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ISSN 00978078, Water Resources, 2010, Vol. 37, No. 4, pp. 437–445. © Pleiades Publishing, Ltd., 2010. Original Russian Text © T.S. Gubareva, B.I. Gartsman, 2010, published in Vodnye Resursy, 2010, Vol. 37, No. 4, pp. 398–407. WATER RESOURCES AND THE REGIME OF WATER BODIES

Estimating Distribution Parameters of Extreme Hydrometeorological Characteristics by L Method T. S. Gubareva and B. I. Gartsman Pacific Institute of Geography, Far East Division, Russian Academy of Sciences, ul. Radio 7, Vladivostok, 690041 Russia

Abstract—Lmoment method is briefly described and compared with classical methods of moments and maximal likelihood. Algorithms are given for calculating parameters of some distributions widely used in engineering hydrology, meteorology, etc. The advantages of Lmoment method relative to alternative meth ods are analyzed for several examples.

Keywords: Lmoments, coefficients of Lvariation, Lasymmetry, L, distribution parameters, GEV, GPD, lognormal distribution, IIItype . DOI: 10.1134/S0097807810040020

INTRODUCTION Russian publications give few references to this Parameter estimation for analytical probability dis method [2, 5, 6], and its descriptions are incomplete tributions based on empirical is an important and insufficient for practical use. It has not been used problem in engineering hydrometeorology. For a long in engineering calculation practice in Russia. The time, it has been actively discussed and innovated, at objective of this paper is to give a systematic descrip least, in what regards the distributions of extreme val tion of the theoretical basis of Lmoments method and ues. The main issue is due to the fact that estimating to compare it with conventional methods, as well as to extreme characteristics involves the extrapolation of describe the algorithms of parameter estimation for distribution laws into the domain of events with small the most frequently used distribution laws. The dem probabilities, in many cases, far beyond the of onstration of the advantages of Lmoments method is available observational data. Theoretically, such aimed primarily to extending its use by Russian extrapolation, based on purely statistical approach, researchers and engineers. cannot be reliable; however, such estimates are of crit ical importance for economic practice. A number of statistical methods are available for THE ADVANTAGES OF LMOMENTS evaluating distribution parameters by sample data. METHOD Their development is in progress in The advantages of moments as compared with and mathematical , and the results are more L ordinary distribution moments are shown in several or less actively assimilated in hydrometeorology. An studies [13, 14, 18]. First, moments always exist important point is the identification of the perspec L where the value exists for the probability distri tives of a method for the practical evaluation of param bution. This extends to the cases for which ordinary eters of various characteristics. Hereafter, the method higher order moments may not exist as, for example, is presented as applied to the issue currently most the third and fourth moments of distribution with complicated and urgent—evaluating maximal dis GEV a heavy tail and the distribution ≤ charges with low occurrence. k ⎯ 1/3 and k ≤ –1/4 do not exist. The second central The methods of moments and maximal likelihood moment for GEV distribution does not exist when the are in wide use in Russia. Their description can be shape parameter k ≤ –0.5. However, Lmoments and found in textbooks on statistics, while some aspects of their ratios exist in all ranges mentioned above. their application to hdyrometeorological variables are described in special publications [7, 8]. A considerable Second, unlike ordinary moments, only linear achievement of the late XX century in statistical esti functions of sample values of the variable are used to mation was the method of Lmoments proposed by estimate Lmoments based on an observational series. J.R.M. Hosking [13]. The basic methodological The result is that the sample estimates of Lmoments aspects of the method’s application were published by are unbiased and more effective; they are also less sen him in coauthorship with J.R. Wallis in 1997. Nowa sitive to random outliers and gross observational errors days, this method is in wide use for estimating various than the sample estimates based on ordinary hydrometeorological variables. moments.

437 438 GUBAREVA, GARTSMAN

Lmoments, as well as ordinary moments, are used by contrast to ordinary moments defined as as the first step of distribution parameter estimation 1 procedures based on available measurement data. The r r second step of such procedure is to obtain the param EX()==∫xF()dF, r 012,,,…, (2) eters based on Lmoment estimates. This procedure is 0 much more convenient than the maximum likelihood method, since it allows one to draw data on the shape where x(F) is the inverse to the dis of the distribution (based on Lmoment estimates) tribution function F(x). λ and, as a rule, to obtain the parameters by relatively It has been shown that Lmoments r of a random simple direct calculations. variable X can be defined in terms of the probability The application of the maximum likelihood weighted moments, in particular, as method is based on the assumed knowledge of the true λ = β , (3) distribution law, which significantly reduces the 1 0 potentialities of its practical application. The solutions λ = 2β – β , (4) by the maximum likelihood method for most distribu 2 1 0 tions used in hydrology, even in the approximate form, λ = 6β – 6β + β , (5) are systems of transcendent equations, whose solu 3 2 1 0 tions are sought for by numerical optimization proce λ = 20β – 30β + 12β – β , (6) dures. The major difficulties here are due to the likeli 4 3 2 1 0 hood function having many local maximums and to or in the general form the computational problems in the search for maxi mum. In some situations, the computation process is r λ ()r * β unpredictable, i.e., it can oscillate, diverge, inter r + 1 = –1 ∑ prk, k, (7) rupted, etc. The existence of numerous local maxi k = 0 mums for the IIItype Pearson distribution was shown * in studies [12, 17, 20]. where prk, are coefficients defined as A significant fact is that in Russian regulatory doc ⎛⎞⎛⎞()rk– () uments, the maximum likelihood method is imple ()rk– r rk+ –1 rk+ ! prk*, ==–1 ⎜⎟⎜⎟. (8) mented in an approximate form and recommended ⎝⎠k ⎝⎠k ()k! 2()rk– ! only for Kritskii–Menkel distribution. The experience in the application of maximum likelihood method in Clearly, the first Lmoment is equivalent to the the form recommended in SNIP 2.01.1483 shows expectation of the distribution. The second L that it is often impossible to obtain estimates for max moment serves as a scale of the distribution. The L imaldischarge series because of computational prob moments of higher order, converted to this scale, were lems [3], i.e., the only universal method for engineer called Lmoment ratios ing–hydrological practice available in Russia is the τ λ λ ,,… method of moments. r ==r/ 2, r 34 . (9) The Lmoment ratios determine the shape of the LMOMENTS OF ANALYTICAL distribution irrespective of the scale of the variables DISTRIBUTION FUNCTIONS AND THEIR being measured. J.R.M. Hosking proposed the follow SAMPLE ESTIMATES ing terms: λ1 is Llocation, corresponding to the ordi nary first moment or the norm; λ is scale, analo The brief description of the theoretical principles 2 L gous to the ordinary ; τ = λ /λ is and algorithms of moments method follows the 2 1 L L variation coefficient ( – v), an analogue of the ordi work [14]. Its authors regard the method they propose L C nary v; τ is ( – as an alternative for representing the shape of proba C 3 L L ), and analogue of the ordinary coefficient of skew bility distribution curves and emphasize that, histori Cs ness ; τ is kurtosis coefficient, an analogue of the cally, moments appeared as a modification of Cs 4 L L ordinary coefficient of kurtosis. “probability weighted moments,” proposed by J. Greenwood et al. [9]. Two special cases of probabil In practice, we commonly deal with a sample α β including a finite number of observations n. First, ity weighted moments of the rth order, r and r, exist for a distribution function represented in an integral unbiased sample estimates of probability weighted form: moments of distribution br [15] for a sample arranged ≤ ≤ ≤ in ascending order x1:n x2:n … xn:n with a size of n, 1 1 which can be represented as α ==xF()()1 – Fx()rdF, β xF()Fx()rdF, r ∫ r ∫ (1) n –1 0 0 b0 = n ∑xj:n, (10) ,,,… r = 012 , j = 1

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n The types of distribution laws are chosen from –1 ()j – 1 b1 = n ∑ xj:n, (11) those used in hydrometeorology from the viewpoint of ()n – 1 their perspective, by which we mean the possible ade j = 2 quacy of the description of extreme hydrological val n ues. The mathematical form of expressions and –1 ()j – 1 ()j – 2 b2 = n ∑ xj:n. (12) parameter denotations are taken from [14]. Table 1 ()n – 1 ()n – 2 j = 3 gives the distributions in generalized form, which, with certain values of parameters, pass into other The general expression for br has the form known distributions. n Thus, the generalized extreme distribution ()()…() GEV –1 j – 1 j – 2 jr– with the shape parameter k = 0 coincides with Gumbel br = n ∑ xj:n. (13) ()n – 1 ()…n – 2 ()nr– distribution; with k < 0, it yields an extreme IItype jr= + 1 distribution, known as Freshet distribution; while with By analogy with expressions (3)–(7), sample L k > 0, we obtain an extreme distribution of the III type moments are defined as (). Similarly, the generalized Pareto GPD with k = 0 passes into exponential distri , (14) l1 = b0 bution; with k = 1, into uniform distribution, bounded ξ ξ α l = 2b – b , (15) by the shift value from below and by + from 2 1 0 above. The lognormal distribution, parameterized by l3 = 6b2 – 6b1 + b0, (16) parameters of shift, shape, and scale (Table 1), in l4 = 20b3 – 30b2 + 12b1 – b0. (17) J.R.M. Hoskings’ opinion, has some advantages over the accepted parameterization with the use of ordinary In the general form moments μ and σ and the shift parameter ξ. The major r advantage is the more stable estimate of the parame ters used here at the nearzero skewness of distribu l ==p*, b ; r 01,,… ,n – 1, (18) r + 1 ∑ rk k tions. The generalized lognormal distribution can have k = 0 either positive skewness (lower bound, the shape where coefficients p* are defined by (8). As can be parameter k < 0) or negative skewness (upper bound, rk, k > 0). With k = 0, the distribution coincides with nor seen from the expressions given here, the sample L mal law. moments are linear functions of unbiased estimates of probability weighted moments; therefore, they are also Table 3 also gives calculation formulas for Pearson unbiased estimates of λ . III distribution of PIII type, which is widely used both r in Russia and abroad. Kritskii–Menkel distribution the sample Lmoment ratios of the rth order are (threeparameter ) has no analyti defined by the expressions cal expression; the authors of [1] recommend that L tl= /l , (19) moments based on a procedure of numerical optimi 2 1 zation of solution and the construction of special dia t ==l /l , r 34,,…, (20) grams be used for this distribution. r r 2 The procedures for parameter estimation can be whence t is the sample coefficient of Lvariation, t3 is implemented in any computation environment. A the sample coefficient of Lasymmetry, t4 is the sample convenient instrument is “RProject for Statistical Lcurtosis. The sample Lmoment ratios are biased Calculations” [16], which contains practically all estimates. major components required for the construction of various procedures for statistical estimation. Of partic ular use is Imomco package, including parameter esti PARAMETER ESTIMATION AND SHAPE mation algorithms by Lmoment method for almost ANALYSIS OF DISTRIBUTIONS BASED all distributions in use. However, the widely used ON LMOMENTS Microsoft Excel environment enables practically any For most distributions, formulas for parameter variant of calculations with the use of Lmoments. estimation based onLmoments can be found in the By now, the authors have implemented algorithms appropriate literature in the explicit form. Table 1 gives for the construction of 13 analytical curves with the algorithms for mutual calculations of Lmoments and use of Excel builtin mathematical functions and mac parameters for several threeparameter distribution rocommands, including algorithms for normal, expo laws, characterized by shift, shape, and scale parame nential, GEV, GPD, Gumbel, two and threeparame ters. Parameter estimation based on samples can be ter lognormal, generalized logistic, PIII, Pareto, made by substitution into these algorithms of appro Weibull, Frechet, and twoparameter gamma distribu priate sample estimates of Lmoments and Lmoment tion. Most these distributions are in wide use in ratios. hydrometeorology. For most these distribution laws,

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Table 1. Parameter calculation for distribution laws by Lmoments [14] (ξ, α, k are shift, scale, and shape parameters of distribu tions, except for Pearson III distribution, where ξ is shift parameter, β is , and α is shape parameter; Γ is the symbol of gamma function, Φ is integral function of standard , G is incomplete gamma function, I is incomplete beta function; Z is a with standard normal distribution; the values of coefficients E0 = 2.0466534, E1 = –3.654437, E2 = 1.8396733, E3 = –0.20360244, F1 = –0.20182173, F2 = 1.2420401, F3= –0.21741801) Distribution type Moments Parameter

Generalized extreme GEV For k > 1 k ≈ 7.8590c + 2.9554c2

–y λ ξ α Γ – ()1 – k y – e 1 = + {1 – (1 + k)}/k 2 log2 f(x) = α–1 e c = – –k τ λ2 = a(1 – 2 ) Γ(1 + k)/k 3 + 3 log3 ⎧ –1 {}()ξ α ≠ –k log 1 – kx– / , k 0 τ = 2(1 – 3–k)/(1 – 2–k) – 3 λ y = ⎨ 3 α 2k ⎩()ξ /α, 0 = x – k = 5(1 – 4–k) –10(1 – 3–k) + 6 (1 – 2–k) ()Γ–k () τ4 = 12– 1 + k –k ⎧ξα{}1 ()1 F k /k, k ≠ 0 (1 – 2 ) ξ = λ – α{1 – Γ(1 + k)}/k xF()= ⎨ + – – 1 ⎩ξα– log()1 – F , k = 0

Generalized ParetoGPD

–1 –(1 – k)y f(x) = α e λ1 = ε + α/(1 + k) k = (1 – 3τ3)/(1 + τ3) λ = α/{(1 + k)(2 + k)} α = (1 + k)(2 + k)λ ⎧ k–1log{}1 kx()ξ /α , k ≠ 0 2 2 y = ⎨– – – τ3 = (1 – k)/(3 + k) ξ = λ1 – (2 + k)λ2 ⎩()x – ξ /α, k = 0 τ ()1 – k ()2 – k 4 = ⎧ξα{}1 ()1 F k /k, k ≠ 0 ()3 + k ()4 + k xF()= ⎨ + – – ⎩ξα– log()1 – F , k = 0

Generalized lognormal 2 2 4 6 λ ξα()k /2 E +++E τ E τ E τ 1 = + 1 – e /k τ 0 1 3 2 3 3 3 k = – 3 ky– y2/2 2 4 6 α 2 1 +++F τ F τ F τ () e λ k /2{}Φ() 1 3 2 3 3 3 f x = , 2 = e 12– –k/ 2 α 2π k λ –k/2 2ke 2 4 6 α = ⎧ {}()ξ α A +++A k A k A k Φ() ⎪log 1 – kx– / ≠ τ ≈ –k0 1 2 3 12– –k/ 2 , k 0 3 2 4 6 y = ⎨ –k 1 +++B k B k B k 1 ⎪ 1 2 3 x – ()ξ α Φ() ()π 2 ⎛⎞1 2 () ⎩ x – / , k = 0 2 4 6 x = ∫ 2 exp⎝⎠– x d x τ ≈ τ0 2C0 +++C1k C2k C3k 2 4 4 + k –∞ –kZ 2 4 6 ⎧ξα()1 / , ≠ 0 1 +++D1k D2k D3k α 2 X ⎨ + – e k k ξλ ()k /2 = = 1 – 1– e ⎩ξα+ Z, k = 0 k

Pearson III

α – 1 –()x – ξ /β λ = ξ + αβ if 0 < |τ | < 1/3, ()x – ξ e 1 3 f()x = α λ π–1/2βΓ() α Γα() 10.2906+ z β Γα() 2 = + 1/2 / α ≈ 2 3 τ α α z ++0.1882z 0.0442z ξ 3 = 6I1/3( , 2 ) – 3 () ⎛⎞α, x – Γα() Fx = G⎝⎠ / (), πτ2 β Ix pq z = 3 3 x x if 1/3 ≤ |τ3| < 1, ()α, α – 1 –t Γ()pq+ p – 1()q – 1 G x = ∫t e dt = ∫t 1 – t dt 2 3 Γ()Γp ()q 0.36067z – 0.59567z + 0.25361z 0 α = 0 2 3 if a ≥ 1, 1– 2.78861z + 2.56096z – 0.77045z |τ α–1 α–2 α–3 z = 1 – 3| τ ≈ C0 +++C1 C2 C3 4 α–1 α–2 1 ++D1 D2 1 α α2 α3 τ ≈ ++G1 G2 +C3 if a < 1, 4 α α2 α3 1 ++H1 H2 +H3

WATER RESOURCES Vol. 37 No. 4 2010 ESTIMATING DISTRIBUTION PARAMETERS 441 three methods of parameter estimation by sample were Table 2. Lmoments of maximal discharge series implemented: the methods of moments, Lmoments, Length, and maximum likelihood. The example in Fig. 1 gives River–point l1 l2 tt3 t4 different variants of empirical and analytical distribu years tion curves. Ussuri R., 54 785 384 0.49 0.51 0.34 Examples of calculated Lmoments and their Koksharovka Settl. ratios— Lvariation, Lskewness, Lcurtosis—for Arsen’evka R., 62 599 348 0.58 0.53 0.29 several series of maximal discharges are given in Yakovlevka V. Table 2. Estimates of Lmoments can be used, in par Malinovka R., 69 504 259 0.51 0.47 0.33 ticular, in the substantiation some distribution law as Rakitnoe V. an approximation [21]. For this purpose, sample esti Bira R., 52 300 203 0.67 0.65 0.43 mates of Lskewness and Lcurtosis for an observa Lermontovka V. tional series are used as coordinates of point (t3, t4) in the field of Lmoment diagram, which represent a Avvakumovka R., 75 386 232 0.60 0.52 0.32 Vetka Settl. series of dependencies τ4 = f(τ3) for different theoreti cal distributions. Lmoment diagram, constructed in Margaritovka R., 58 357 210 0.59 0.55 0.38 Margaritovo V. the variation ranges 0 < τ3 < 0.7 and 0 < τ4 < 0.6, which are of greatest practical interest, is given in Fig. 2. Twoparametric distributions are represented in the diagram by a point. Threeparametric distribu Table 3. Coefficients of polynomial approximation of func τ τ tions, characterized by the parameters of shape, shift, tions 4 = f( 3) (dash the absence of coefficients) and scale, are represented in the diagram by curves, Distribution which can be approximated by the polynomial func Coefficient tion [14] GEV GPD LN3 PIII 8 τ τk A0 0.10701 0 0.12282 0.12240 4 = ∑ Ak 3, (21) k = 0 A1 0.11090 0.20196 – – where Ak are coefficients, which are different for dif A2 0.84838 0.95924 0.77518 0.30115 ferent distribution laws (Table 3). A3 –0.06669 –0.20096 – – The position of the point with coordinates (t3, t4) for the observational series under consideration in the A4 0.00567 0.04061 0.12279 0.95812 diagram field in Fig. 2 may suggest which distribution A5 –0.04208 – – – law will be appropriate as an approximation of the empirical distribution. In this case, the diagram pre A6 0.03763 – –0.13638 –0.57488 sents calculated values for maximal discharge series A7 –––– given in Table 2. A8 – – 0.11368 0.19383 EXAMPLES OF USE OF LMOMENTS IN CALCULATIONS bility can be seen even from the comparison of formu As mentioned above, an important advantage of the las for evaluating simple moments and Lmoments. method is, first, the unbiasedness of the Lmoments For sample estimates, this property manifests itself calculated from the sample and the weak biasedness of in the degree of change in their values caused by elim Lmoment ratios for different sample series length. ination of extreme values from the series. Estimates of tr and t for moderatelength and long Let us consider an example of a series of maximal samples have very small bias and variations; for exam annual discharges in the Zima River at Zulumai Set ple, according to data of [13, 14], the asymptotic bias tlement (57 years) with characteristics of Cv = 0.77, –1 t3 for is 0.19n , and that for the Cs = 2,89, L – Cv = 0.35, and L – Cs = 0.37. Once the –1 normal distribution t4 is 0.03n . The bias in small sam extreme (maximal) value is excluded, Cv and Cs will ples can be estimated by modeling; however, in the decrease to 0.62 and 1.66, respectively; and L – Cv and general case for the sample size of 20 and more, the L – Cs, to 0.31 and 0.28, respectively; expressed in per bias of the coefficient of Lvariation is small. cent, the decrease will be 20% in Cv, 42% in Cs, 10% It is also important that the extreme (large or small) in L – Cv, and 25% in L – Cs. As shown in [13], the terms of sample, containing important information ordinary coefficient of skewness in samples with size of about the tails of theoretical distribution, have small 5000 and more can be largely determined by its weight in Lmoment parameter estimates. Such esti extreme terms, while the coefficient L – Cs will not. mates are stable, which is in principle characteristic of One more example is the series of maximal dis the maximum likelihood method. The property of sta charges of rain freshets in the Avvakumovka River at

WATER RESOURCES Vol. 37 No. 4 2010 442 GUBAREVA, GARTSMAN

x (а) 2000 1 2 1500 3 4 1000

500

0 (b) 2000

1500

1000

500

0 (c) 2000

1500

1000

500

0 0.1 1 20 50 70 98 99.9 P, %

Fig. 1. Examples of analytical distributions of maximal annual discharges in the Izvilinka R. at Izvilinka V. (a) PIII, (b) LN3, (c) GEV, whose parameters were calculated by (1) moments method, (2) maximum likelihood, (3) Lmoments; (4) empirical curve.

Vetka Settlement (75 years in length). The largest value number of terms in the sample. Overall, Cs varies from in the series exceeds the second largest by 1.6 times. 0.5 to 1.5; whereas t3, from 0.8 to 1.4; the of We for subsamples with a length of 20 to 74 by random the series of calculated Cs and t3 (the total number of elimination of terms from the initial series and calcu subsamples is 55) is 0.046 and 0.009, respectively. The late the skewness and Lskewness coefficients for each variance of Cs estimates for subsamples from 20 to subsample. Standardized results of calculations are 46 years is 0.078, and that for subsamples from 47 to given in Fig. 3, where sample estimates of parameters 74 years is 0.016; while the variance of t3 estimates is show changes of different character with increasing 0.017 and 0.003, respectively.

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homogeneous material. The analysis involves 36 series Lэkurtosis, τ4 0.6 GEV of maximal annual river discharges in Primorskii Krai LN3 with a length of 40 to 76 years, prechecked for the GPD absence of statistically significant linear trends. The 0.5 rivers of this region feature the predominance of rain 4 summer freshets in their regime and high specific val 0.4 6 PIII ues of maximal freshet discharges, large variance and 1 skewness values for maximal runoff series. The param 3 0.3 5 eter estimation methods are compared on PIII, GEV, 2 and LN3 distributions. The two latter distributions are most adequate for the conditions under consideration 0.2 G (among distributions that are in wide use in hydrol N E ogy), as can be seen from both the analysis of vast lit 0.1 erature in this field and the results of the authors’ own studies [10, 11]. U Three different criteria are used to assess the 0 0.1 0.2 0.3 0.4 0.5 0.5 0.7 χ2 τ approximation quality. The first is the criterion Lskewness, 3 widely used for the analysis of the agreement between sample data and the given distribution law. The other Fig. 2. Diagram of Lmoments ratios. Twoparameter dis two criterions, denoted by ω and S, were used by the tributions: U is uniform, N is normal, G is Gumbel, E is exponential; threeparameter distributions: GEV is gener authors before to compare calculation schemes against alized extreme, LN3 is lognormal, GPD is generalized mass data on maximal discharges of Northern Eurasia Pareto, PIII is Pearson III type. (1 ⎯ 6) Points, correspond rivers. ing to river numbers in Table 2. The criterion omega is used in accordance with recommendations of Yu.B. Vinogradov and represent The visual analysis of plots in Fig. 3 shows insignif the total relative deviation between the abscissas of the icant bias and relatively rapid decrease in the variance empirical and analytical occurrence curves. It is calcu of skewness estimate by Lmoments method with lated as increasing sample length n relative to the ordinary method of moments. For larger subsamples, the value n ω ⎛⎞p* – p** of Lskewness coefficient remains almost unchanged, = ∑⎝⎠ , (22) p* – p* unlike the ordinary Cs, whose value depends on i = 1 b a i whether an outlier enters the subsample. According to data [14], when the deviation of the outlier is very where p* is the empirical occurrence, p** is analytical large, the diagram of estimates by the method of occurrence, pa*, pb* are the boundaries of the confi moments distinctly separates into two fields of points, dence interval of empirical occurrence; all these values while this is not the case for Lmoments estimates. refer to each ith value from the observation sample This suggests the relative stability of Lmoments esti with the total volume of n. Individual formulas [2, p. mates for small samples. 241] are recommended for evaluating pa*, pb* . The Let us analyze an example of correlating the main minimal value of criterion ω corresponds to the best method for parameters’ assessment on mass, relatively approximation quality by probability.

Cs (а)t3 (b) 2.0 2.0

1.5 1.5

1.0 1.0

0.5 0.5

0 20 40 60 80 n 0 20 40 60 80 n

Fig. 3. Variations in (a) skewness and (b) Lskewness coefficients vs. the number of terms in the subsample. An example of the Avvakumovka R., Vetka Settl.

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Table 4. Comparison of approximation quality of sample data in assessing distribution law parameters by different methods (MM is method of moments, LM is Lmoments method, ML is maximum likelihood method)

Distribution

Estimation characteristic P III LN3 GEV

MM LM ML MM LM ML MM LM ML

The number of cases where null hypothesis was accept 23 33 34 18 35 35 8 35 35 ed by χ2 criterion (5% confidence level) The number of cases with the best value of criterion ω 6246 2132141022 Mean value of criterion ω 20.4 11.4 17.0 11.8 6.06 6.01 14.1 6.36 7.60 The number of cases with the best value of criterionÿ S 6228 3249 –2016 Mean value of criterion S 3.27 3.28 4.63 3.98 2.73 13.2 13.8 3.53 72.8

Criterion S is an estimate by the least square low bias and of Lmoment estimates; the method, and its minimal value corresponds to the best possibility to substantiate the type of the distribution approximation quality by the values of the variable. It law by Lasymmetry and Lkurtosis. is calculated by the wellknown formula The comparison of the Lmoments method with n the maximum likelihood method shows that the small 1 ()2 bias, the consistency and efficiency of parameter esti S = ∑ kp* – kp** i , (23) n mates are the common advantages of both methods. i = 1 However, Lmoment method can be implemented by relatively simple and more stable computation proce where * and ** are quantiles, with the same occur kp kp dures and a priori does not require one to know the rence, of the empirical and analytical distribution actual distribution law. As compared with the ordinary curves, respectively. method of moments, the Lmoments method has The characteristics used in the case of comparison considerable advantages regarding all major require by 36 samples for each of the three methods included ments imposed to distribution parameter estimates. A the number of cases when the zero hypothesis was longrun study perspective is the analysis of coordinate accepted by χ2 criterion with 5% significance level; the estimates for distribution curves of hydrometeorologi number of cases with the best values of ω and S crite cal variables obtained with the help of this method. ria; the mean weighted values of ω and S criteria. The results of comparison are given in Table 4. They show the generally comparable quality of the results of ACKNOWLEDGMENTS applying Lmoments and maximum likelihood method and the much worse results of the ordinary The authors are grateful to R. van Noien (Delft moment method. Technical University) and A. G. Kolechkina (ARON WIS Consulting Bureau, the Netherlands) for meth odological and technical help in the preparation of the CONCLUSIONS manuscript. This paper briefly describes the uptodate method This study was supported by grants of RF President of estimation of probability dis to Young Scientists, MK343.2007.5, and Russian tributions—the Lmoments method. Algorithms for Federation for Basic Research, proj. no. 06–05– the calculation of Lmoments estimates by observa 90625. tional series are given. Relationships between alterna tive characteristics of distribution shape (L – Cv, L – Cs, and Lkurtosis) with the parameters of distribution REFERENCES functions, and the algorithms for determining param 1. Bolgov, M.V., Mishon, V.M., and Sentsova, N.I., eters by Lmoments method for a number of distribu Sovremennye problemy otsenki vodnykh resursov i tion functions used in the practice of engineering– vodoobespecheniya (Current Problems in Estimating hydrological calculations are described. The advan Water Resources and Water Availability), Moscow: tages of the method are shown: the existence of L Nauka, 2005. moments for some variants of distributions with heavy 2. Vinogradov, Yu.B., Matematicheskoe modelirovanie tails, for which the ordinary moments do not exist; the protsessov formirovaniya stoka (Mathematical Model

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