Appled Mathematcs ad Computato 215 (2010) 4198 4202 Cotets lsts avalable at SceceDrect Appled Mathematcs ad Computato joural homepage: www.elsever.com/locate/amc Improvemet estmatg the populato mea smple radom samplg usg formato o auxlar attrbute A.M. Abd-Elfattah *, E.A. El-Sherpe, S.M. Mohamed, O.F. Abdou Isttute of Statstcal Studes ad Research, Caro Uverst, Dokk, Gza 12613, Egpt artcle fo abstract Kewords: Rato-tpe estmator Smple radom samplg Auxlar attrbute Effcec Ths paper proposes some estmators for the populato mea b adaptg the estmator Sgh et al. (2008) [5] to the rato estmators preseted Kadlar ad Cg 2006 [2]. We obta mea square error (MSE) equato for all proposed estmators, ad show that all proposed estmators are alwas more effcet tha rato estmator Nak ad Gupta (1996) [3], ad Sgh et al. (2008) [5]. The results have bee llustrated umercall b takg some emprcal populato cosdered the lterature. Ó 2009 Elsever Ic. All rghts reserved. 1. Itroducto Cosder a sample of sze draw b smple radom sample wthout replacemet from a populato of sze N. Let ad u deoted the observato o varable ad u, respectvel, for th ut ð ¼ 1; 2; 3;...; NÞ. Suppose there s a complete dchotom the populato wth respect to the presece or absece of a attrbute, sa u, ad t s assumed that attrbute u takes ol the two values 0 ad 1 accordg as u ¼ 1; f th ut of the populato possesses attrbute u ¼ 0; f otherwse: Let A ¼ P N ¼1u ad a ¼ P ¼1u deoted the total umber of uts the populato ad sample possessg attrbute u, respectvel. Let P ¼ A ad P b ¼ a deoted the proporto of uts the populato ad sample, respectvel, possessg attrbute u. Takg to cosderato the pot bseral correlato coeffcet betwee auxlar attrbute ad stud varable, N Nak ad Gupta [3] defed rato estmator of populato mea whe the pror formato of populato proporto of uts, possessg the same attrbute s avalable, as follows: t NG ¼ P b P ; ð1:1þ where s the sample mea of stud varable. The MSE of t NG up to the frst order of approxmato s MSEðt NG Þ¼ 1 f h S 2 þ R2 1 S2 u 2R 1S u ; ð1:2þ where f ¼ ; s the sample sze; N s the umber of uts the populato; R N 1 ¼ Y ; P S2 u s the populato varace of auxlar attrbute u, ad S u s the populato covarace betwee varable of terest ad auxlar attrbute u. * Correspodg author. Address: Departmet of Statstcs, Facult of Scece, Kg Abdul Azz Uverst Box 80203, Jedda 21589, Sauda Araba. E-mal address: a_afattah@hotmal.com (A.M. Abd-Elfattah). 0096-3003/$ - see frot matter Ó 2009 Elsever Ic. All rghts reserved. do:10.1016/j.amc.2009.12.041
A.M. Abd-Elfattah et al. / Appled Mathematcs ad Computato 215 (2010) 4198 4202 4199 Sgh et al. [5] suggested the followg rato estmators for estmatg the populato mea Y of the stud varable smple radom samplg usg kow parameters of auxlar attrbute u, such as, coeffcet of varato C P, coeffcet of kurtoss B 2 ðuþ, ad pot bseral correlato coeffcet q Pb as: t 1 ¼ þ b uðp PÞ b P; ð1:3þ t 2 ¼ þ b uðp PÞ b ½P þ B 2 ðuþš; þ B 2 ðuþ ð1:4þ t 3 ¼ þ b uðp PÞ b ½P þ C P Š; þ C P ð1:5þ t 4 ¼ þ b uðp b PÞ B 2 ðuþþc P ½PB 2 ðuþþc P Š; ð1:6þ t 5 ¼ þ b uðp b PÞ C P þ B 2 ðuþ ½PC P þ B 2 ðuþš; where C P ad B 2 ðuþ are the populato coeffcet of varato ad the populato coeffcet of kurtoss of auxlar attrbute, respectvel, ad b u ¼ su s the regresso coeffcet. Here s 2 s 2 u s the sample varace of auxlar attrbute ad s u s the sample covarace betwee the auxlar attrbute ad the stud varable. u I Sgh et al. [5], mea square error MSE equato of these rato estmators were gve b MSEðt Þ¼ 1 f h R 2 S2 u þ S2 ð1 q2 Pb Þ ; ð ¼ 1; 2; 3;...; 5Þ; ð1:8þ where R 1 ¼ Y ; R P 2 ¼ Y ; R PþB 2 ðuþ 3 ¼ Y PþC P ; R 4 ¼ YB 2ðuÞ PB 2 ðuþþc P ad R 5 ¼ YC P. PC P þb 2 ðuþ Sgh et al. [5] cocluded that the rato estmators t ð ¼ 1; 2;...; 5Þ whch uses some kow value of populato proporto were more effcet tha the sample mea ad rato estmator Nak ad Gupta [3]. I the ext secto, we develop ew estmators combg rato estmators Sgh et al. [5] ad obta the MSE equatos of these ew estmators. I Secto 3, we compare the effceces, theoretcall, based o MSE equatos, betwee the proposed estmators ad the rato estmators preseted Sgh et al. [5]. I Secto 4, we also dscuss the comparso amog all the suggested estmators umercall. I Secto 5, we gve a ht to obta dfferet estmators b a smlar method preseted ths stud. 2. Suggested estmators We propose the estmator usg the procedure preseted Kadlar ad Cg 2006 [2] combg rato estmators (1.3) ad (1.4) as follows: ð1:7þ t pro1 ¼ m 1 P þ m b 2 ðp þ B 2 ðuþþ; þ B 2 ðuþ ð2:1þ where m 1 ad m 2 are weghts that satsf the codto m 1 þ m 2 ¼ 1. The MSE of ths estmator ca be foud usg the frst degree approxmato the Talor seres method defed b MSEðt pro1 Þffd X d 0 ; ð2:2þ where d s a vector defed as d ¼ @hða;bþ @hða;bþ @a @b ; P s the varace covarace matrx as P " # ¼ 1 f S 2 S u Y;P Y;P S u S 2 (see Wolter [7]). Here hða; bþ ¼hð; PÞ¼t b pro1. Accordg to ths defto, we obta d for the proposed estmator as u follows: d ¼ 1 m 1 ðr 1 þ B u Þ m 2 ðr 2 þ B u Þ ; where B u ¼ Su ¼ q Pb S. Note that we omt the dfferece : b B (Cochra [1]). S 2 u Su We obta the MSE of the proposed estmator usg (2.2) as MSEðt pro1 Þ¼ 1 f S 2 2gS u þ g 2 S 2 u ; ð2:3þ where g ¼ m 1 ðr 1 þ B u Þþm 2 ðr 2 þ B u Þ: ð2:4þ We also propose the estmator combg rato estmators (1.3) ad (1.5) as t pro2 ¼ m b 1 P þ m b 2 ðp þ C P Þ: ð2:5þ þ C P The MSE of ths estmator s the same as (2.3) but R 2 (2.4) s replaced wth R 3.
4200 A.M. Abd-Elfattah et al. / Appled Mathematcs ad Computato 215 (2010) 4198 4202 I addto, we propose the followg estmator combg rato estmators (1.3) ad (1.6) as t pro3 ¼ m 1 P þ m 2 B 2 ðuþþc P ½PB 2 ðuþþc P Š: ð2:6þ The mea square error of ths estmator s aga the same as (2.3) but R 2 (2.4) s replaced wth R 4. Lastl, we propose the followg estmator combg rato estmators (1.3) ad (1.7) as t pro4 ¼ m b 1 P þ m b 2 C P þ B 2 ðuþ ½PC P þ B 2 ðuþš: ð2:7þ The mea square error of ths estmator s aga the same as (2.3) but R 2 (2.4) s replaced wth R 5. The optmal values of m 1 ad m 2 to mmze (2.3) ca easl be foud as follows: m 1 ¼ R 2 R 2 R 1 ad m 2 ¼ R 1 R 1 R 2 ; ð2:8þ whe we use m 1 ad m 2 stead of m 1 ad m 2 (2.4), we get g ¼ B u.asgs depedet of R 2, all proposed estmators have the same mmum MSE as follows: MSE m ðt pro Þ¼ 1 f S 2 2B us u þ B 2 u S2 u ; ¼ 1; 2; 3; 4: We ca also wrte ths expresso as MSE m ðt pro Þ¼ 1 f S2 ð1 q2 PbÞ: ð2:9þ 2.1. New rato estmators We suggest followg estmator: t pro ¼ ðm m 1P b 1 P þ m 2 Þ; þ m2 where m 1 ad m 2 are ether real umber or the fucto of the kow parameter of auxlar attrbute such as C P ; B 2 ðuþ ad q Pb, ote that the sum of m 1 ad m 2 ot ecessarl equal to oe. The followg scheme presets some of the mportat estmators of the populato mea, whch ca be obtaed b sutable choce of costats m 1 ad m 2 : ð2:10þ Estmator Values of m 1 m 2 t pro1sd ¼ b PþCP PþCP 1 C P t pro2sk ¼ PþB 2ðuÞ b PþB2ðuÞ 1 B 2 ðuþ t pro3us1 ¼ PB 2ðuÞþC P b PB2ðuÞþC P B 2 ðuþ C P t pro3us2 ¼ PCPþB 2ðuÞ b PCPþB 2ðuÞ C P B 2 ðuþ t pro4st ¼ Pþq Pb b PþqPb 1 q Pb We obta the MSE equato for these proposed estmators as MSEðt pro Þ¼ 1 f h Y2 C 2 þ C2 P w ðw 2K Pb Þ ¼ 1; 2;...; 5; where w 1SD ¼ P PþC P ; w 2SK ¼ P ; w PþB 2 ðuþ 3US1 ¼ PB 2ðuÞ PB 2 ðuþþc P ; w 3US2 ¼ PC P ad w PC P þb 2 ðuþ 4ST ¼ Pþq P. Pb ð2:11þ 3. Effcec comparso I ths secto, frstl, we compare MSE of proposed estmators, gve (2.9), wth the MSE of rato estmator preseted Sgh et al. [5], gve (1.8). As we obta the followg codto b these comparso: R 2 S2 u > zero: ð3:1þ
A.M. Abd-Elfattah et al. / Appled Mathematcs ad Computato 215 (2010) 4198 4202 4201 Table 1 Percet relatve effceces of ; t NG; t ð ¼ 1; 2;...; 5Þ ad t pro wth respect to. Estmator PREs (.,Þ Populato I II 100 100 t NG 7.124 7.804 t 1 4.931 5.695 t 2 237.766 161.873 t 3 221.45 155.231 t 4 72.098 69.341 t 5 215.271 152.612 t pro 242.19 163.61 We ca fer that all proposed estmators are more effcet tha all rato estmators preseted Sgh et al. [5] all codtos, because the codto gve (3.1) s alwas satsfed. Secodl, we compare the MSE of the ew estmators gve (2.11) wth the varace of sample mea, so we have the followg codto: MSEðt pro Þ < VðÞ; ¼ 1; 2;...; 5; f; w 2q Pb C C P < zero; C 2q Pb > w C ; P ) q Pb > 1 C P w 2 C ; ¼ 1; 2;...; 5: ð3:2þ Whe ths codto s satsfed, proposed estmators are more effcet tha the sample mea. 4. Emprcal stud We ow compare the performace of varous estmators cosdered here usg the two data sets as prevousl used b Shabbr ad Gupta [4]. Populato I (Source: Sukhatme ad Sukhatme [6], p. 256). = Number of vllages the crcles. u = A crcle cosstg more tha fve vllages. N ¼ 89; Y ¼ 3:36; P ¼ 0:124; q Pb ¼ 0:766; C ¼ 0:601; C P ¼ 2:678; ¼ 23; B 2 ðuþ ¼ 6:162; R 1 ¼ 27:18; R 2 ¼ 0:534; R 3 ¼ 1:199; R 4 ¼ 6:019; R 5 ¼ 1:386. Populato II (Source: Sukhatme ad Sukhatme [6], p. 256). = Area ( acres) uder wheat crop the crcles. u = A crcle cosstg more tha fve vllages. N ¼ 89; Y ¼ 1102; P ¼ 0:124; q Pb ¼ 0:624; C ¼ 0:65; C P ¼ 2:678; ¼ 23; B 2 ðuþ ¼6:162; R 1 ¼ 8915; R 2 ¼ 175:31; R 3 ¼ 393:31; R 4 ¼ 6:019; R 5 ¼ 454:468. We have computed the percet relatve effceces (PREs) of ; t NG ; t ð ¼ 1; 2;...; 5Þ ad t pro wth respect to usual ubased estmator ad dsplaed Table 1. From Table 1 t ca be cocluded that all proposed estmators t pro ð ¼ 1; 2; 3; 4Þ are more effcet tha the usual ubased estmator, rato estmators of Nak ad Gupta [3], ad the rato estmators preseted Sgh et al. [5]. 5. Cocluso We have developed ew estmators combg rato estmators cosdered Sgh et al. [5] ad obtaed the mmum MSE equato for the proposed estmators. Theoretcall, we have demostrated that all proposed estmators are alwas more effcet tha rato estmators. I addto, we support ths theoretcal result umercall usg the data used b Shabbr ad Gupta [4].
4202 A.M. Abd-Elfattah et al. / Appled Mathematcs ad Computato 215 (2010) 4198 4202 Some other estmators ca also be derved combg rato estmators gve (1.4) (1.7) the form (2.1), but all these estmators have aga the same mmum MSE equato gve (2.9). We would lke to recall that R 1 ad R 2 (2.4) ad (2.8) should be chaged accordg to rato estmators that are combed. Ackowledgemets The authors are deepl grateful to the referee ad the edtor of the joural for ther extremel helpful commets ad valued suggestos that led to ths mproved verso of the paper. Refereces [1] W.G. Cochra, Samplg Techques, Joh Wle ad Sos, New York, 1977. [2] C. Kadlar, H. Cg, Improvemet estmatg the populato mea smple radom samplg, Appled Mathematcs Letters 19 (2006) 75 79. [3] V.D. Nak, P.C. Gupta, A ote o estmato of mea wth kow populato of a auxlar character, Joural of Ida Socet Agrcultural Statstcs 48 (2) (1996) 151 158. [4] J. Shabbr, S. Gupta, O estmatg the fte populato mea wth kow populato proporto of a auxlar varable, Paksta Joural of Statstcs 23 (1) (2007) 1 9. [5] R. Sgh, P. Chauha, N. Sawa, F. Smaradache, Rato estmators smple radom samplg usg formato o auxlar attrbute, Paksta Joural of Statstcs ad Operato Research IV (1) (2008) 47 53. [6] P.V. Sukhatme, B.V. Sukhatme, Samplg Theor of Surves wth applcatos, Iowa State Uverst Press, Ames, USA, 1970. [7] K.M. Wolter, Itroducto to Varace Estmato, secod ed., Sprger-Verlag, 1985.