 † and  ˆ  ˆ d=…1 ‡ c†; COV ˆ p ; ˆ COV  …1 ‡ 3c† …1 ‡ 3c†…1 ‡ 4c† 1 ‡ 2c

…A:4†

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79

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