Physics and Chemistry of the Earth

Physics and Chemistry of the Earth 34 (2009) 626 634 Contents lists available at ScienceDirect Physics and Chemistry of the Earth journal homepage: www.elsevier.com/locate/pce Performances of some parameter estimators of the generalized Pareto distribution over rounded-off samples Roberto Deidda *, Michelangelo Puliga Dipartimento di Ingegneria del Territorio, Facoltá di Ingegneria, Universitá di Cagliari, Piazza d Armi, I-09123 Cagliari, Italy article info abstract Article history: Received 25 July 2008 Received in revised form 26 November 2008 Accepted 1 December 2008 Available online 24 December 2008 Keywords: Generalized Pareto distribution Estimators Rounded-off records Root mean square errors Bias Recent analyses on some daily rainfall time series highlighted the presence of records with anomalous rounding (1 and 5 mm), while the standard resolution should be 0.1 or 0.2 mm. Assuming that the generalized Pareto distribution (GPD) can reliably represent the distribution of daily rainfall depths, this study investigates how such discretizations can affect the inference process. The performances of several GPD estimators are compared using the Monte Carlo approach. Synthetic samples are drawn by GPDs with shape and scale parameters in the ranges of values estimated on daily rainfall depth time series. Results show how the relative efficiency of estimators could be very different for continuous or rounded-off samples. Moreover, when the rounding-off magnitude becomes larger than a few millimeters, all the considered estimators reveal very poor performances. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction Several statistical distributions have been applied and sometimes specifically proposed to characterize the ordinary and the extreme behaviour of rainfall and discharge observations, as well as other meteo-climatological variables. E.g., the normal, lognormal, exponential, gamma, Pearson Type III, log-pearson Type III, Gumbel, Weibull, two component extreme value (TCEV), generalized extreme value (GEV) and generalized Pareto distribution (GPD) are all distributions which have received consideration in statistical hydrology. Referring the reader to Chow et al. (1988) and Stedinger et al. (1993) for more details on these distributions and their applications in hydrology, we want to stress here the importance of the last two distributions, which appear particularly attractive for their asymptotic statistical properties: namely the GPD and the GEV. Defining the extreme values as the maxima within time blocks (usually assumed a year long in Earth Sciences), it can be proved that, if a limit distribution of these maxima exists, this distribution belongs to the GEV family (Fisher and Tippett, 1928; Gnedenko, 1943). Conversely, when looking at the exceedances above a threshold, it can be proved that the GPD is the expected distribution (Pickands, 1975). Moreover, if a process follows a GPD with a given shape parameter, the block maxima follow a GEV distribution with the same shape parameter (Balkema De Haan Pickands theorem). The reader is referred to Gumbel (1958), Castillo (1988), * Corresponding author. Tel.: +39 070 6755324; fax: +39 070 6755310. E-mail address: rdeidda@unica.it (R. Deidda). and Coles (2001) and references therein for more details on these important properties of GPD and GEV. The advantage in adopting the couplet of distributions GPD and GEV is thus straightforward. GPD has only two parameters which we can estimate with good accuracy on the large samples of ordinary rainfall time series, while the three GEV parameters must be estimated only on annual maxima and are affected by large estimation errors. Thus one of the reasons why we want to focus on the problem of fitting the GPD on ordinary rainfall time series is that it should be possible, according to theoretical arguments, to estimate first the GPD parameters using a large amount of data and then to use the results to determine the GEV distribution which describes the extremes. Besides these important theoretical features, there are also empirical arguments for adopting the GPD to describe ordinary rainfall time series: an analysis based on the L-moments ratio diagram (Hosking, 1990) performed on 200 daily rainfall time series by Deidda and Puliga (2006) showed that the GPD is the best candidate to describe these datasets. The GPD, first introduced by Pickands (1975) to describe the exceedances over high thresholds, is, because of its nature, sometimes also referred to as points over threshold (or peaks over threshold ) (POT) distribution. Adopting hydrological language, the GPD can describe the partial duration series (PDS) which are obtained by letting a threshold select the highest values from ordinary (continuous) hydrological time series. On the other hand, the block maxima are often referred to as annual maxima series (AMS) because the time block length is usually assumed to be one year. A comparison of the relative performance of the AMS 1474-7065/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pce.2008.12.002

R. Deidda, M. Puliga / Physics and Chemistry of the Earth 34 (2009) 626 634 627 and PDS estimation methods is discussed in Madsen et al. (1997a,b) for rainfall and flood time series: they show that the POT approach in estimating the statistical distribution of rain records for a network of hydrological stations can be an advantageous alternative to block maxima methods. The generalized Pareto distribution GPD has the following equation: ( x u 1=n 1 1 þ n F u ðx; a; nþ ¼PrfX 6 xjx > ug ¼ a n 0 ð1þ 1 exp x u a n ¼ 0 where n is the shape parameter, a the scale parameter, while u is the threshold value. For n ¼ 0 the GPD becomes a simple exponential distribution, for n > 0 is characterized by a heavy tail, while for n < 0 becomes a bounded distribution. The determination of the optimal threshold for GPD fitting is still an open problem. Graphical and numerical methods have been proposed and applied by several authors to detect the threshold (e.g., Davison and Smith, 1990; Smith, 1994; Lang et al., 1999; Dupuis, 1999; Choulakian and Stephens, 2001). More recently, on the basis of a preliminary study on the threshold estimation by Guillou and Hall (2001), a paper by Peng and Qi (2004) proposes an alternative approach to the selection of an optimal threshold: rather than searching for the correct threshold, Peng and Qi (2004) look for powerful and robust estimators with moderated also for too low (or too high) values of the threshold. Deidda and Puliga (2006) highlighted how the presence of rounded-off values in the sample can affect the determination of the optimal threshold, sometimes also masking the presence of the threshold: they proposed to overcome this problem by using a modification of the failure-to-reject method (Choulakian and Stephens, 2001). Deidda and Puliga (2006) suggested computing the regions of acceptance of the goodness of fit tests on roundedoff samples generated via Monte Carlo techniques, adopting the same rounding-off rule as the observed sample. The presence of rounded-off values in rainfall records and the related problems in the inference of probability distributions was then investigated in depth by Deidda (2007): a systematic analysis on 340 daily rainfall time series collected by the rain gauge network of the Sardinian Hydrological Survey (Italy) revealed the presence of significant percentages of roughly rounded off measurements, even at 1 and 5 mm resolutions, rather than at the standard 0.1 or 0.2 mm discretization. Nevertheless, the presence of rounded-off values not only affects the determination of the GPD threshold, but also induces errors in the estimation of the shape n and the scale a parameters. This paper investigates these errors and compares the performances of different estimators when applied to estimate n and a on synthetic samples generated by a GPD with zero threshold and then rounded off at different discretizations. A review of GPD estimators is presented in Section 2. The evaluation of the performances of selected GPD estimators is discussed in Section 3, while the conclusions are drawn in Section 4. 2. A review of GPD estimators Over the last years many estimators have been developed to improve the efficiency and the robustness of the fitting techniques. The estimators can be of several classes: tail index (as Hill or De Haan), maximum likelihood functions, moments, probability weighted moments, medians and goodness of fit based. Every estimator class has some drawbacks and some advantages. The maximum likelihood estimator is based on the maximization of the likelihood function L: this method is very important because it is asymptotically (i.e. for large samples) the best. Nevertheless, when the sample is small or is contaminated by spurious data, the method could provide unrealistic estimates. In a classical study, Hosking and Wallis (1987) compare the performances with those of the methods of moments and of probability weighted moments: they show how the could lead to inaccurate results for small samples. Juárez and Schucany (2004) analyze the performances of the estimator on contaminated GPD samples showing that the method lacks robustness: they introduce a robustified method known as minimum divergence power density estimator. Another estimator class is that of the tail index estimators. The basic idea is to estimate the average slope of the distribution tail using a plot position rule of ranked data in log diagrams. The method is useful for a large variety of distributions like gamma, student and GPD. By combining the tail index with other order statistics it is possible to estimate the GPD parameters as shown in Pickands (1975). The estimator gives generally good performances for large samples of pure data, but must be tuned using a suitable threshold. Moreover, the numerical algorithms for GPD parameters estimation are often affected by convergence problems. The accuracy of the GPD fit depends on the kind of data and thus it must be evaluated for specific cases. In fact, as shown by Rosbjerg et al. (1992), it is possible that, for moderate tails (values of the shape parameter near zero) and small datasets, the description with the ordinary exponential distribution may be more accurate than the GPD one. This is an expected result because the exponential distribution requires the estimation of a single parameter, while the GPD needs the estimation of two parameters. Obviously similar results can be obtained, more generally, by comparing the GPD performances in the case of n known and a unknown with the case of both parameters unknown. Nevertheless, the reader should be careful in adopting these approaches because they are advantageous only if the shape parameter is a priori known. In this study we compare the performances of several estimators of GPD parameters on rounded-off samples. We discarded the estimators based on tail index because we found that the numerical algorithms are very unstable and often fail. The estimators compared here are listed and discussed in the following. For the numerical implementation of the first three widely applied methods the reader is referred to Hosking and Wallis (1987) and Stedinger et al. (1993). For the other (and more recent) methods, the references are provided in the text of each description. Maximum likelihood estimator (). The is a standard and widely adopted estimation technique that can be applied to any statistical distribution. It is based on the idea of finding the set of h parameters which maximize the likelihood function LðX; hþ evaluated on the sample X. We remark that the location parameter u of the GPD can not be obtained by the, in fact the score function ol=ou is unbounded. The maximum likelihood estimates of the remaining shape n and scale a parameters can be obtained in different ways leading to slightly different results: the function can be formalized in the classical bivariate way or in an univariate and computationally more efficient way by introducing a smart substitution for the scale and shape parameters, as suggested by Grimshaw (1993). Many authors have proved that the is the best estimator in the presence of large samples, the asymptotic behaviour of this method is known to be the best possible one. But for small samples (6 100 values) the fit is not always good and the method can be outperformed by other techniques (Hosking and Wallis, 1987), moreover, the numerical algorithms used to estimate the maxima sometimes fail to converge to local maxima, thus robust and powerful computational methods must be used to find the maxima by avoiding convergence problems. Moments estimator (). It represents the simplest method: estimates of the shape n and scale a parameters are

628 R. Deidda, M. Puliga / Physics and Chemistry of the Earth 34 (2009) 626 634 Bias for ξ: all estimators Bias 0.04 0.02 0.00 0.02 0.4 0.2 0.0 0.2 0.4 ξ Fig. 1. Bias of shape parameter n for different GPD estimators. The Bias is computed with Monte Carlo techniques over continuos samples of size 500, generated by a GPD with threshold u ¼ 0, a ¼ 7 and n in the range ð 0:5; 0:5Þ. The final result is filtered by a robust gaussian kernel smooth function (Nadaraya, 1964). obtained as simple functions of the mean and the variance of the distribution. The method is theoretically applicable only for values of n < 1=2 because for n! 1=2 the variance tends to be infinite. Hosking and Wallis (1987) suggest using the estimator for n < 1=4. When n 0 the accuracy of the method is close to the estimator. Nevertheless, we must note that the moments method is very sensitive to outliers, in fact the mean and the variance are statistical quantities lacking robustness: a single outlier could dramatically change all these quantities. Probability Weighted Moments estimators ( and ). The probability-weighted moments PWM were introduced by Greenwood et al. (1979) and represent an alternative to the ordinary moments. As for the estimator, parameters can be expressed as a function of PWMs. The PWM estimator is particularly advantageous for small datasets because the probability weighted moments have a smaller uncertainty than the ordinary moments. The best performance is reached for n 0:2, for positive shape values performances are very close to ones, while for n < 0 PWM performances become a little worse than those of. Hosking and Wallis (1987) give two definitions of PWM, uned () and ed (), but the difference can be detected only for small samples. Maximum penalized likelihood (). Coles and Dixon (1999) introduced a weight function for the maximum likelihood function LðX; hþ for n > 0. This estimator corrects the tendency of to diverge for small samples. Minimum density power divergence (). This robust estimator has been introduced by Juárez and Schucany (2004) and was derived from the by using a special function of divergence between the fitted function and the data. A constant is introduced to control the trade-off between robustness and efficiency. This property could be very attractive when dataset are contaminated. Likelihood moment estimator (). This method has recently been proposed by Zhang (2007) as a replacement for the PWM and moments method. This estimator should be efficient and robust but it is slow and computational intensive. for ξ: all estimators 0.04 0.06 0.08 0.10 0.12 0.4 0.2 0.0 0.2 0.4 ξ Fig. 2. Same as Fig. 1, but for of the shape parameter n.

R. Deidda, M. Puliga / Physics and Chemistry of the Earth 34 (2009) 626 634 629 Median estimator (). This estimator is the most CPU time intensive, it was designed to be resistant to outliers (Peng and Welsh, 2001). Nevertheless, for pure GPD data its performances are very poor as shown by Juárez and Schucany (2004). Numerical packages containing the implementation of these estimation techniques can be found in the R package POT created and maintained by Ribatet (2007). ξ = 0. α = 7 3. Performances of GPD estimators The performances of a parameter estimator depend on many factors: the sample size, the shape of the sample distribution, the presence of spurious, multivariate or trended (not stable) data, the internal dependence and the algorithmic stability and precision of the estimation technique. For instance the is efficient only for big samples while the PWM estimator is able to estimate ξ = 0. α = 12 ξ = 0.15 α = 7 ξ = 0.15 α = 12 ξ = 0.3 α = 7 ξ = 0.3 α = 12 Fig. 3. Bias for the shape parameter n estimated with different techniques on rounded-off samples. Results are presented as a function of rounding-off magnitude which ranges from 0 to 5 mm. Subplots refer to different couples of shape and scale parameters (see subtitles) selected in the range of representative values of daily time series.

630 R. Deidda, M. Puliga / Physics and Chemistry of the Earth 34 (2009) 626 634 the GPD parameters for small samples, but it has a large for values of n < 0. In order to compare the performances of the estimating techniques described in the previous Section, two groups of tests were carried out using Monte Carlo techniques. The first group aims to compare the performances of estimators on continuous samples, while the second one aims to evaluate the performances on rounded-off records. The performances are evaluated by the Bias and the root mean square error : Bias ¼ Eðh est h true Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ¼ E ðh est h true Þ 2 ð2þ ð3þ ξ = 0. α = 7 ξ = 0. α = 12 ξ = 0.15 α = 7 ξ = 0.15 α = 12 ξ = 0.3 α = 7 ξ = 0.3 α = 12 Fig. 4. Same as Fig. 3, but for Bias of the scale parameter a.

R. Deidda, M. Puliga / Physics and Chemistry of the Earth 34 (2009) 626 634 631 where h est ; h true are the estimated and the true (i.e. used to generate the synthetic samples) values of the parameter respectively. In our case h can be the n and/or the a parameter of the GPD. Bias and are computed on synthetic samples generated by the GPD given by Eq. (1) with threshold u ¼ 0 and ðn; aþ couples of parameters which can be considered representative of daily rainfall records, while the size of each synthetic sample is 500. In addition, in the second group of tests, samples are then rounded off with different discretization magnitudes. Before continuing with the presentation of the results, we highlight that in order to reduce the sampling variability in the computation of the Bias and the, a large number of samples would be necessary for each test (i.e. for each couple of n and a parameters and each discretization). Thus, as a compromise with the computational time required by the amount of carried out tests, on one hand each Monte Carlo simulation was limited to 10,000 synthetic samples, on the other hand a Gaussian kernel smoother (Nadaraya, 1964) was applied to smooth lines plotted in the following Figures, for ξ : rounded data ξ = 0. α = 7 for ξ : rounded data ξ = 0. α = 12 for ξ : rounded data ξ = 0.15 α = 7 for ξ : rounded data ξ = 0.15 α = 12 for ξ : rounded data ξ = 0.3 α = 7 for ξ : rounded data ξ = 0.3 α = 12 Fig. 5. Same as Fig. 3, but for of the shape parameter n.

632 R. Deidda, M. Puliga / Physics and Chemistry of the Earth 34 (2009) 626 634 filtering out the artificial noise due to the residual sampling variability of Bias and estimates. 3.1. Tests over continuous GPD samples In this group of tests, Bias and are computed with Monte Carlo techniques on continuous samples generated by GPD with threshold u ¼ 0, parameters a ¼ 7 and n 2ð 0:5; 0:5Þ. In the plots of the Bias (Fig. 1) and of the (Fig. 2) we can evaluate the performances of the considered methods. We remark again that, at least for large samples, the is expected to be the best estimator, thus lines close to the one give evidence of good performances. For instance, the estimator has a severe breakdown for n > 0:3 and a good accuracy for n 0. The estimator fails for n < 0:3, but this region is of scarce importance in rainfall and flood time series applications since distributions are usually unbounded. The performance of is very good for negative shape values, on the contrary the estimator, as highlighted by Juárez and Schucany (2004), has a poor efficiency (less than 20% of the estimator) and a high Bias: it becomes com- for α : rounded data ξ = 0. α = 7 for α : rounded data ξ = 0. α = 12 for α : rounded data ξ = 0.15 α = 7 for α : rounded data ξ = 0.15 α = 12 for α : rounded data ξ = 0.3 α = 7 for α : rounded data ξ = 0.3 α = 12 Fig. 6. Same as Fig. 3, but for of the scale parameter a.

R. Deidda, M. Puliga / Physics and Chemistry of the Earth 34 (2009) 626 634 633 petitive only on contaminated samples. The PWM estimators (ed and uned) perform well for n 0:1 where the efficiency is the best possible. Nevertheless below this value the is worse than for the other methods. Finally for the penalized maximum likelihood we have good performances close to ones. 3.2. Tests over rounded-off GPD samples This group of tests investigates the performances of the estimators over rounded-off samples. Recent analyses on daily rainfall time series (Deidda and Puliga, 2006; Deidda, 2007) revealed the presence of rounded-off records with a mixture of different resolutions (0.1, 0.2, 0.5, 1.0 and 5.0 mm). To simplify the interpretation of the results of our analysis, here we consider a single rounding off d for each test. The magnitude of d explores the entire range from 0 to 5 mm with increments of 0.1 mm. We expect that the Bias and the increase with the magnitude of the rounding off. This result could be explained with geometrical arguments: the empirical cumulative distribution function (ecdf) of a rounded-off sample displays a step-like behaviour which produces a big uncertainty in the estimate. To explore the performances of the considered GPD estimators for parameters values in the domain of interest of daily rainfall records, we evaluated the Bias and the for n 2f0; 0:15; 0:30g and a 2f7; 12g that are representative of parameters values estimated over the 200 longest datasets of the Sardinia Region, see e.g. Fig. 5 in Deidda (2007). The Biases in n and a estimates are presented in Figs. 3 and 4, respectively, as a function of the rounding-off resolution: each subplot is obtained by Monte Carlo generations of GPD samples generated with a fixed couple of parameters (n; a), obtained by the combination of the values reported above. The Bias of both n and a estimates increases with the magnitude of the discretization. A similar behaviour is displayed by Figs. 5 and 6, where the of n and a is plotted again versus the rounding-off resolution. The comparison of and Bias for rounded-off samples shows noticeable differences with respect to the continuous case. The estimator has an evident failure in the n estimates for rounding off more than 1 mm while in the continuous case it has very good performances close to (or even better than) the ones. Similar problems affect the PWM estimators: we highlight that the tests on rounded samples are performed with the values n 2f0; 0:15; 0:3g where the PWM performances in the continuous case are very good and close to. Moreover and unexpectedly, looking again at displayed in Figs. 5 and 6, we can observe that the estimator performs better for high n values showing an opposite behaviour with respect to the continuous case where the efficiency becomes worse for large n values. The efficiency in some cases is even better than the one. However, besides the above considerations on the relative performances of a given estimator with respect to the others, we highlight that all the considered methods provide parameter estimates affected by unacceptable errors. In some cases the Bias and the are larger than the 100% of the parameter value of the parent distribution, even when we select the best estimator for the specific case. For instance, looking at the (which is the most significant index because it also includes the Bias), we can observe in Fig. 5 that even for the best estimator the error is often about (or larger than) 0.2 for the n parameter which usually assumes values between 0.0 and 0.3 in hydrological applications. Similar considerations also hold for Fig. 6, where of a is often about 4 mm and more, while a estimates for daily rainfall depths are usually about 10 mm, see again Fig. 5 in Deidda (2007). 4. Conclusions The main objective of this study was the investigation of the performances of some estimators of the GPD shape and scale parameters on rounded-off samples. With this aim, some widely used GPD estimators (such as the maximum likelihood, the moments, and the probability-weighted moments methods) and some other recently proposed ones (such as the maximum penalized likelihood, minimum density power divergence, likelihood moment estimator, median estimator) were considered and compared. Performances were computed with Monte Carlo techniques on synthetic samples generated by GPD with parameter values which can be considered representative of daily rainfall distributions. A first group of tests was performed on continuous samples. Results showed that the maximum likelihood and the maximum penalized likelihood estimators outperform the other estimators. The likelihood moment estimator has a good performance, although slightly worse than the first two previous ones. The simple moment estimator performs quite well for moderated values of the shape parameter (n 0), while the effects of moment divergence become relevant for the shape parameter approaching 0.5. A second group of tests was performed on rounded-off samples, exploring a range of discretizations from 0 to 5 mm detected in daily rainfall time series by Deidda and Puliga (2006) and Deidda (2007). Comparisons among the performances of the considered estimators showed some surprising and unexpected results, revealing that some estimators display a very different behaviour on discretized samples with respect to the continuous case. For instance, the performances of the maximum penalized likelihood estimator slowly worsens on samples rounded at resolutions larger than 1 mm, while the same estimator performs better than the maximum likelihood one for continuous samples. Performances of the probability-weighted moments estimator are very bad with respect to the other estimators, while it shows good performances in the continuous case for positive values of the shape parameter. On the contrary, performances of the moments estimator are relatively better than the other ones, although its relative performances are not good for large positive values of the shape parameter in the continuous case. But in conclusion, although we can speculate on which estimator could perform better than another one for a specific case, it is very important to observe that none of the considered estimators can give acceptable estimates on sample data rounded off at a resolution of a few millimeters. Indeed, computed Bias and are often of the same magnitude as the parameter value to be estimated. Thus the determination of efficient estimators for rounded-off sample is still an open problem. 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