Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 10, Issue 1 (June 015), pp. 106-113 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) Exponental Tpe Product Estmator for Fnte Populaton Mean wth Informaton on Auxlar Attrbute Abstract Monka San and Ashsh Kumar Department of Mathematcs and Statstcs Manpal Unverst Japur Japur (Rajasthan) - 303007 drmnksan4@gmal.com Receved: November 4, 014; Accepted: March 10, 015 The man objectve of the present stud s to develop a new modfed unbased exponental tpe product estmator for the estmaton of the populaton mean. The proposed estmator possesses the characterstc of a b-seral negatve correlaton between the stud varable and ts auxlar attrbute. Effcenc comparson has been carred out between the proposed estmator and the exstng estmators theoretcall and numercall. Kewords: Fnte populaton; Smple random samplng; Product estmator, Auxlar attrbute; Effcenc MSC 010 No.: 6D05, 6D99 1. Introducton In the exstng lterature of samplng theor, auxlar nformaton s generall used to mprove the effcenc of estmators. Man estmators such as rato, product, dfference and regresson estmators are good examples n ths reference. When the stud and auxlar varables are postvel correlated then the rato estmator s used n ths context. On the other hand, f both varables are negatvel correlated then the product estmator s used to estmate the populaton parameter. The frst attempt was made b Cochran (1940) to nvestgate the problem of estmaton of the populaton mean when auxlar varables are present and he proposed the usual rato estmator of populaton mean. Robson (1957) and Murth (1964) worked ndependentl on the usual product estmator of the populaton mean. Recent developments n the rato and product methods of estmaton along wth ther varet of modfed forms b Sngh et al. (010) proposed a rato-cum-dual to rato estmator for the estmaton of the fnte populaton mean of 106
AAM: Intern. J., Vol. 10, Issue 1 (June 015) 107 the stud varable. Yadav (011), Pande et al. (011), Shukla et al. (01), Oneka (01) etc. have proposed man estmators utlzng auxlar nformaton. There are man real lfe crcumstances where auxlar nformaton s qualtatve n nature, that s auxlar nformaton s avalable n the form of an attrbute, whch s hghl correlated wth a stud varable, for example the sex and heght of a person, amount of mlk produced and a partcular breed of the cow, amount of eld of wheat crop and a partcular varet of wheat etc. [see Jhajj et al. (006)]. In such stuatons, takng the advantage of the pont b-seral correlaton between the stud varable and the auxlar attrbutes the estmators of the populaton parameter of nterest can be constructed b usng pror knowledge of the populaton parameter of auxlar attrbute. The man objectve of ths stud s to suggest a modfed unbased exponental tpe product estmator for estmatng the populaton mean of the varable under stud when the stud varable and ts auxlar attrbute are negatvel correlated. The outlnes of the present paper are as follows: n secton, The samplng procedure and notatons are gven for constructng varous estmators exstng as well as proposed. The exstng estmators and ther propertes are dscussed n secton 3. In secton 4, the suggested estmator and ts propertes are developed. Effcenc comparson between proposed and exstng estmators s carred out n secton 5. Secton 6 s devoted to a numercal stud. In secton 7, concludng remarks are gven.. Samplng Procedure and Notatons Consder a fnte populaton whch conssts of N dentfable unts (1 N). Suppose that there s a complete dchotom n the populaton wth respect to the presence or absence of an the attrbute, sa, and t s assumed that the attrbute takes onl two values 0 and 1 accordng as assumes value 1 when the assumes the value zero. th unt of the populaton possesses attrbute, otherwse Assume that a sample of sze n drawn b usng smple random samplng wthout replacement (SRSWOR) from a populaton of sze N. Let and denote the observatons on the varable and respectvel for defne smbolcall. Let th unt (=1,,..., N). Accordng to the above samplng scheme, we n / n and p / n be the sample means of varable of nterest and auxlar attrbute and n be the correspondng populaton means, where Y N / N and P / N N
108 Monka San and Ashsh Kumar P N 1 / N and p n 1 / n denote the proporton of unts n the populaton and sample respectvel possessng attrbute. We take the stuaton when the mean of the auxlar attrbute (P) s known. Let be the sample varance and n 1 s ( ) / n 1and s ( p) / n 1 n 1 N 1 S ( Y) / N 1 and S ( P) / N 1 be the correspondng populaton varance. Let C S / Y and C S / P. N 1 Fnall let S S S be the pont b-seral correlaton coeffcent between and. In order to determne the characterstc of the proposed estmators and exstng estmators consdered here, we defne the followng terms, ( Y) / Y and ( p P) / P such that E [ ] 0, for (, ), E( ) S Y, E( ) S P and E( ) S Y S P, where (1/ n) (1/ N ). 3. Exstng Product Estmator In ths secton, exstng product estmators are consdered for the estmaton of populaton mean.
AAM: Intern. J., Vol. 10, Issue 1 (June 015) 109 3.1. Exponental Tpe Product Estmator [Bahl and Tuteja (1991)] Below Bahl and Tuteja (1991) suggested the exponental tpe product estmator b usng auxlar attrbute: p P ( BTP) exp p P. (3.1) The mean square error (MSE) of BTP, s gven b 1 MSE( BTP ) Y C C C C 4. (3.) 3.. Product Estmator [ Nak and Gupta (1996) ] Nak and Gupta (1996) suggested the product estmator when the same attrbute s avalable s as follows: NGP p. (3.3) P The mean square error (MSE) of NGP up to the frst order of approxmaton are gven b MSE( NGP ) S R S R S S. (3.4) 4. Suggested Estmator and ther Propertes Under the same samplng desgn, we propose the modfed exponental tpe product estmator for the estmaton of the populaton mean as: where k s an constant and [ k ( t 1)], (4.1) t exp NP np P N n. To obtan the unbasedness and Mean square error (MSE) of (4.1) n terms of 's, we expend the equaton
110 Monka San and Ashsh Kumar NP np(1 ) ( ) 1 {1 Y k P...} 1. N n Expandng the rght hand sde of equaton (4.1) up to the frst order of approxmaton n terms of 's we wll have: NP np(1...) ( ) 1... Y k {1 P...} 1. N n Rewrtng up to the frst order of approxmaton, we have np ( ) Y 1 k N n. (4.) Takng expectaton on both sdes of equaton (4.), we can easl prove that E( ) Y,.e., s an unbased estmator of the populaton mean Y. Now varance of equaton (4.) can be obtaned as Var( ) E ( ) E(( )), NP np(1...) Var( ) [ 1... Y k {1 P...} 1 Y]. N n Up to the frst order of approxmaton, the varance of can be wrtten as: NPk npk(1 ) Var( ) Y Pk N n n E Y kp N n (neglectng the hgher order terms) n n Y E( ) k P ( ) E( ) k YP E( ). N n N n After smplfcaton, varance of wll be: n n Var( ) S k ( ) S k S S N n N n. (4.3)
AAM: Intern. J., Vol. 10, Issue 1 (June 015) 111 Theorem: 4.1. The estmator s an unbased estmator of populaton mean Y, up to the frst order of approxmaton and ts varance n n Var( ) S k ( ) S k S S N n N n. 5. Effcenc of Comparson Now we compare the suggested estmator defned n secton (3) wth exstng estmators defned n secton (). We derve the followng condton n whch proposed estmators are better than the exstng estmators: () Proposed Estmator vs. Exponental Tpe Product Estmator [Bahl and Tuteja (1991)]. Condton (): From equaton (3.) and (4.3) MSE( ) Var( ) BTP 1 n n 4 N n N n [ RS S R S k ( ) S k S S] 0, f S kn R kn ( ) R S N n 4 N n. () Proposed Estmator vs. Nak & Gupta (1996) Rato Estmator. Condton (): From equaton (3.4) and (4.3) MSE( ) Var( ) NGP n n R S R S S k S k S S 0 N n N n S kn f R S N n. From condtons () and (), we arrve at the followng theorems:
11 Monka San and Ashsh Kumar Theorem 5.1. The estmator s more effcent than BTP, f S kn R kn ( ) R S N n 4 N n. Theorem: 5.. The estmator s more effcent than NGP, f S kn R S N n. 6. Numercal Stud Now we compare the performance of the suggested and exstng estmators consdered here b usng the data sets as prevousl used b Shabbr and Gupta (010). Populaton: (Source: Sukhatme and Sukhatme (1970), pp. 56). = Number of vllages n the crcles. = A crcle consstng more than fve vllages. Table 1. Values of Parameters N 89 Y 3.36 P 0.14 0.766 n 3 C 0.604 C.19 Table. Percent Relatve Effcenc of Proposed Estmator vs. Exstng Product Estmators k Percent Relatve Effcenc w.r.t BTP Percent Relatve Effcenc w.r.t NGP BTP NGP 4 100 499 100 946 100 46 100 876 0 100 45 100 808 100 391 100 74 4 100 358 100 680
AAM: Intern. J., Vol. 10, Issue 1 (June 015) 113 6. Concluson In the present stud, a new modfed unbased exponental tpe product estmator and ts characterstcs are obtaned. Theoretcall, we obtan the condtons for whch the proposed estmator s more effcent than the exponental tpe product estmator (Bahl and Tuteja (1991)) and product estmator (Nak and Gupta (1996)) alwas. We support the theoretcal results b a numercal stud wth the help of nformaton used b Shabbr and Gupta (010). The results of numercal stud reveals that proposed estmator s alwas better than exstng estmators proposed b Bahl & Tuteja (1991) and Nak and Gupta (1996). REFERENCES Bahl, S. and Tuteja, R.K. (1991). Rato and product tpe exponental estmators, Informaton and Optmzaton Scences, Vol. 1 (1), pp. 159 163. Cochran, W.G. (1940). The estmaton of the elds of the cereal experments b samplng for the rato of gran to total produce, The journal of agrcultural scence, Vol. 30, pp. 6-75. Jhajj, H.S., Sharma, M.K. and Grover, L.K. (006). A faml of estmators of populaton means usng nformaton on auxlar attrbutes, Pakstan journal of Statstcs, Vol. (1), pp. 43-50. Murth, M.N. (1964): Product method of estmaton, Sankha A, Vol.6, pp. 69-74. Nak, V.D. and Gupta, P.C. (1996). A note on estmaton of mean wth known populaton proporton of an auxlar character, Jour. Ind. Soc. Agr. Stat., Vol. 48(), pp. 151-158. Oneka, A. C. (01). Estmaton of populaton mean n post stratfed samplng usng known value of some populaton parameter(s), Statstcs In Transton-new seres, Vol. 13, No. 1, pp. 65-78. Pande, H., Yadav, S.K. and Shukla, A.K. (011). An Improved General Class of Estmators Estmatng Populaton Mean usng Auxlar Informaton, Internatonal Journal of Statstcs and Sstems, Vol. 6(1), pp. 1 7. Robson, D.S. (1957). Applcaton of multvarate polkas to the theor of unbased rato tpe estmators, Jour. Amer. Stat. Assoc., Vol. 5, pp. 511-5. Shukla, D., Pathak, S. and Thakur, N.S. (01). Estmaton of populaton mean usng two auxlar sources n sample surve, Statstcs In Transton-new seres, March 01, Vol. 13, No. 1, pp. 1-36. Sukhatme, P.V., and Sukhatme, B.V. (1970). Samplng Theor of Surves wth Applcatons, Lowa State Unverst Press, Ames, USA. Shabbr, J. and Gupta, S. (010). Estmaton of the fnte populaton mean n two phase samplng when auxlar varables are attrbutes, Hecettepe Journal of Mathematcs and Statstcs, Vol. 39(1), pp. 11-19. Sngh, H.P., Talor, R., and Talor, R. (010). On rato and product methods wth certan known populaton parameters of auxlar varable n sample surves, Statstcs and operatons research trans., Vol. 34, pp. 157-180. Yadav, S.K. (011), Effcent Estmators for Populaton Varance usng Auxlar Informaton, Global Journal of Mathematcal Scences: Theor and Practcal. Vol. 3(4), pp. 369-376.