An Almost Unbiased Estimator for Population Mean using Known Value of Population Parameter(s)

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1 J. a. Al. ro., o., -6 (04) Journal of aisics Alicaions & robabili An Inernaional Journal h://dx.doi.org/0.785/jsa/ahare An Almos Unbiased Esimaor for oulaion Mean using nown Value of oulaion arameer(s) Rajesh ingh,.b. Gua and achin Malik, * Dearmen of aisics, Banaras Hindu Universi, Varanasi-005, India Dearmen of ommuni Medicine, RM Insiue of Medical ciences, Bareill- 40, India Received: Feb. 04, Revised: Ma. 04, Acceed: 5 Ma 04 ublished: Jul. 04 Absrac: In his aer we have roosed an almos unbiased esimaor using known value of some oulaion arameer(s) wih known oulaion roorion of an auxiliar variable. A class of esimaors is defined which includes [], [] and [] esimaors. Under simle random samling wihou relacemen (RWOR) scheme he exressions for bias and mean square error (ME) are derived. umerical illusraions are given in suor of he resen sud. e words: Auxiliar informaion, roorion, bias, mean square error, unbiased esimaor.. Inroducion I is well known ha he recision of he esimaes of he oulaion mean or oal of he sud variable can be considering imroved b he use of known informaion on an auxiliar variable x which is highl correlaed wih he sud variable. Ou of man mehods raio, roduc and regression mehods of esimaion are good illusraions in his conex. Using known values of cerain oulaion s arameers several auhors have roosed imroved esimaors including [4, 5, 6, 7, 8, 9, 0,,, ]. In man racical siuaions, insead of exisence of auxiliar variables here exi some orresonding auhor sachinkurava999@gmail.com c 04 aural ciences ublishing or.

2 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) auxiliar aribues (sa), which are highl correlaed wih he sud variable, such as i. Amoun of milk roduced () and a aricular breed of cow ( ). ii. ex ( ) and heigh of ersons () and iii. Amoun of ield of whea cro and a aricular varie of whea ( ) ec. (see [4]). Man more siuaions can be encounered in racice where he informaion of he oulaion mean Y of he sud variable in he resence of auxiliar aribues assumes imorance. For hese reasons various auhors such as [5, 6, 7, 8, 9] have aid heir aenion owards he imroved esimaion of oulaion mean Y of he sud variable aking ino consideraion he oin biserial correlaion beween a variable and an aribue. Le A i and i n i i a denoe he oal number of unis in he oulaion and samle A a ossessing aribue resecivel, and denoe he roorion of unis in he n oulaion and samle, resecivel, ossessing aribue. To esimae Y, he usual esimaor is given b Var f () Define, e Y Y e, e 0, i, E i f, Ee f, E e e e fρb. E c 04 aural ciences ublishing or.

3 J. a. Al. ro., o., -4 (04) / Where, f,,, n Y and b is he oin bi-serial correlaion coefficien. Here, i Y i, i i and ii Y, f, i n' In order o have an esimae of he sud variable, assuming he knowledge of he oulaion roorion, [] roosed he following esimaor GR () G () Following [], we roose he following esimaor α (4) The Bias and ME exressions of he esimaor u o he firs order of aroximaion are, resecivel, given b c 04 aural ciences ublishing or.

4 4 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) α(α)v Yf αv B (5) ME Y f α V V α (6) Also following [], we roose he following esimaor β exλ (7) The Bias and ME exressions of he esimaor u o he firs order of aroximaion are, resecivel, given b B Yf λvβ β β λλ 8 V β λv (8) λ V λv Y f β βλv β ME 4 (9), and are suiable chosen consans. Also,, 4, 5 are eiher real numbers or funcion of known arameers of he auxiliar aribues such as, is an ineger which akes values + and - for designing he esimaors and V V 4 4 5, b and. We see ha he esimaor s and are biased esimaors. In some alicaions bias is disadvanageous. Following hese esimaors we have roosed almos unbiased esimaor of Y. c 04 aural ciences ublishing or.

5 5 J. a. Al. ro., o., -4 (04) / Almos unbiased esimaor uose, 0, α, β exλ uch ha 0,, W, where W denoes he se of all ossible esimaors for esimaing he oulaion mean Y. B definiion, he se W is a linear varie if wii W i 0 (0) uch ha, w i0 i and w i R () where, i 0,, w i denoes he consans used for reducing he bias in he class of esimaors, W denoes he se of hose esimaors ha can be consruced from i i 0,, and R denoes he se of real numbers. Exressing equaion (0) in erms of e s, we have Y e w V e V e V e e e V e V e V e Ve e w e e e () 8 ubracing Y from boh sides of equaion () and hen aking execaion of boh sides, we ge he bias of he esimaor u o he firs order of aroximaion, as c 04 aural ciences ublishing or.

6 6 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) B( ) Yf w V V V Yf w V () From (), we have 8 V ( Y) Ye w αve w βe λve (4) quaring boh sides of (4) and hen aking execaion, we ge he ME of he esimaor u o he firs order of aroximaion, as ME Y f Q Q (5) Which is minimum when Q (6) where, V w V w (7) Q uing he value of Q in (5), we have oimum value of esimaor as (oimum). Thus he minimum ME of is given b min.me Y f b (8) Which is same as ha of radiional linear regression esimaor. c 04 aural ciences ublishing or.

7 7. J. a. Al. ro., o., -4 (04) / from () and (7), we have onl wo equaions in hree unknowns. I is no ossible o find he unique values for w i 's, i=0,,. In order o ge unique values of 's, we shall imose he linear resricion i0 w B (9) i i 0 where, B i denoes he bias in he i h esimaor. w i Equaions (), (7) and (9) can be wrien in he marix form as 0 0 w 0 λv αv β w k (0) B B w 0 Using (0), we ge he unique values of w i 's, i=0,, as w w w A A X X 0 V V V A AX X A X V V A AX V V A A X V A V X V A A A X A A X V V V V A Use of hese 's, c 04 aural ciences ublishing or. X 8 V V V w i i=0,, remove he bias u o erms of order n A o a (0).

8 8 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s).emirical sud For emirical sud we use he daa ses earlier used b [0] (oulaion ) and [] (oulaion ) o verif he heoreical resuls. Daa saisics: oulaion n Y b oulaion I oulaion II Table.: Values of w i 's, w i 's, oulaion oulaion w w w c 04 aural ciences ublishing or.

9 9. J. a. Al. ro., o., -4 (04) / Table.: RE of differen esimaors of Y wih resec o hoice of scalars w 0 w w 4 Esimaor RE 5 (OI) RE (OII) GR G (,0) (,0) (,) 0 - (, ) (0,) (0, ) w 0 w w oimum c 04 aural ciences ublishing or.

10 0 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) 4. roosed esimaors in wo hase samling In some racical siuaions when is no known a riori, he echnique of wo-hase samling is used. Le 'denoe he roorion of unis ossessing aribue in he firs hase samle of size n' ; denoe he roorion of unis ossessing aribue in he second hase samle of size n' n and denoe he mean of he sud variable in he second hase samle. In wo-hase samling he esimaor will ake he following form d hi i0 id H () uch ha, h i0 i and h i R () Where, 0d, d ' m, d ' q exγ 4' ' The Bias and ME exressions of he esimaor aroximaion are, resecivel, given b d and d u o he firs order of B d m(m)r f Y m(m)r f m R f mrfk () ME Y f m R f mr k f d (4) c 04 aural ciences ublishing or.

11 . J. a. Al. ro., o., -4 (04) / B d q(q )f Y q(q )f qf k q f f γr k fγrq (5) ME Y f L f d (6) Where, R R L q γa (7) Exressing () in erms of e s, we have d w Y e qe q w R e mm m m mr e q e qq mr e e mr e e R e' mr e qe' q ee' γre' e γre e ee' qe e e' e' ubracing Y from boh sides of he above equaion and hen aking execaion of boh sides, we ge he bias of he esimaor d u o he firs order of aroximaion, as B( d Also, B ) Y B (8) d d d 0 mr e' mr e w qe qe' γr e' γr e ( Y) Y e w (9) c 04 aural ciences ublishing or

12 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) quaring boh sides of (9) and hen aking execaion, we ge he ME of he esimaor d u o he firs order of aroximaion, as ME Y f L f L f k d (0) Which is minimum when L () Where L h mr h q γr () uing he value of L in (0), we have oimum value of esimaor as d (oimum). Thus he minimum ME of d is given b min.me Y f f ρ d b () which is same as ha of radiional linear regression esimaor. from () and (), we have onl wo equaions in hree unknowns. I is no ossible o find he unique values for h i ' s, =0,,. In order o ge unique values of ' s, he linear resricion i0 h i we shall imose h ib id 0 (4) where, B id denoes he bias in he i h esimaor. Equaions (), () and (4) can be wrien in he marix form as 0 0 h 0 mr q γr h k (5) B d B d h 0 c 04 aural ciences ublishing or.

13 . J. a. Al. ro., o., -4 (04) / Using (5), we ge he unique values of s, ' h i i=0,, as 0 γr mr q h γr mr q γr q mr k h h h h where, q γr f k γr f f q k qf )f q(q )f q(q k mr f f R m )R f m(m )R f m(m Use of hese s, ' h i i=0,, remove he bias u o erms of order n o a (). 5. Emirical ud For emirical sud we use he daa ses earlier used b [0] (oulaion ) and [] (oulaion ) o verif he heoreical resuls. Daa saisics: o. n Y ' b n' o.i o.ii c 04 aural ciences ublishing or.

14 4 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) Table 5.: RE of differen esimaors of Y wih resec o hoice of scalars w 0 w w 4 m q γ Esimaor RE 5 (OI) RE (OII) GR G d(,0) - 0 d(,0) d(,) d(, ) d(0,) d(0, ) w 0 w w d oimum c 04 aural ciences ublishing or.

15 5. J. a. Al. ro., o., -4 (04) / 6. onclusion In his aer, we have roosed an unbiased esimaor and d using informaion on he auxiliar aribue(s) in case of single hase and double hase samling resecivel. Exressions for bias and ME s of he roosed esimaors are derived u o firs degree of aroximaion. From heoreical discussion and emirical sud we conclude ha he roosed esimaors and d under oimum condiions erform beer han oher esimaors considered in he aricle. c 04 aural ciences ublishing or.

16 6 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) Aendix A. ome members of he roosed famil of esimaors - ome members (raio-e) of he class When w 0 0, w, w 0, Esimaors ( ) Esimaors ( RE RE a b ( ) ( ) a b ( ) ( ) a ( ) ( ) b ( ) ( ) a4 ( ) ( ) b4 ( ) ( ) b a5 b b b5 b b a6 b F f f a7 b7 f f c 04 aural ciences ublishing or.

17 7. J. a. Al. ro., o., -4 (04) / b a8 b b b8 b b b a9 b b b9 b b b0 a a b b a b b b b b a b F f f b4 f f a g=-f g g b5 g g a b a6 b b b6 b b.5.0 b a7 n n b b b7 n n b b a8 n n b b b8 n n b b F n f nf b9 n f nf a9.8.0 c 04 aural ciences ublishing or.

18 8 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) g=-f n g ng b0 n g ng a b a n n b b b n n b b a b a b b4 a4.. n n b5 n n a Aendix B. ome members (roduc-e) of he class When w 0 0, w, w 0, Esimaors ( ) Esimaors ( ) RE RE c d c 04 aural ciences ublishing or.

19 9. J. a. Al. ro., o., -4 (04) / ( ) ( ) c d ( ) ( ) c ( ) ( ) d ( ) ( ) c4 ( ) ( ) d4 ( ) ( ) b c5 b b d5 b b c6 d6.5. f f f c7 f f d7.0.5 b c8 b b d8 b b.8.68 b c9 b b d9 b b.89.6 c0 d0.7.0 ( ) ( ) c d ( ) ( ) c ( ) ( ) d ( ) ( ) c4 ( ) ( ) d4 ( ) ( ) c 04 aural ciences ublishing or.

20 0 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) c d.50. b c b b d b b c d.6.44 f c4 f f d4 f f g=-f g g d5 g g c5.4. b c6 b b d6 b b.9.60 b c7 n n b b d7 n n b b c8 n n b b d8 n n b b.65.4 f c9 n f n f d9 nf nf.06.5 g=-f n g ng d0 n g ng c b c n n b b d n n b b.88.6 c 04 aural ciences ublishing or.

21 . J. a. Al. ro., o., -4 (04) / c d c d.7.0 c4 d4..47 c5 n n d5 n n.49. Aendix. ome members (roduc-e) of he class When w 0 0, w 0, w 4 5 Esimaors (, ) ex RE.4 ex.9 ex ex.5 c 04 aural ciences ublishing or.

22 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) b 5 ex b.86 6 ex F 7 ex f b 8 ex b b 9 ex b ex 54.0 ex b ex b ex F 4 ex f c 04 aural ciences ublishing or.

23 . J. a. Al. ro., o., -4 (04) / g=-f 5 ex g b 6 ex b b n 7 ex n b n 8 ex n F n 9 ex n f g=-f n 0 ex n g b n ex n b ex ex 4 ex n 5 ex n c 04 aural ciences ublishing or.

24 4 R. ingh,.b. Gua and. Malik: An almos unbiased oulaion arameer(s) In addiion o above esimaors a large number of esimaors can also be generaed from he roosed esimaors jus b uing differen values of consans w i ' s, h i ' s,,,, 5,, and. References [] aik, V.D., Gua,.. (996): A noe on esimaion of mean wih known oulaion roorion of Agriculural aisics 48() an auxiliar characer. Journal of he Indian ocie of [] ingh, H.., olanki, R.. (0): Imroved esimaion of oulaion mean in simle random samling using informaion on auxiliar aribue.alied Mahemaics and omuaion 8 (0) [] ahai A.,.. Ra.. (980) An efficien esimaor using auxiliar informaion, Merika, vol. 7, no. 4, [4] ingh, H.. and Tailor, R. (00): Use of known correlaion coefficien in esimaing he finie oulaion mean. aisics in Transiion, 6,4, [5] adilar,. and ingi,h. (00): Raio Esimaors in raiified Random amling. Biomerical Journal 45 (00), 8-5. [6] habbir, J. and Gua,. (007): A new esimaor of oulaion mean in sraified samling, ommun. a. Theo. Meh.5: [7] Gua,. and habbir, J. (008) : On imrovemen in esimaing he oulaion mean in simle random samling. Jour.Of Alied aisics, 5, 5, [8] hoshnevisan, M. ingh, R., hauhan,., awan,. and marandache, F. (007): A general famil of esimaors for esimaing oulaion mean using known value of some oulaion arameer(s). Far eas journal of heoreical saisics, (), 8-9. [9] ingh, R.,auhan,., awan,. and marandache,f. (007): Auxiliar informaion and a rior values in consrucion of imroved esimaors. Renaissance High ress. 4 c 04 aural ciences ublishing or.

25 5. J. a. Al. ro., o., -4 (04) / [0] ingh, R., umar, M. and marandache, F. (008): Almos Unbiased Esimaor for Esimaing oulaion Mean Using nown Value of ome oulaion arameer(s). ak. J. a. Oer. Res., 4() [] ouncu,. and adilar,. (009) : Famil of esimaors of oulaion mean using wo auxiliar variables in sraified random samling. omm. In a. Theor and Meh., 8:4, [] Diana G., Giordan M. erri.(0): An imroved class of esimaors for he oulaion mean. a Mehods Al (0) 0: 40 [] Uadhaa L, ingh H, haerjee, Yadav R (0) Imroved raio and roduc exonenial e esimaors. J a Theor rac 5():85 0 [4] Jhajj, H.., harma, M.. and Grover, L.. (006): A famil of esimaors of oulaion mean using informaion on auxiliar aribue. akisan journal of saisics, (), 4-50 (006). [5] Abd-Elfaah, A.M. El-herien, E.A. Mohamed,.M. Abdou, O. F. (00): Imrovemen in esimaing he oulaion mean in simle random samling using informaion on auxiliar aribue. Al. Mahe. and om. doi:0.06/j.amc [6] Grover L.,aur.(0): An imroved exonenial esimaor of finie oulaion mean in simle random samling using an auxiliar aribue.alied Mahemaics and omuaion 8 (0) [7] Malik,. and ingh, R. (0a) : An imroved esimaor using wo auxiliar aribues. Ali. Mah. om., 9, [8] Malik,. and ingh, R. (0b) : A famil of esimaors of oulaion mean using informaion on oin bi-serial and hi correlaion coefficien. In. Jour. a. Econ. 0(), [9] Malik,. and ingh, R. (0c) : Dual o raio cum roduc esimaors of finie oulaion mean using auxiliar aribue(s) in sraified random samling. WAJ, 8(9), 9-98 [0] ukhame,.v. and ukhame, B.V.,(970) : amling heor of surves wih alicaions. Iowa ae Universi ress, Ames, U..A. [] Mukhoadhaa,.(000): Theor and mehods of surve samling. renice Hall of India, ew Delhi, India. c 04 aural ciences ublishing or.

Almost Unbiased Estimator for Estimating Population Mean Using Known Value of Some Population Parameter(s)

Almost Unbiased Estimator for Estimating Population Mean Using Known Value of Some Population Parameter(s) Almos Unbiased Esimaor for Esimaing Populaion Mean Using Known Value of Some Populaion Parameers Rajesh Singh Deparmen of Saisics, Banaras Hindu Universi U.P., India rsinghsa@ahoo.com Mukesh Kumar Deparmen