Is daily precipitation Gamma-distributed?

Atmospheric Research 93 (2009) 759–766

Contents lists available at ScienceDirect

Atmospheric Research j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a t m o s

Is daily precipitation Gamma-distributed? Adverse effects of an incorrect use of the Kolmogorov–Smirnov test Ondřej Vlček ⁎, Radan Huth Institute of Atmospheric Physics, Boční II 1401, 141 31 Praha 4, Czech Republic

a r t i c l e

i n f o

Article history: Received 22 August 2008 Received in revised form 10 January 2009 Accepted 10 March 2009 Keywords: Daily precipitation Gamma distribution Kolmogorov–Smirnov test Lilliefors test

a b s t r a c t The applicability of the Gamma distribution for the description of daily precipitation amounts within a single season is tested on 90 European stations. The test is carried out separately for winter, spring, summer, and autumn seasons. The Lilliefors (LI) modification of the Kolmogorov– Smirnov (KS) test is used to assess the goodness-of-fit of the Gamma distribution to daily precipitation in every year and season of the reference period. The possible rejection of the Gamma distribution for a particular station and season is based on the number of years in which the distribution is rejected. The results are compared with those obtained from the non-modified KS test, whose application is incorrect when the fit to the theoretical distribution is tested on the same data from which its parameters were estimated. We confirm that the two tests yield considerably different results. The validity of the Gamma distribution is not as wide as it is commonly believed: in winter, precipitation at more than 40% of the examined stations is not Gamma-distributed. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Probability density function (PDF) is a useful and efficient tool for a comprehensive description of a distribution of any climatological variable, including precipitation amounts. Several theoretical distributions have been used to characterize daily precipitation amounts, for example, Weibull (e.g. Castellvi et al., 2004), lognormal (Shoji and Kitaura, 2006), and mixed exponential (Schoof, 2008). Probably the most popular one is the Gamma distribution (e.g., Katz, 1999; Groisman et al., 1999; Semenov and Bengtsson, 2002; Zolina et al., 2004; Husak et al., 2007), especially for its capability to describe a wide range of distribution shapes. Examples of its use in stochastic weather generators can be found in studies by Semenov et al. (1998) and Kyselý and Dubrovský (2005). Unfortunately, some studies using theoretical distributions as approximations for daily precipitation amounts suffer from various deficiencies: (i) the validity of the selected theoretical distribution for a given location is not tested at all; (ii) a sufficient guidance is not provided to the reader on which testing procedure was exactly employed, usually as a consequence of ambiguous descriptions of statis⁎ Corresponding author. Tel.: +420 272016010. E-mail address: [email protected] (O. Vlček). 0169-8095/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.atmosres.2009.03.005

tical procedures in software packages (cf. Steinskog et al., 2007); or (iii) an incorrect testing procedure is used. The goodness-of-fit test, perhaps most popular in climatological and meteorological literature, is that of Kolmogorov– Smirnov. The Kolmogorov–Smirnov test is often employed for testing the null hypothesis that a dataset has a distribution described by a fully determined (theoretical) distribution function. The testing statistic is a maximum difference between the empirical and theoretical distribution functions. The null hypothesis is rejected at a given significance level if the testing statistic exceeds the critical value. It is important to realize that a non-rejection of the null hypothesis does not imply that it can be accepted. However, to keep the text easy-to-read, we claim in such circumstances that the hypothesis was accepted. Cautionary notes on limitations of the use of a classical Kolmogorov–Smirnov test have been repeatedly published in meteorological and climatological literature (Crutcher, 1975; Wilks, 1995; Steinskog et al., 2007). As early as in 1975, Crutcher drew attention of the meteorological community to the fact, which had already been widely recognized at that time, that it is not possible to use the Kolmogorov–Smirnov test when the parameters of the distribution function are estimated from the same dataset on which they are tested. This, however, is usually the case in any practical application. This approach leads to type

760

O. Vlček, R. Huth / Atmospheric Research 93 (2009) 759–766

Table 1 Stations used in this study, their geographical coordinates in degrees (Lat = latitude, Long = longitude), elevation (Hght) in meters above the sea level, and the year when precipitation record ends if before 2000. Station

Lat (°)

Long (°)

Hght (m)

Arad Armavir Axel Bamberg Barkestad Basel Beja Berlin Bjørnøya Bologna Bourges Bragança Bremen Bucharest Bulken Buzău Călăraşi Châteauroux Cluj-Napoca De Bilt Dresden Drobeta-Turnu Severin Dublin Eskdalemuir Falun Ferrara Groenback Groningen Halden Hamburg Heerde Helsinki Hohenpeißenberg Holen Hoofddorp Hull Innsbruck Jyväskylä Karlsruhe Kerkwerve Krasnoufimsk Kremsmünster Landbohjskolen Lien i Selbu Linköping Lisbon Ljubljana Lugano Luxembourg Madrid Malaga Mantova Mestad Milan Moscow Navacerrada Nord-Odal Orenburg Oudenbosch Perpignan Potsdam Prague Prilep Roermond Salamanca San Sebastián Säntis

46.13 44.98 51.00 49.88 68.82 47.55 38.02 52.45 74.52 44.48 47.07 41.80 53.05 44.42 60.65 45.13 44.20 46.86 46.78 52.10 51.12 44.63 53.36 55.32 60.62 44.82 56.28 53.18 59.12 53.55 52.38 60.17 47.80 52.63 52.31 53.77 47.27 62.40 49.02 51.67 56.65 48.05 55.68 63.22 58.40 38.72 46.07 46.00 49.62 40.41 36.67 45.15 58.22 45.47 55.83 40.78 60.38 51.68 51.57 42.74 52.38 50.09 41.33 51.17 40.95 43.31 47.25

21.35 41.12 4.00 10.88 14.80 7.58 − 6.13 13.30 19.02 11.25 2.37 − 5.27 8.78 26.10 6.22 26.85 27.33 1.72 23.57 5.19 13.68 22.63 − 5.68 − 2.80 15.62 11.50 9.62 6.60 11.38 9.97 6.05 24.95 11.02 5.07 4.70 0.37 11.40 25.68 8.38 3.85 57.78 14.13 12.53 11.12 15.53 − 8.85 14.52 8.97 6.22 − 2.34 − 3.52 10.75 7.88 9.00 37.62 − 2.01 11.57 55.10 4.52 2.87 13.07 14.42 21.57 5.97 − 4.53 − 1.97 9.35

117 159 2 282 3 316 246 55 16 60 161 690 4 82 323 97 19 155 410 2 246 77 68 242 160 15 25 1 8 26 6 4 977 −1 −3 2 577 137 114 7 206 383 9 255 93 77 299 273 376 667 7 20 151 122 156 1890 147 117 1 42 81 191 673 20 790 259 2490

End year 1997 1995 1999 1999 1999 1999 1998 1999

Table 1 (continued) Station

Lat (°)

Long (°)

Hght (m)

Sarajevo Schwerin Sonnblick Stensele Ter Apel Tîrgu Jiu Torrevieja Toulouse Turnu Măgurele Uccle Ufa Valencia Valentia Växjö Verstervig Vienna West Ters Wick Winterswijk Zagreb Zaragoza Zugspitze Zurich

43.85 53.65 47.05 65.07 52.87 45.03 37.98 43.62 43.75 50.80 54.72 39.48 51.94 56.87 56.77 48.23 53.37 58.45 51.97 45.82 41.66 47.42 47.38

18.38 11.38 12.95 17.15 7.05 23.27 0.69 1.38 24.88 4.35 55.83 0.35 − 9.78 14.80 8.32 16.35 5.22 − 2.92 6.70 15.98 0.99 10.98 8.57

577 59 3106 325 12 203 1 151 31 100 104 11 9 166 18 198 7 36 34 156 247 2960 556

End year

1998

1999 1997

1997 1997

1999

1999

1999 1998 1996

1997 1995

1999 1998 1999 1999

1997 1999 1998 1997 1999 1997

II error (failing to reject the null hypothesis when it is false). The Lilliefors modification of the Kolmogorov–Smirnov test, consisting in adjusting the Kolmogorov–Smirnov critical values (commonly referred to as the Lilliefors test), should be applied instead. Lilliefors (1967, 1969, 1973) determined critical values for, among others, the Gamma distribution with the scale parameter estimated and the shape parameter either estimated and equal to 1, 2, 3, 4, ≥8 or known and ≥3. The critical values can be found in Crutcher (1975). In situations when the distribution parameters have been estimated and there is no available modification of the Kolmogorov–Smirnov test, the critical values for the particular case can be determined using a resampling-based bootstrap technique. We are aware of two recent studies (Semenov and Bengtsson, 2002; Zolina et al., 2004) where a validity of the Gamma distribution for precipitation was tested by means of a classical Kolmogorov–Smirnov test, which, in the light of the above cited cautionary papers, must have led to a considerable overestimation of the validity of Gamma distribution. The goal of this paper is twofold: first, we examine whether daily precipitation amounts at selected stations across whole Europe are Gamma-distributed. More specifically, we examine whether it is possible to use Gamma distribution for description of daily precipitation amounts in a single year of a particular season. Second, we illustrate consequences of the improper choice of the unmodified (classical) Kolmogorov–Smirnov test by a comparison with the correct Lilliefors test. 2. Data and methods 2.1. Data and overview of the methodology

1999

In this study, non-blended series (i.e. series that are not completed by infilling from nearby stations) of daily data at 90 European stations in the 1951–2000 period from the ECA database (Klein Tank et al., 2002) are used. Data series are shorter at 31 stations where they finish during the second half of 1990s (see Table 1).

O. Vlček, R. Huth / Atmospheric Research 93 (2009) 759–766

Data were divided into four three-month seasons: winter (DJF), spring (MAM), summer (JJA), and autumn (SON). Parameters of the Gamma distribution were estimated separately for each season, station, and year. Its validity was tested using the local test at significance level αloc = 0.1. The validity of the Gamma distribution for each station and season was tested by the global test at significance level αglob = 0.1. The decision whether to reject the null hypothesis, on the global level was based on the number of years when it was rejected by the local test. Here and hereinafter the term null hypothesis (H0) refers to the hypothesis that daily precipitation has Gamma distribution either on local or global level, depending on the context. 2.2. Estimation of distribution parameters The two-parameter Gamma distribution is used to describe daily precipitation amounts. Its probability density function f(x) is given by

f ðxÞ =


γ − 1 − x = β x e ; β β  CðγÞ

where γ and β are parameters of shape and scale, respectively, and Γ(γ) is the Gamma function. A number of 10 days with precipitation was subjectively set as a minimum sample size for the estimation of parameters since the estimation of distribution parameters from fewer than 10 values would be highly unreliable. According to the available data, days with precipitation under 0.1 mm were considered as dry. Two different methods were used for the estimation of parameters: maximum likelihood estimate (Greenwood and Durand, 1960) and L-moments (Hosking, 1990); the latter is expected to work better for small-size samples and be less influenced by outliers in the data. To compare the performance of the maximum likelihood method with L-moments, we simulated ten thousand times the Gamma-distributed datasets of the length of 10 and 45, with parameters γ = 1 and β = 30. (These values can be considered as typical of a daily precipitation distribution.) In both cases, L-moments gave in average better estimate of the shape parameter; especially for the short datasets of the length of 10. Nevertheless, we apply both methods for comparison.

761

tabulated only for discrete values of the shape parameter γ = 1, 2.... Therefore the critical value for the tabulated parameter value closest to the estimated one was used. The estimated values of γ were usually close to 1. The classical (unmodified) Kolmogorov–Smirnov (KS) test at the significance level αloc = 0.1 was also applied to demonstrate how the incorrect use of the KS test affects the results. The critical values for the KS and Lilliefors tests are plotted in Fig. 1. 2.4. Global test If the local test is conducted at the significance level αloc = 0.1 then, on average, H0 will be falsely rejected in 10% of years. Therefore, if the hypothesis is to be rejected on the global level, we need H0 to be rejected in more than 10% of years. (Please note that ‘global’ is not used here in a geographical context; it has the meaning of ‘over all years at a single station for a single season.’). In this paper we use a simple counting test for the global hypothesis testing. We are aware of more sophisticated tests (e.g. Walker test), which could be used instead, but in this way we avoid the difficulty of evaluating p values of local tests, still having a good enough tool for global hypothesis testing. Let Y denote the number of years when the parameters were estimated and H0 locally tested. It is reasonable to suppose that the results from different years are independent from each other. Suppose further that H0 was rejected in R years at local significance level αloc. Under these presumptions the probability that H0 will be falsely rejected in at least R years is given by (e.g. Wilks, 2006):

Prðk z RÞ =

Y X i=R

Y! i Y −i ðα Þ ð1 −α loc Þ : i!  ðY −iÞ! loc

The global hypothesis will be rejected at the significance level αglob if Pr(k ≥ R) ≤ αglob.

2.3. Local test The validity of the Gamma distribution for a given station, season, and year was tested by the Lilliefors test on the significance level αloc = 0.1. Critical values for the Lilliefors test can be found in Crutcher (1975). They are tabulated for the sample sizes of 25 and 30; for larger samples they are given by coef(γ) ∙ n− 0.5, where n is the sample size and coef depends on the estimated shape parameter γ. In this paper we use the relationship for large n to derive critical values also for n ≤ 30. (The results of local tests are the same for the above approximation of critical values as for critical values given by a ∙ nb, where a and b result from the interpolation between tabulated critical values for n = 25 and 30 and critical values given by coef(γ) ∙ n− 0.5 for 40 ≤ n ≤ 90.) There is one more difficulty with the Lilliefors test — critical values are

Fig. 1. Critical values for testing the hypothesis that data have Gamma distribution: Kolmogorov–Smirnov (KS) test (Owen, 1962; solid line); Lilliefors (LI) test for γ estimated within (0; 1.5) and β estimated: tabulated critical values for N = 25 and 30 (crosses) and values estimated (see text; dashed line). The difference of the critical values between the KS and LI test is shown by a dot–dot–dashed line.

762

O. Vlček, R. Huth / Atmospheric Research 93 (2009) 759–766

Fig. 2. Results of global goodness-of-fit tests for winter: (a) Lilliefors test, parameter estimation by L-moments; (b) Lilliefors test, maximum likelihood; (c) Kolmogorov–Smirnov test, L-moments. Stations where the Gamma distribution is/is not rejected are indicated by red crosses/black plus signs. The size of signs is proportional to the number of rejected years for stations where the global hypothesis was rejected and to the number of not rejected years for stations where the global hypothesis was not rejected. In other words, stations where the data conform to the Gamma distributions are displayed in black, and the larger black plus sign, the better conformity. Rectangles in panels (a) and (b) mark stations with different results obtained by the two parameter estimation methods.

O. Vlček, R. Huth / Atmospheric Research 93 (2009) 759–766

3. Results First we discuss the effect of the choice of the parameter estimation method. As can be seen for winter by comparing panels (a) and (b) in Fig. 2, which shows geographical distribution of stations where the Gamma distribution is/is not rejected, the effect is small but not negligible. The number of stations not having Gamma distribution of precipitation is approximately the same for L-moments and maximum likelihood, but the test output differs at 10% of stations. Results are analogous in other seasons when, however, the number of stations with a test result different between the two parameter estimation methods is larger (see Table 2). We examined potential causes for the sensitivity to the estimation method; we did not find any effect of the number of wet days per season (one could expect a larger sensitivity where the number of wet days, and hence the size of the dataset on which the parameters are estimated, is small) nor any peculiarities in the precipitation characteristics, such as precipitation intensity or high quantiles (not shown). In the following, we limit ourselves to discussing results obtained by L-moments only; the reason for this selection is their better performance on artificial datasets, described in Section 2.2. Next we demonstrate the effect of the incorrect use of the KS test, which can be seen for winter by comparing panels (a) and (c) of Fig. 2. The difference is fundamental: when the correct local Lilliefors test is applied, the Gamma distribution is globally rejected at 42% of stations, whereas with the local KS test we receive global rejection at 14% of stations only. A deeper insight into the differences between the two tests can be gained from Table 3. It contains the numbers of local rejections of the Gamma distribution by Lilliefors and KS local test respectively. Only the results for the stations where the Gamma distribution was rejected globally (using the local Lilliefors test) are shown: At all stations, the KS test yields lower (and, at most stations considerably lower) numbers of seasons not having Gamma distribution. Results for other seasons are analogous (Table 4), with the numbers of global acceptances being larger for both tests. We can compare our results with Zolina et al. (2004) who claim, respectively, 90% and 92% of stations in Europe to be Gammadistributed in winter (January to March) and summer (July to September), using the non-corrected KS test. It is obvious that if a correct procedure were applied in their study, there would have been considerably fewer stations with Gamma distribution. The results obtained with the correct local Lilliefors test are as follows: The Gamma distribution is more frequently accepted in spring and summer, while less often in winter and autumn (Table 4); the range is from 58% in winter to 76% in

763

Table 3 Percentage of years (of those for which parameters were estimated, i.e., of those with at least 10 wet days) in which the local hypothesis of Gamma distribution was rejected; for DJF season and only the stations where the global hypothesis by the Lilliefors (LI) test was rejected. Station

Percentage of years, in which H0 was rejected LI / KS

Arad Axel Bjørnøya Bologna Bragança Buzău Călăraşi Cluj-Napoca De Bilt Drobeta-Turnu Severin Dublin Falun Ferrara Groenback Groningen Hamburg Helsinki Holen Hull Jyväskylä Karlsruhe Kerkwerve Krasnoufimsk Linköping Mantova Moscow Orenburg Oudenbosch Perpignan Prague Prilep Sarajevo Schwerin Torrevieja Ufa Valencia Vienna West Ters

22/8 43/⁎27 65/⁎42 31/16 38/17 29/⁎20 29/14 49/⁎27 18/14 18/12 39/⁎18 31/12 36/⁎21 20/8 31/14 20/8 49/⁎33 29/16 29/13 65/⁎51 20/6 33/⁎18 48/⁎24 18/6 19/9 26/9 41/⁎22 22/8 52/⁎33 33/15 18/12 29/⁎18 27/10 27/6 28/11 22/13 35/14 18/6

Asterisks indicate stations where the global hypothesis was rejected by the Kolmogorov–Smirnov (KS) test. Only stations with global rejection by the LI test are shown.

spring. The maps displaying the rejection or acceptance of the Gamma distribution for spring, summer, and autumn are in Fig. 3. There are considerable seasonal differences in some regions: for example, precipitation in the lowland areas of the Netherlands, Germany, and Denmark tends to be Gamma-

Table 4 Percentage of stations where the global hypothesis was accepted by the Lilliefors (LI) and Kolmogorov–Smirnov (KS) test. Table 2 Percentage of stations where the global test output is different between the L-moments and maximum likelihood parameter estimation methods. DJF

MAM

JJA

SON

10

14

13

23

Lilliefors test is used as a local test.

Season

DJF MAM JJA SON

% stations passed LI

KS

58 76 73 63

86 98 96 92

764

O. Vlček, R. Huth / Atmospheric Research 93 (2009) 759–766

Fig. 3. Results of global goodness-of-fit tests for: (a) spring; (b) summer; (c) autumn, all for L-moments and Lilliefors test. See the caption to Fig. 2 for symbol description.

distributed in summer, whereas in winter, the rejection of the Gamma distribution prevails. A similar tendency can be seen in Romania. Notable in all seasons is the lack of any clear geo-

graphical structure in the stations being/not being Gammadistributed: Stations having Gamma distribution are typically interspersed with those not being Gamma-distributed, and

O. Vlček, R. Huth / Atmospheric Research 93 (2009) 759–766

765

Fig. 4. Summary over all seasons. Stations where the Gamma distribution is not rejected in any season are denoted by black plus signs; stations where the hypothesis is rejected in all seasons are denoted by red crosses, remaining stations being indicated by dots. All for the Lilliefors test and L-moments parameter estimation.

vice versa. The spatially organized exceptions are e.g. the areas with Gamma-distributed precipitation in the Netherlands, northern Germany, Denmark, and Scandinavia in summer; the Alpine region and the eastern Adriatic coast in spring and summer; and the continental region where the Gamma distribution is rejected in winter, covering Finland, Russia, Romania, and central Europe (Czech Republic, Austria). Results are summarized over seasons in Fig. 4 where the stations with the same test output (rejection or acceptance) in all seasons are highlighted. The map is determined by a high spatial and interseasonal variability of test results; nevertheless, there is an observable tendency for precipitation to be Gamma-distributed throughout the year in the Alps and their vicinity, in western Scandinavia, and the central Iberian Peninsula. There are hints for a potentially coherent area of the rejection in central to Eastern Europe, which cannot however be confirmed due to the lack of stations in this area. 4. Conclusion Our conclusions can be summarized as follows: – The choice of the method to estimate the parameters of the Gamma distribution has a small, but not negligible effect; of the two options examined here, the L-moments are better than maximum likelihood. – The use of the uncorrected Kolmogorov–Smirnov test strongly exaggerates the acceptance of the Gamma distribution for daily precipitation amounts. – The areas of acceptance and rejection of the Gamma distribution are geographically only little coherent; there is no obvious candidate physical mechanism that might be responsible for precipitation being or not being Gammadistributed. – The validity of the Gamma distribution for daily precipitation is much less than what is usually thought. For example, winter daily precipitation does not have the Gamma distribution at more than 40% of examined stations.

– A future study should be devoted to the examination of the applicability of other theoretical distributions to characterize daily precipitation. – Results of this study may have implications e.g. in stochastic weather generators, which usually assume daily precipitation to be Gamma-distributed. One should use the Gamma distribution with knowledge of its limited validity if there is no better distribution on hand. In the end, we stress again that whenever the validity of a theoretical distribution is tested on the same sample from which the parameters of the distribution have been derived, one should avoid using the classical Kolmogorov–Smirnov test and to use its Lilliefors modification instead. Acknowledgements This study was conducted within the EU project ENSEMBLES (ENSEMBLE-based Predictions of Climate Changes and their Impacts, contract GOCE-CT-2003-505539). Support from the Grant Agency of the Czech Academy of Sciences, project A300420806, is also acknowledged. References Castellvi, F., Mormeneo, I., Perez, P.J., 2004. Generation of daily amounts of precipitation from standard climatic data: a case study for Argentina. Journal of Hydrology 289, 286–302. Crutcher, H.L., 1975. A note on the possible misuse of the Kolmogorov– Smirnov test. Journal of Applied Meteorology 14, 1600–1603. Greenwood, J.A., Durand, D., 1960. Aids for fitting the Gamma distribution by maximum likelihood. Technometrics 2, 44–56. Groisman, P.Ya., Karl, T.R., Easterling, D.R., Knight, R.W., Jamason, P.F., Hennessy, K.J., Suppiah, R., Page, C.M., Wibig, J., Fortuniak, K., Razuvaev, V.N., Douglas, A., Førland, E., Zhai, P.-M., 1999. Changes in the probability of heavy precipitation: important indicators of climatic change. Climatic Change 42, 243–283. Hosking, J.R.M.,1990. L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society. Series B (Methodological) 52, 105–124. Husak, G.J., Michaelsen, J., Funk, C., 2007. Use of the Gamma distribution to represent monthly rainfall in Africa for drought monitoring applications. International Journal of Climatology 27, 935–944.

766

O. Vlček, R. Huth / Atmospheric Research 93 (2009) 759–766

Katz, R.W., 1999. Extreme value theory for precipitation: sensitivity analysis for climate change. Advances in Water Resources 23, 133–139. Klein Tank, A.M.G., Wijngaard, J.B., Können, G.P., Böhm, R., Demarée, G., Gocheva, A., Mileta, M., Pashiardis, S., Hejkrlik, L., Kern-Hansen, C., Heino, R., Bessemoulin, P., Müller-Westermeier, G., Tazanakou, M., Szalai, S., Pálsdóttir, T., Fitzgerald, D., Rubin, D., Capaldo, M., Maugeri, M., Leitass, A., Bukantis, A., Aberfeld, R., van Engelen, A.F.V., Forland, E., Mietus, M., Coelho, F., Mares, C., Razuvaev, V., Nieplova, E., Cegnar, T., Antonio López, J., Dahlström, B., Moberg, A., Kirchhofer, W., Ceylan, A., Pachaliuk, O., Alexander, L.V., Petrovic, P., 2002. Daily dataset of 20th-century surface air temperature and precipitation series for the European Climate Assessment. International Journal of Climatology 22, 1441–1453. Kyselý, J., Dubrovský, M., 2005. Simulation of extreme temperature events by a stochastic weather generator: effects of interdiurnal and interannual variability reproduction. International Journal of Climatology 25, 251–269. Lilliefors, H.W.,1967. On the Kolmogorov–Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association 62, 399–402. Lilliefors, H.W., 1969. On the Kolmogorov–Smirnov test or the exponential distribution with mean unknown. Journal of the American Statistical Association 64, 387–389. Lilliefors, H.W. (1973) The Kolmogorov–Smirnov and other distance tests for the Gamma distribution and for the extreme-value distribution when parameters must be estimated. George Washington University, supported in part by U. S. Department of Commerce Contract 1-35214.

Owen, D.B., 1962. Handbook of Statistical Tables. Pergamon Press, London. Semenov, V., Bengtsson, L., 2002. Secular trends in daily precipitation characteristics: greenhouse gas simulation with a coupled AOGCM. Climate Dynamics 19, 123–140. Semenov, M.A., Brooks, R.J., Barrow, E.M., Richardson, C.W., 1998. Comparison of the WGEN and LARS-WG stochastic weather generators for diverse climates. Climate Research 10, 95–107. Shoji, T., Kitaura, H., 2006. Statistical and geostatistical analysis of rainfall in central Japan. Computers & Geosciences 32, 1007–1024. Schoof, J.T., 2008. Application of the multivariate spectral weather generator to the contiguous United States. Agricultural and Forest Meteorology 148, 514–521. Steinskog, D.J., Tjøstheim, D.B., Kvamstø, N.G., 2007. A cautionary note on the use of the Kolmogorov–Smirnov test for normality. Monthly Weather Review 135, 1151–1157. Wilks, D.S., 1995. Statistical Methods in the Atmospheric Sciences. An Introduction. Academic Press. 467 pp. Wilks, D.S., 2006. On “field significance” and the false discovery rate. Journal of Applied Meteorology and Climatology 45, 1181–1189. Zolina, O., Kapala, A., Simmer, C., Gulev, S.K., 2004. Analysis of extreme precipitation over Europe from different reanalyses: a comparative assessment. Global and Planetary Change 44, 129–161.