TL-moments of the exponentiated generalized extreme value distribution

TL-moments and LQ-moments of the exponentiated generalized extreme value distribution (EGEV) will be obtained and used to estimate the unknown parameters of the EGEV distribution. Many special cases may be obtained such as the L-moments, LH-moments and LL-moments. The estimation of the EGEV distribution parameters is studied in numerical simulations where the method for obtaining TL-moments is compared with other estimation methods (L-moments estimators, LQ-moment estimators and the method of moment estimators). The true formulae for the rth classical moments and the probability weighted moments for the EGEV distribution will be obtained to correct the Adeyemi and Adebanji [1] results.

Journal of Advanced Research (2010) 1, 351–359 Cairo University Journal of Advanced Research ORIGINAL ARTICLE TL-moments of the exponentiated generalized extreme value distribution Noura A.T Abu El-Magd * Institute of Statistical Studies and Research, Cairo University, Egypt Received November 2009; revised 16 February 2010; accepted March 2010 Available online 24 July 2010 KEYWORDS Exponentiated generalized extreme value distribution; TL-moments; L-moments; LH-moments; LL-moments; LQ-moments Abstract TL-moments and LQ-moments of the exponentiated generalized extreme value distribution (EGEV) will be obtained and used to estimate the unknown parameters of the EGEV distribution Many special cases may be obtained such as the L-moments, LH-moments and LL-moments The estimation of the EGEV distribution parameters is studied in numerical simulations where the method for obtaining TL-moments is compared with other estimation methods (L-moments estimators, LQ-moment estimators and the method of moment estimators) The true formulae for the rth classical moments and the probability weighted moments for the EGEV distribution will be obtained to correct the Adeyemi and Adebanji [1] results ª 2010 Cairo University Production and hosting by Elsevier B.V All rights reserved * Tel.: +20 182828396 E-mail address: Shon_Stat@hotmail.com distribution The GEV distribution is often used to model extremes of natural phenomena such as river heights, sea levels, stream flows, rainfall and air pollution in order to obtain the distribution of daily or annual maxima Additionally, in a reliability context, analogous analyses are performed where the interest is in sample minima strengths and failure times The GEV distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fre´chet and Weibull families also known as type I, II and III extreme value distributions, respectively The cumulative distribution function of the EGEV distribution is: 2090-1232 ª 2010 Cairo University Production and hosting by Elsevier B.V All rights reserved Fxị ẳ Introduction A new two-parameter distribution, called the exponentiated generalized extreme value distribution (EGEV), has been introduced by Adeyemi and Adebanji [1] The EGEV distribution is a generalized version of the generalized extreme value (GEV) Peer review under responsibility of Cairo University doi:10.1016/j.jare.2010.06.003 Production and hosting by Elsevier ( expb1 kxị1=k ị; expbexpxịịị; k0 kẳ0 where < x < k1 for k > 0; k1 < x < for k < and b > The corresponding density function will be: ( bð1 À kxÞ1=kÀ1 expb1 kxị1=k ị; k fxị ẳ b expbexpxịịị expxị; kẳ0 352 N.A.T Abu El-Magd In practice, the shape parameter usually lies in the range À1/ < k < 1/2 Hosking et al [2] for the GEV distribution Adeyemi and Adebanji [1] studied the EGEV distribution with k „ They studied some of its mathematical properties and obtained the rth classical moments and the probability weighted moments of the EGEV distribution The first aim of this paper is to introduce the TL-moments and LQ-moments of the EGEV distribution The TL-moment estimators (TLMEs), L-moments estimators (LMEs), LQ-moment estimators (LQMEs) and the method of moment estimators (classical estimators) (MMEs) for the EGEV distribution will be obtained A numerical simulation compares these methods of estimation mainly with respect to their biases and root mean squared errors (RMSEs) will be obtained The second aim of this paper is to derive the true formulae for the rth classical moments and the probability weighted moments (PWMs) for the EGEV distribution to correct the Adeyemi and Adebanji [1] formulae for the EGEV The remaining sections are as follows In section two, the TL-moments and the LQ-moments with different special cases of the EGEV distribution will be derived In section three, the TL-moments estimators (TLMEs) and the LQ-moments estimators (LQMEs) will be obtained for the EGEV distribution In section four, the true formulae for the rth classical moments and the probability weighted moments (PWMs) for the EGEV distribution will be obtained In section five, the method of moment estimators (MMEs) and the L-moments estimators (LMEs) of the EGEV distribution will be derived In section six, material and methods of a numerical simulation to compare the properties of the TLMEs, LMEs, LQMEs and the MMEs of the EGEV distribution will be obtained Results are discussed in the experimental results, and final section is the conclusion TL-moments and LQ-moments In this section, the TL-moments and LQ-moments of the EGEV distribution will be obtained From the TL-moments with generalized trimmed, many special cases can be obtained such as the TL-moments with the first trimmed, L-moments, LH-moments and LL-moments for the EGEV distribution TL-Moments Let X1,X2, .,Xn be a conceptual random sample (used to define a population quantity) of size n from a continuous distribution and let X(1:n) X(2:n) Á Á Á X(n:n) denote the corresponding order statistics Elamir and Seheult [3] defined the rth TL-moment kðs;tÞ as follows: r   rÀ1 X k r1 ks;tị ẳ 1ị EXrỵsk:rỵsỵtị ị; r ẳ 1; 2; r r k¼0 k The TL-moments reduce to L-moments (see Hosking [4]) when s ¼ t ¼ They considered the symmetric case (s = t) Hosking [5] obtained some theoretical results for the TL-moments with generalized trimmed for s and t (symmetric case (s = t) and asymmetric case (s „ t)) and obtained the TL-moments coefficient of variation TL-CV, the TL-skewness and the TLkurtosis as follows: s;tị s;tị ss;tị ẳ k2 =k1 ; s;tị s3 s;tị s;tị ẳ k3 =k2 ; s;tị and s4 s;tị s;tị ẳ k4 =k2 : Hosking [4] concluded that L-moments have the following theoretical advantages over classical moments: For L-moments of a probability distribution to be meaningful, we require only that the distribution has a finite mean; no higher-order moments need be finite For standard errors of L-moments to be finite, we require only that the distribution has a finite variance; no higherorder moments need be finite L-moments, being linear functions of the data, are less sensitive than are classical moments to sampling variability or measurement errors in the extreme data values The boundedness of L-moments ratios: s3 is constrained to lie within the interval (À1, 1) compared with classical skewness, which can take arbitrarily large values L-moments sometimes yield more efficient parameter estimates than the maximum likelihood estimates L-moments provide better identification of the parent distribution that generated a particular data sample A distribution may be specified by its L-moments even if some of its classical moments not exist Asymptotic approximations to sampling distributions are better for L-moments than for classical moments Maillet and Me´decin [6] introduced the relation between the rth TL-moments and the first TL-moments with generalized trimmed for s and t (symmetric case (s = t) and asymmetric case (s „ t)) Indeed, it is sufficient to compute TL-moments of order one to obtain all TL-moments They obtained the following rth TL-moments:   rÀ1 1X rỵsj1;tỵjị j r1 ks;tị k1 ẳ 1ị ; r ¼ 1; 2; ;3; r r j¼0 j where s; t ¼ 0; 1; 2; This relation is very important and helped to enable easier calculations for the rth TL-moments with any trimmed and L-moments as particular cases of the rth TL-moments with generalized trimmed for s and t They underlined that the TL-moments approach is a general framework that encompasses the L-moments, LH-moments and the LL-moments Elamir and Seheult [3] concluded that TL-moments have the following theoretical advantages: TL-moments are more resistant to outliers TL-moments assign zero weight to the extreme observations They are easy to compute Moreover, a population TL-moments may be well defined where the corresponding population L-moments (or central moment) not exist: for example, the first population TLmoment is well defined for a Cauchy distribution, but the first population L-moment, the population mean, does not exist TL-moments ratios are bounded for any trimmed for s and t ¼ 0; 1; 2; Their sample variance and covariance can be obtained in closed form The method of TL-moments is not intended to replace the existing robust methods but rather to complement them According to the above relations, the first four TLmoments with generalized trimmed for s and t ðs; t ¼ 0; 1; 2; ; Þ of the EGEV distribution for k P À will be: TL-moments of the exponentiated generalized extremevalue distribution " #   t sỵtỵ1ị! X j t kỵ1ị ; 1ị sỵjỵ1ị 1bk Ckỵ1ị 1ị k sị!tị! jẳ0 j " sỵtỵ2ị! sỵ1ịk s;tị k2 ẳ bk Ckỵ1ị 2k sỵ1ị! tỵ1ị! # j k t X 1ị sỵjỵ2ị ; 2ị jỵ1ị!tjị! jẳ0 s;tị k1 ẳ k sỵtỵ3ị! sỵ3ị sỵ2ị k k b Ckỵ1ị sỵ2ị sỵ1ị 3k sỵ2ị! tỵ1ị! tỵ2ị! # t X 1ịj sỵjỵ3ịk fsỵjỵ4ịg ; 3ị jỵ2ị!tjị! jẳ0 s;tị k3 ẳ and s;tị k4 "   k s ỵ t ỵ 4ị! s ỵ 3ịk ỵ s ỵ 2ịs ỵ 7ị ẳ b Ck ỵ 1ị t ỵ 1ị! 4k s ỵ 3ị! s þ 3Þðs þ 4Þ ðs þ 3Þðs þ 2Þ ðs þ 2ÞÀk þ ðs þ 1ÞÀk À ðt þ 2Þ! t ỵ 3ị! # t X 1ịj s ỵ j þ 4ÞÀk ðs þ j þ 5Þðs þ j þ 6ị : j ỵ 3ị!t jị! jẳ0 4ị From these results we can obtain the TL- coefficient of variaðs;tÞ ðs;tÞ tion s(s,t), TL-skewness s3 and TL-kurtosis s4 for the EGEV distribution For b = 1, we obtain the TL-moments for the GEV distribution as a special case from the TL-moments of the EGEV distribution as per Maillet’s and Me´decin’s [6] results Special cases The TL-Moments with the first trimmed (s = t = 1) By substituting, s ¼ and t ¼ and in Eqs (1)–(4), the first four TL-moments with the first trimmed for the EGEV distribution will be: ð1Þ  i 1h À 6bÀk Ck ỵ 1ị 2ịkỵ1ị 3ịkỵ1ị ; k h i ẳ bk Ck ỵ 1ị 2ịkỵ1ị 3ịk ỵ 2ị2kỵ1ị ; k h i 20 ẳ bk Ck ỵ 1ị 23ịk 2ịkỵ1ị 52ị2kỵ1ị ỵ 5ịk : 3k k1 ẳ 1ị k2 1ị k3 and 1ị k1 ẳ bk Ck ỵ 1ịị; k k2 ẳ bk Ck ỵ 1ị1 2k ị; k k3 ẳ bk Ck ỵ 1ị32ịk 23ịk 1ị; k h 15 k b Ck ỵ 1ị 152ị2kỵ1ị 103ịkỵ1ị ỵ 2ịkỵ1ị 2k i 75ịk ỵ 72ịk 3ịkỵ1ị : From these results we can obtain the TL- coefficient of variation, TL-skewness and TL-kurtosis with the first trimmed for the EGEV distribution For b = 1, we obtain the TL-moments with the first trimmed for the GEV distribution as a special case of the EGEV distribution as per Maillet’s and Medecins [6] results 5ị 6ị 7ị and k4 ẳ bk Ck ỵ 1ị1 54ịk ỵ 103ịk 62ịk Þ: k ð8Þ Also, we can obtain the L-coefficient of variation s = k2/k1, Lskewness s3 = k3/k2 and L-kurtosis s4 = k4/k2 for the EGEV distribution For b = 1, we can obtain the first four L-moments for the GEV distribution as a special case from the L-moments of the EGEV distribution as per Hosking’s [4] results The LH-Moments (t = 0) The LH-moments are linear functions of the expectations of the highest order statistic and introduced by Wang [7] as a modified version of L-moments, to characterize the upper part of a distribution When one wants to put more emphasis on extreme events, the LH-moment approach allows us to give more weight to the largest items When, the LH-moment corresponds to the L-moments As s increases, LH-moments reflect more and more the characteristics of the upper part of the data Wang [7] found that the method of LH-moments resulted in large sampling variability for high s and recommended not to use values of s higher than By substituting t ¼ in Eqs (1)–(4), the first four LH-moments with generalized trimmed for s for the EGEV distribution will be: ! s ỵ 1ị! s;0ị k1 ẳ bk Ck ỵ 1ị s ỵ 1ịkỵ1ị k sị! k s ỵ 2ị! s;0ị ẵs ỵ 1ịk s ỵ 2ịk ; k2 ẳ b Ck ỵ 1ị 2k s ỵ 1ị! s ỵ 3ị! s;0ị k3 ẳ bk Ck ỵ 1ị 3k s ỵ 2ị! ! 1 s ỵ 3ịs ỵ 2ịk s ỵ 2ịs ỵ 1ịk s ỵ 4ịs ỵ 3ịk ; 2 and s;0ị k4 ẳ 353 k4 ẳ   k s ỵ 4ị! b Ck ỵ 1ị s ỵ 3ịk ỵ s ỵ 2ịs ỵ 7ị 4k s ỵ 3ị! 1 k s ỵ 3ịs ỵ 4ịs ỵ 2ị ỵ s ỵ 2ịs ỵ 3ịs ỵ 1ịk k s ỵ 5ịs ỵ 6ịs ỵ 4ị From these results we can obtain the LH-coefficient of variaðs;0Þ s;0ị s;0ị s;0ị s;0ị tion ss;0ị ẳ k2 =k1 , LH-skewness s3 ẳ k3 =k2 and s;0ị s;0ị s;0ị LH-kurtosis s4 ¼ k4 =k2 with generalized trimmed for s for the EGEV distribution Also, for s ¼ 1; 2; 3; , the LHmoments can be obtained with any trimmed s for the EGEV distribution The L-Moments (s = t = 0) By substituting, s ¼ and t ¼ in Eqs (1)–(4), we can obtain the first four L-moments for the EGEV distribution as a special case from the TL-moments for the EGEV distribution The first four L-moments for the EGEV distribution for k P À will be: The LL-Moments (s = 0) The LL-moments are linear functions of the expectations of the lowest order statistic and were introduced by Bayazit and Oănoăz [8] L-moments are a special case for, and if t increases the weight of the lower part of the data will be increased By 354 N.A.T Abu El-Magd substituting in Eqs (1)–(4), the first four LL-moments with generalized trimmed for t can be obtained for the EGEV distribution as follows: " #   t t t ỵ 1ị! X 0;tị k1 ẳ bk Ck ỵ 1ị j ỵ 1ịkỵ1ị ; 1ịj k tị! jẳ0 j " # t X k 1ịj j ỵ 2ịk 0;tị k2 ẳ b Ck ỵ 1ịt ỵ 2ị! ; 2k t ỵ 1ị! jẳ0 j ỵ 1ị!t jị! " k t ỵ 3ị! 32ịk 0;tị k3 ẳ b Ck ỵ 1ị 3k 2ị! t ỵ 1ị! t þ 2Þ! # t X ðÀ1Þj ðj þ 3ÞÀk fðj ỵ 4ịg ; j ỵ 2ị!t jị! jẳ0 and 0;tị k4 " k t ỵ 4ị! 103ịk 122ịk ẳ b Ck ỵ 1ị 4k 3ị! t þ 1Þ! ðt þ 2Þ! # t X ðÀ1Þj j ỵ 4ịk ỵ j ỵ 5ịj ỵ 6ị : t ỵ 3ị! jẳ0 j ỵ 3ị!t jị! From these results we can obtain the LL-coefficient of variað0;tÞ 0;tị 0;tị 0;tị 0;tị tion s0;tị ẳ k2 =k1 , LL-skewness s3 ẳ k3 =k2 and LL0;tị 0;tị 0;tị kurtosis s4 ¼ k4 =k2 with generalized trimmed for t for the EGEV distribution Also, for t ¼ 1; 2; 3; , the LL-moments can be obtained with any trimmed t for the EGEV distribution for the population by g3 = f3/f2 and g4 = f4/f2, respectively; it may be used for identifying the population and estimating the parameters The LQ-skewness takes the value of zero for symmetrical distributions Mudholkar and Hutson [10] concluded that LQ-moments have the following theoretical advantages: LQ-moments are often easier to evaluate and estimate than L-moments LQ-moments always exist are unique Their asymptotic distributions are easier to obtain In general behave similarly to the L-moments when the latter exist The LQ-moments with the three cases (median, trimean and Gastwirth) will be obtained for the EGEV distribution Using the median case (p = 0.5, d = 0.5) and the quantile function for the EGEV distribution, the first four LQ-moments for the EGEV distribution will be: n1 ẳ ẵQo 0:5ị; k n2 ẳ ẵQo 0:707ị Qo 0:293ị; 2k n3 ẳ ẵQo 0:794ị 2Qo 0:5ị ỵ Qo 0:206ị; 3k and n4 ¼ Results for the generalized extreme value distribution (b = 1) By putting b = in Eqs (1)–(4), the first four TL-moments with generalized trimmed for s and t can be obtained for the GEV distribution and these results are the same as the Maillet and Me´decin [6] results and by putting s ¼ t ¼ 1, s ¼ t ¼ 0, t ¼ and s ¼ in the TL-moments for the GEV distribution; the results for the TL-moments with the first trimmed Maillet and Me´decin [6], L-moments Hosking [9], LH-moments Wang [7] and LL-moments Maillet and Me´decin [6], respectively, will be obtained for the GEV distribution LQ-Moments Let X1,X2, .,Xn be a random sample from a continuous distribution function F(x) with quantile function QX uị ẳ F1 X uị and let X(1:n) X(2:n) Á Á Á X(n:n) denote the order statistics Mudholkar and Hutson [10] defined the rth population LQmoments fr of X, as:   rÀ1 X rÀ1 1ịk fr ẳ r1 sp;d Xrk:rị ị; r ẳ 1; 2; k k¼0 where  k Qo uị ẳ lnuị : b Using the trimean case (p = 1/4, d = 1/4), the first four LQmoments for the EPD will be obtained as follows: ẵQ 0:25ị ỵ 2Qo 0:5ị ỵ Qo 0:75ị; 4k o n2 ẳ ẵ2Qo 0:707ị 2Qo 0:293ị ỵ Qo 0:866ị Qo 0:134ị; 8k ẵQ 0:909ị ỵ 2Qo 0:794ị 2Qo 0:674ị n3 ẳ 12k o þ Qo ð0:630Þ À 4Qo ð0:5Þ þ Qo ð0:370Þ n1 ẳ 2Qo 0:326ị ỵ 2Qo 0:206ị ỵ Qo 0:091ị; and n4 ¼ where d 1/2, p 1/2, and The linear combination sp,d is a ‘quick’ measure of the location of the sampling distribution of the order statistic X(rÀk:r) The candidates for sp,d include the function generating the common quick estimators by using the median (p = 0.5, d = 0.5), the trimean (p = 1/4, d = 1/4) and the Gastwirth (p = 0.3, d = 1/3) They introduced the LQ-skewness and LQ-kurtosis ẵQ 0:931ị ỵ 2Qo 0:841ị 3Qo 0:757ị ỵ Qo 0:707ị 16k o 6Qo 0:614ị ỵ 3Qo 0:544ị 3Qo 0:456ị ỵ 6Qo 0:386ị Qo 0:293ị ỵ 3Qo ð0:243Þ À 2Qo ð0:159Þ À Qo ð0:069ފ: sp;d ðXðrÀk:rÞ ị ẳ pQXrk:rị dị ỵ 2pịQXrk:rị 1=2ị ỵ pQXrk:rị dị: ẵQ 0:841ị 3Qo 0:614ị þ 3Qo ð0:386Þ À Qo ð0:159ފ: 4k o Using the Gastwirth case (p = 0.3, d = 1/3), the first four LQmoments for the EGEV distribution will be: ½3Qo 0:333ị ỵ 4Qo 0:5ị ỵ 3Qo 0:667ị; 10k n2 ẳ ẵ3Qo 0:816ị ỵ 4Qo 0:707ị ỵ 3Qo 0:577ị 20k À 3Qo ð0:423Þ À 4Qo ð0:293Þ À 3Qo ð0:184ފ; n1 ¼ TL-moments of the exponentiated generalized extremevalue distribution n3 ¼ ẵ3Qo 0:874ị ỵ 4Qo 0:794ị ỵ 3Qo 0:693ị 30k 6Qo 0:613ị 8Qo 0:5ị 6Qo 0:387ị ỵ 3Qo 0:307ị ỵ 4Qo 0:206ị ỵ 3Qo 0:126ị; and n4 ẳ ẵ3Qo 0:904ị ỵ 4Qo 0:841ị ỵ 3Qo 0:760ị 9Qo 0:709ị 40k 12Qo 0:614ị ỵ 9Qo 0:514ị 9Qo 0:486ị ỵ 12Qo 0:386ị ỵ 9Qo 0:291ị 3Qo ð0:240Þ À 4Qo ð0:159Þ À 3Qo ð0:096ފ: Then, the LQ-skewness and the LQ-kurtosis for each case (median, trimean and Gastwirth) for the EGEV distribution can be obtained for the EGEV distribution TL-moments and LQ-moments estimators In this section, the use of the TL-moments and the LQ-moments for estimating the unknown parameters of the EGEV distribution will be derived solve (because the gamma function is a function of k) So, these equations will be solved numerically As a special case, ^ for the TL-moments by putting s ¼ t ¼ 1, the TLMEs k^ and b with the first trimmed and for s ¼ t ¼ 0, the L-moments estimates (LMEs) can be obtained for the EGEV distribution Also, for b = 1, and by putting s ¼ t ¼ 1, the TLME k^ for the TL-moments with the first trimmed and for s ¼ t ¼ 0, the LME for k can be obtained for the GEV distribution LQ-Moments estimators To estimate the unknown parameters k and b for the EGEV distribution using the LQ-moments, the first and the second sample LQ-moments for the EGEV distribution will be obtained by using the following definition of the rth sample LQ-moments:   r1 X r1 ^fr ẳ r1 ^sp;d Xrk:rị Þ; r ¼ 1; 2; ðÀ1Þk k kẳ0 where bX bX ^sp;d Xrk:rị ị ẳ p Q dị ỵ 2pị Q 1=2ị rk:rị rk:rị TL-Moments estimators bX ỵ pQ dị: rk:rị The TL-moment estimators (TLMEs) for the unknown parameters of the EGEV distribution can be obtained by equating   ðs;tÞ ðs;tÞ the first two population TL-moments k1 ; k2 to the corre  ðs;tÞ ðs;tÞ sponding sample TL-moments l1 ; l2 for the EGEV distribution Hosking [5] obtained the first two sample TL-moments to be:   nÀt  X jÀ1 nÀj s;tị  l1 ẳ  xj:nị ; n s t jẳsỵ1 sỵtỵ1 and nt X s;tị  l2 ẳ  n jẳsỵ1 sỵtỵ2  n j tị xj:nị t ỵ 1ị  j1 s  nÀj t  ðj À s À 1Þ ðs þ 1Þ Clearly, sample TL-moments reduce to sample L-moments when s ¼ t ¼ Now, we can obtain the TL-moment estima^ of the EGEV distribution by solving tors (TLMEs) ðk^ and bÞ the following two equations: " #   t ^ s ỵ t ỵ 1ị! X s;tị j t kỵ1ị k^ ^ ^ ; l1 ẳ b Ck ỵ 1ị 1ị s ỵ j ỵ 1ị sị!tị! jẳ0 j k^ 9ị ^sp;d Xrk:rị ị is the quick estimator of the location for the distrib X ð:Þ debution of X(rÀk:r) in a random sample of size r, and Q notes the linear interpolation estimator of Quị given by: b X uị ẳ eịXẵn0 u:n ỵ eXẵn0 uỵ1:n; Q where e ẳ n0 u ẵn0 u0n ẳ n ỵ and [n u] denote the integral part of n u Then, the first two sample LQ-moments will be: ^f1 ¼ ^sp;d X1:1ị ị; and ^f2 ẳ ^sp;d X2:2ị Þ À ^sp;d ðXð1:2Þ Þ : By equating the first two population LQ-moments with the first two sample LQ-moments for the EGEV distribution for each case (median, trimean, and Gastwirth), the LQ-moments estimators for the two unknown parameters will be obtained for each case Now, the unknown parameters k and b for the EGEV distribution using the LQ-moments with the median case (LQMEm) will be estimated Since, the first sample LQ-moments is a function of k and b and the second sample LQ-moments is a function also of k and b, then by numerically solving ^ the equations for and to obtain the LQ-moments estimates k^ ^ ^ then: and b, ^1 ẳ ẵ Q b 0:5ị; n ^ h k^ and and ðs;tÞ l2 355 ^Àk^ ^ s ỵ t ỵ 2ị! ẳ b Ck ỵ 1ị ^ s ỵ 1ị! 2k " # ^ k^ t X s ỵ 1ị 1ịj s ỵ j ỵ 2ịk : t ỵ 1ị! j ỵ 1ị!t jị! jẳ0 10ị The Eqs (9) and (10) are valid for any trimmed s and t and To solve these equations, determine the value of trimmed or the value of s and t; but the resulting equations are difficult to i hb ^ b h ð0:293Þ ; Q h 0:707ị Q n2 ẳ ^ 2k^ !k^^ ¼ À À lnðuÞ : ^ ^ b b h ðuÞ where Q ^ For the trimean case the LQ-moments estimates (LQMEt) k^ ^ ^ will be obtained by solving the following two equations: and b 356 N.A.T Abu El-Magd i hb ^ b h 0:5ị ỵ Q b h 0:75ị ; n1 ẳ Q 0:25ị ỵ Q ^^ h 4k   r r Àkj X lr ẳ r b Ckj ỵ 1ị; 1ịj k j¼0 j and In particular, the first four moments of the EGEV distribution will be: l ¼ ð1 À bk Ck ỵ 1ịị; k P 1; k 1 l2 ẳ 2bk Ck ỵ 1ị ỵ b2k C2k ỵ 1ịị; k P ; k Àk À2k l3 ¼ ð1 À 3b Cðk þ 1Þ þ 3b Cð2k þ 1Þ k À b3k C3k ỵ 1ịị; k P ; i h b ^ b h 0:293ị ỵ Q b h 0:866ị Q b h 0:134ị ; n2 ẳ Q h ð0:707Þ À Q ^ 8k^ and, for the Gastwirth case the LQ-moments estimates ^ ^^ (LQMEg) k^ and b will be obtained by solving the following two equations: i h b ^ b h ð0:5Þ ỵ Q b h 0:667ị ; n1 ẳ Q h 0:333ị ỵ Q ^ 10k^ and h b ^ b h 0:707ị ỵ Q b h 0:577ị n2 ẳ Q h 0:816ị ỵ Q ^ 20k^ i b h ð0:293Þ À Q b h ð0:184Þ b h ð0:423Þ À Q À3 Q For b = 1, the unknown parameter k will be estimated for the GEV distribution using the LQ-moments Each of the three cases can be obtained as a special case from the LQ-moments for the EGEV distribution By equating the first population LQ-moments with the first sample LQ-moments for the trimean case as follows: i 1h b ^ b b b h uị n1 ẳ Q where Q h 0:25ị ỵ Q h 0:5ị ỵ Q h 0:75ị À 1;  ! ¼ À À lnðuÞ ^^ b ^^ to obtain the LQ-moments estimator k, the equation for for the GEV distribution is solved numerically; and the same for the other two cases (median and Gastwirth) for the GEV distribution The classical moments and L-moments In this section, the true formulae for the rth classical moments and the probability weighted moments (PWMs) for the EGEV distribution will be obtained Also, we will obtain the L-moments for the EGEV distribution by using the PWMs The classical (traditional) moments of the EGEV distribution The rth moments for the EGEV distribution can be obtain from: Z 1=k lr ¼ xr fðxÞdx kPÀ : j ð11Þ and l4 ẳ 1 4bk Ck ỵ 1ị ỵ 6b2k C2k ỵ 1ị 4b3k C3k ỵ 1ị k4 þ bÀ4k Cð4k þ 1ÞÞ; k P À : From these results we can obtain the coefficient of skewness and the kurtosis for the EGEV distribution The L-moments of the EGEV distribution Let, X be a real valued random variable with cdf FðxÞ and quantile function Q(u) Greenwood et al [11] defined the PWMs of X to be the following quantities: Mp;r;s ẳ EẵXp fFXịgr f1 FXịgs where p, r and s are real numbers; we can obtain the PWMs of the EGEV to be: R1 Mp;r;s ¼ ðQðuÞÞp ur ð1 À uÞs du;   k p R1 ¼ k1p À À b1 lnðuÞ ur ð1 À uÞs du;     k p s R1 P s s 1 b lnuị urỵj du ẳ kp 1ị j jẳ0 Let  k v ẳ b1 lnuị ) du ẳ bk v1=k1 expbv1=k ịịdv and u = (exp(v)1/k)Àb then:   s P s s R1 p 1ị vị v1=k1 expbr ỵ j ỵ 1ịv1=k ịdv; j jẳ0   p s s s P i b P 1ị 1ị ẳ kpỵ1 j iẳ0 jẳ0   p R 1=kỵi1 v expbr ỵ j ỵ 1ịv1=k ịdv: i b Mp;r;s ẳ kpỵ1 ẳ Z 1=k xr b1 kxị1=k1 expðÀbð1 À kxÞ1=k Þdx: À1   k  Let v = b(1 À kx)1/k ) vkÀ1dv = À bkdx and x ¼ k1 À bv then we have:   k !r Z 1 v lr ¼ r 1À expðÀvÞdv k b   Z r X j r 1ị vkj expvịdv: ẳ r bÀkj j k j¼0 Then, the rth moments for the EGEV distribution is: Now, let y = v1/k ) yk = v and dv = kykÀ1dy then we have:   p s P s P ðÀ1Þs ðÀ1Þi j iẳ0 jẳ0   p R k 1=kỵi1 y ị expbr ỵ j ỵ 1ịyịkyk1 dy; i   p s s P P ẳ kbp 1ịs 1ịi j iẳ0 jẳ0   p br ỵ j ỵ 1ịịkiỵ1ị Cki ỵ 1ị; k > 1i : i b Mp;r;s ẳ kpỵ1 TL-moments of the exponentiated generalized extremevalue distribution Putting s ¼ r ¼ 0, and p ¼ r, we will obtain the rth moments (11) for the EGEV distribution as a special case from the PWMs for the EGEV distribution as follows:   r r b X Mr;0;0 ẳ r bịkiỵ1ị Cki ỵ 1ị; k > : 1ịi k iẳ0 i i One possible approach is to work with the moments into which X enters linearly and in particular with the quantities: Z br ẳ M1;r;0 ẳ EẵXfFXịgr ẳ Quịur du: Using the PWMs, Hosking [9] introduced the L-moments of order r ỵ as follows: krỵ1 ẳ r X 357 the GEV distribution as a special case from the br for the EGEV distribution as per the Hosking et al [2] and Hosking [9] results The L-moments of order r ỵ for the EGEV distribution will be:    r X r rỵj krỵ1 ẳ 1ịrj j j jẳ0 1 bj ỵ 1ịịk Ck ỵ 1ịị; k P 1: kj ỵ 1ị Putting r ¼ 0; 1; 2; 3, the results are the same as the results in Eqs (5)–(8) for the first four L-moments for the EGEV The classical moments and L-moments estimators Cr;j bj ; r ¼ 0; 1; where    r rỵj Cr;j ẳ 1ị : j j In this section, we will introduce the method of moment estimators (MMEs) (classical estimators) and the L-moment estimators (LMEs) for the EGEV distribution Hence, we have: The classical estimators of the EGEV distribution j¼0 rÀj br ¼ M1;r;0 b ẳ br ỵ 1ịị1 br ỵ 1ịịkỵ1ị Ck ỵ 1ịị k 1 br ỵ 1ịịk Ck ỵ 1ịị; k > ẳ kr ỵ 1ị when k À 1,b⁄ (the mean of the distribution) and the rest of the br does not exist (Hosking et al [2]) For b = 1, we obtain br for Table Biases and RMSEs of the parameter estimators for different types of estimators for k n k 15 À0.4 À0.2 0.2 0.4 25 À0.4 À0.2 0.2 0.4 50 À0.4 À0.2 0.2 0.4 100 Now, we will introduce the method of moment estimators (MMEs) of the parameters k and b of the EGEV distribution For the EGEV distribution, we have two parameters, so we require the first two sample moments: sample mean and variance These sample moments are equated to their population analogues, and the resulting equations are: À0.4 À0.2 0.2 0.4 Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs MME LME TLME LQMEm LQMEt LQMEg 0.794 0.847 0.818 0.827 0.884 0.908 0.558 0.566 0.772 0.820 0.800 0.808 0.903 0.926 0.544 0.548 0.745 0.799 0.784 0.787 0.884 0.885 0.536 0.537 0.731 0.768 0.775 0.777 0.967 0.986 0.535 0.536 0.593 0.595 0.313 0.320 À0.142 0.147 À0.414 0.414 0.595 0.596 0.315 0.321 À0.141 0.146 À0.414 0.414 0.592 0.593 0.318 0.324 À0.136 0.141 À0.414 0.414 0.587 0.587 0.317 0.322 À0.128 0.133 À0.414 0.414 0.017 0.140 0.008 0.106 0.001 0.056 0.0007 0.040 0.010 0.107 0.005 0.081 0.001 0.042 0.0004 0.030 0.005 0.075 0.002 0.056 0.0004 0.029 0.0001 0.021 0.002 0.053 0.001 0.039 0.0002 0.020 0.0007 0.014 À0.048 0.223 À0.031 0.159 À0.012 0.079 À0.007 0.056 À0.027 0.168 À0.018 0.122 À0.007 0.062 À0.004 0.044 À0.012 0.142 À0.008 0.087 À0.003 0.045 À0.002 0.032 À0.006 0.083 À0.004 0.061 À0.001 0.032 À0.001 0.023 À0.066 0.194 À0.040 0.129 À0.015 0.061 À0.009 0.043 À0.038 0.136 À0.024 0.095 À0.009 0.047 À0.005 0.033 À0.018 0.111 À0.011 0.066 À0.004 0.034 À0.002 0.024 À0.008 0.063 À0.005 0.046 À0.002 0.024 À0.001 0.017 À0.051 0.188 À0.034 0.134 À0.013 0.067 À0.008 0.047 À0.031 0.140 À0.021 0.102 À0.008 0.052 À0.005 0.037 À0.045 0.115 À0.010 0.071 À0.004 0.037 À0.002 0.026 À0.008 0.067 À0.005 0.050 À0.002 0.026 À0.001 0.018 358 x ¼ N.A.T Abu El-Magd  1 k Ck ỵ 1ị ; 1b k k P 1; 12ị l2 ẳ and  k 2k  s2 ẳ 2b Ck ỵ 1ị ỵ b C2k ỵ 1ị ; k ð13Þ kà P À where and are the sample mean and the sample variance, respectively Then, the MMEs of k and b, say k* and b*, respectively can be obtained by solving the two Eqs (12) and (13) The L-moments estimators of the EGEV distribution Now, we will introduce the L-moment estimators (LMEs) for the EEGEV distribution If X(1:n) X(2:n) 6Á Á Á X(n:n) denotes the order sample, we have the first and second sample L-moments as: n 1X l1 ẳ xi:nị ; n iẳ1 and l2 ẳ n X ði À 1Þxði:nÞ À l1 : nðn 1ị iẳ1 k l1 ẳ b Ck ỵ 1ịị; k Table k 15 0.4 À0.2 0.2 0.4 À0.4 À0.2 0.2 0.4 À0.4 À0.2 0.2 0.4 100 kÃà P À1: ð15Þ Then, the LMEs of k and b, say k** and b**, respectively, can be obtained by solving the equations for (14) and (15) Methodology In this section, we will introduce a numerical simulation to compare the properties of the TLMEs, LMEs and LQMEs {LQMEm (median), LQMEt (trimean), LQMEg (Gastwirth)} estimation methods with the MMEs of the EGEV distribution mainly with respect to their biases and root mean square errors (RMSEs) The simulation experiments are performed using the Mathcad (2001) software, different sample sizes n ¼ 15; 25; 50 and 100, and different values for the parameter k = À 0.4, À0.2, 0.2, and 0.4 and for b = 15 For each combination of the sample size and the shape parameters values, the experiment will be repeated 50,000 times In each experiment, the biases and RMSEs for the estimates of k and b will be obtained and listed in Tables and Ãà k P À1 ð14Þ It is observed in Table that most of the estimators usually overestimate k except LMEs and LQMEs, which underestimate all times As far as biases are concerned, the TLMEs Biases and RMSEs of the parameter estimators for different types of estimators for b n 50 Ãà k b Ck ỵ 1ị1 2k ị; k Results and discussion Equating the first two population L-moments , to the corresponding sample L-moments l1 , l2 we will obtain: 25 and À0.4 À0.2 0.2 0.4 Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs Bias RMSEs MME LME TLME LQMEm LQMEt LQMEg 393.000 848.100 73.920 21.330 18.178 16.130 À9.892 10.47 911.000 162.300 56.350 44.050 37.330 31.422 À10.278 10.498 952.100 211.500 52.560 30.990 27.720 13.850 À10.591 10.762 865.300 193.200 56.170 20.670 29.285 16.515 À10.751 10.790 À14.981 15.526 À11.364 16.030 À12.301 17.082 À12.914 12.914 À15.000 15.000 À14.583 35.328 À12.264 12.496 À12.903 12.903 À15.000 15.000 À15.000 15.000 À12.364 12.600 À12.900 12.900 À15.000 15.000 À15.000 15.000 À12.785 12.999 À12.900 12.900 4.401 14.702 3.257 11.591 2.045 8.002 1.775 7.033 2.205 8.108 1.644 6.777 1.065 5.110 0.941 4.627 0.973 4.575 0.726 3.966 0.479 3.167 0.429 2.923 0.453 2.939 0.336 2.584 0.223 2.111 0.201 1.965 6.306 37.148 4.482 22.457 2.408 11.930 1.850 9.649 3.463 15.018 2.561 11.740 1.450 7.770 1.137 6.609 1.473 12.018 1.323 6.871 0.769 4.968 0.610 4.348 0.800 4.968 0.606 4.268 0.356 3.218 0.284 2.860 1.233 11.674 0.837 9.412 0.483 6.737 0.382 6.006 0.650 7.739 0.476 6.513 0.297 4.935 0.234 4.464 0.640 4.739 0.312 4.374 0.185 3.410 0.147 3.112 0.197 3.410 0.141 2.956 0.083 2.342 0.066 2.149 2.444 14.511 1.807 11.557 1.049 7.966 0.829 6.933 1.347 9.282 1.022 7.739 0.620 5.689 0.497 5.053 0.307 6.282 0.497 4.817 0.304 3.749 0.245 3.390 0.279 3.630 0.212 3.200 0.131 2.554 0.106 2.331 TL-moments of the exponentiated generalized extremevalue distribution are less unbiased and the minimum RMSEs for all different values of k and n are considered here The RMSEs of the TLMEs are also quite close to the LQMEts and LQMEgs Comparing all the methods, we conclude that for the parameter k, the TLMEs should be used for estimating k Now consider the estimation of b In this case, it is observed in Table that most of the estimators usually overestimate k except LMEs, which underestimate all times As far as biases are concerned, the LQMEts are less unbiased and the minimum RMSEs for all different values of k and for n ¼ 15 and 25 Conclusions Comparing the biases of all the estimators, it is observed that the LQMEts perform the best for most different values of k and n considered here The performance of the LQMEgs and TLMEs is quite close to the LQMEts for all cases considered here As far as RMSEs are concerned, TLMEs are the minimum RMSEs for all different values of kand for n ¼ 50 and 100 Comparing the performance of all the estimators, it is observed that as far as biases or RMSEs are concerned, the TLMEs perform best in most cases considered here Interestingly, while estimating k, the biases and RMSEs of the LQMEt are lower than the other estimators most of the time We recommend using the TLMEs for estimating k and b (n ¼ 50 and 100) and recommend using the LQMEts for estimating b (n ¼ 15 and 25) Acknowledgements The author is deeply grateful to the referee and the editor of the journal for their extremely helpful comments and valued 359 suggestions that led to this improved version of the paper I would like to thank Prof Samir Kamel Ashour for helpful scientific input and for editing this manuscript References [1] Adeyemi S, Adebanji AO The exponentiated generalized extreme value distribution J Appl Funct Differ Equ (JAFDE) 2006;1(1):89–95 [2] Hosking JRM, Wallis JR, Wood EF Estimation of the generalized extreme-value distribution by the method of probability-weighted moments Technometrics 1985;27(3):251–61 [3] Elamir EAH, Seheult AH Trimmed L-moments Comput Stat Data Anal 2003;43(3):299–314 [4] Hosking JRM L-moments: analysis and estimation of distributions using linear combinations of order statistics JR Stat Soc B 1990;52(1):105–24 [5] Hosking JRM Some theory and practical uses of trimmed Lmoments J Stat Plan Inference 2007;137(9):3024–39 [6] Maillet B, Me´decin JP Financial crises, extreme volatilities and L-moment estimations of tail-indexes J Economet, 2010 [7] Wang QJ LH-moments for statistical analysis of extreme events Water Resour Res 1997;33(12):2841–8 [8] Bayazit M, Oănoăz B LL-moments for estimating low ow quantiles [Estimation des quantiles d’e´tiage graˆce aux LLmoments] Hydrolog Sci J 2002;47(5):707–20 [9] Hosking JRM The theory of probability weighted moments 1986 Research Report RC12210 Yorktown Heights, NY; IBM Research Division [10] Mudholkar GS, Hutson AD LQ-moments: analogs of Lmoments J Stat Plan Inference 1998;71(1–2):191–208 [11] Greenwood JA, Landwehr JM, Matalas NC, Wallis JR Probability weighted moments: definition and relation to parameters of several distributions expressable in inverse form Water Resour Res 1979;15(5):1049–54  ... 2.331 TL-moments of the exponentiated generalized extremevalue distribution are less unbiased and the minimum RMSEs for all different values of k and n are considered here The RMSEs of the TLMEs... kpỵ1 TL-moments of the exponentiated generalized extremevalue distribution Putting s ¼ r ¼ 0, and p ¼ r, we will obtain the rth moments (11) for the EGEV distribution as a special case from the. .. section, the use of the TL-moments and the LQ-moments for estimating the unknown parameters of the EGEV distribution will be derived solve (because the gamma function is a function of k) So, these