# Estimation of Gumbel Parameters under Ranked Set Sampling

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1 Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, Sameer A. Al-Subh Mutah Uiversity, Karak, Jorda, Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Yousef, Omar M. ad Al-Subh, Sameer A. (2014) "Estimatio of Gumbel Parameters uder Raked Set Samplig," Joural of Moder Applied Statistical Methods: Vol. 13 : Iss. 2, Article. DOI: /masm/ Available at: This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of

5 YOUSEF & AL-SUBH After takig the derivatives with respect to α ad β equatig to 0, the MLEs are obtaied as ˆ ad ˆ log. (4) ˆ x x w z MLE, S i i MLE, S MLE, S i1 where x i 1 zi zi exp, z zi ad wi. ˆ mle, S i1 z MOMEs The mea ad variace for Gumbel distributio are give by ad. (5) 6 The momet estimators of the two parameters are ˆMOME, S 6 s ad ˆ ˆ MOME, S x MOME, S (6) where s, x are the sample stadard deviatio ad mea, respectively, ad γ = is Euler s costat. REGs x x Let y Fx ( ;, ) exp exp l y= exp x x l y= exp t=l -l y = t a x b 1 where a ad b. 435

6 ESTIMATION OF GUMBEL PARAMETERS The regressio estimators of the two parameters are ˆ 1, ad ˆ ˆ,, ˆ REG S REG S REG S t ax (7) aˆ x x t t i i i1 where aˆ. 2 i1 x i x Parameter Estimatio Uder RSS MLEs Let X(i:m), i = 1,, m ad = 1,, r deote the i th order statistics from the i th set of size m of the th cycle be the RSS data for X with sample size = mr. Usig (1) ad (2), the pdf of X(i:m) is give by (Arold et al.,1992) i-1 m-i 1 fi : m( X ) cf( X ) 1- F( X ) f ( X ), where c B( i, m -i 1), 1 X X f( X ) exp exp ad F X by X ( ) exp exp. r l(, ) f ( X ) m i1 1 r m i1 1 i: m i r m i i1 1 The the likelihood fuctio is give -1 m-i = c F( X ) 1- F( X ) f ( X ) mr -1 m-i c F( X ) 1- F( X ) f ( X ). 436

8 ESTIMATION OF GUMBEL PARAMETERS Table 1. The bias ad MSE of estimators of α Bias MSE (α, b) =mr ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r (1,1) (1,2) (2,1) (0.5,1) (1,0.5) m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r=

9 YOUSEF & AL-SUBH Table 2. The efficiecy of estimators of α (α, b) =mr ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r (1,1) (1,2) (2,1) (0.5,1) (1,0.5) m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r=

10 ESTIMATION OF GUMBEL PARAMETERS Table 3. The bias ad MSE of estimators of β Bias MSE (α, b) =mr ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r (1,1) (1,2) (2,1) (0.5,1) (1,0.5) m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r=

11 YOUSEF & AL-SUBH Table 4. The efficiecy of estimators of β (α, b) =mr ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r (1,1) (1,2) (2,1) (0.5,1) (1,0.5) m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= From Tables 1 to 4, the followig coclusios are put forth i) I geeral, the bias is large for all estimators. Therefore, all the estimators are cosidered as biased estimators for α. ii) From Table 1, it ca be oticed that the REG uder SRS has the smallest bias as compared to the other estimators cosidered i most cases. I geeral, for all estimators of α uder RSS, the bias is less tha the case uder SRS. iii) For fixed α, the MSE of ˆ decreases as the sample size icreases. iv) It is oticed that from Table 2 that MLE uder RSS is the most efficiet tha the MLE based o SRS. 441

13 YOUSEF & AL-SUBH McItyre, G. A. (1952). A method of ubiased selective samplig usig raked sets. Australia Joural of Agricultural Research, 3, Mousa, M. A. M., Jahee, Z. F., & Ahmad, A. A. (2002). Bayesia Estimatio, Predictio ad Characterizatio for the Gumbel Model Based o Records. A Joural of Theoretical ad Applied Statistics, 36(1), Muttlak, H. A., & Al-Saleh, M. F Recet developmets i raked set samplig, Pakista Joural of Statistics, 16, Phie, H. N. (1987). A review of methods of parameter estimatio for the extreme value type-1 distributio. Joural of Hydrology, 90(3 4), Takahasi, K., & Wakitmoto, K. (1968). O ubiased estimates of the populatio mea based o the sample stratified by meas of orderig. Aals of the Istitute of Statistical Mathematics, 20,

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