A Family of Efficient Estimator in Circular Systematic Sampling

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1 olumbia Iteratioal Publishig Joural of dvaced omputig (0) Vol. o. pp doi:0.776/jac.0.00 Research rticle Famil of Efficiet Estimator i ircular Sstematic Samplig Hemat K. Verma ad Rajesh Sigh * Received 8 ugust 0; Published olie September 0 The author(s) 0. Published ith ope access at.uscip.us bstract This paper proposes a famil of epoetial estimators for estimatig the populatio mea of stud variable usig a auiliar variable i circular sstematic samplig desig uder sigle ad to phase samplig. The epressio of the bias ad mea square error of proposed class of estimators are derived i geeral form. It has bee sho that the proposed class of estimators are more efficiet tha ratio, product, regressio ad other estimators cosidered here i circular sstematic samplig uder sigle ad to phase samplig. empirical stud is carried out i support of theoretical stud. Keords: ircular sstematic samplig; Efficiec; Ratio estimator; Regressio estimator; To-phase samplig. Itroductio I sample surves, the utilizatio of auiliar iformatio is frequetl ackoledged to higher the accurac of the estimatio of populatio characteristics uder stud. The auiliar data might either be promptl accessible or ma be made accessible ithout much trouble b occup a part of the surve resources or available from previous eperiece, cesus or admiistrative databases. It is ell ko that he the auiliar iformatio is to be used at the estimatio stage, the ratio, product ad regressio methods are idel emploed. Ma authors icludig Sigh et al. (007), Shabbir ad Gupta (007), Sigh ad Kumar (0), ad Sharma ad Sigh (0, 0) suggested estimators usig auiliar variable. The usual sstematic samplig desig is quite simple ad most commol used i sample surve. Sstematic samplig has a advatage of selectig the hole sample ith just oe radom start. I *orrespodig rsighstat@gmail.com Departmet of statistics, Baaras Hidu Uiversit, Varaasi (U.P), Idia 6

2 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp this method of samplig, the first uit is selected radoml ad remaiig uits are selected automaticall accordig to some predetermied patters. Hasel (9) ad Griffth (9-96) foud sstematic samplig to be efficiet ad coveiet i samplig certai atural populatios like forest areas for estimatig the volume of the timber ad area uder differet tpes of cover. ochra (96) ad Hajeck (99) had stated that i large-scale samplig ork, this procedure provides more efficiet estimators tha those provided b simple radom samplig ad/or stratified radom samplig uder certai coditios. I case of ko populatio mea, of the auiliar variable, Sai (96) ad Shukla (97) have suggested the ratio ad product estimators for the populatio mea of the surve variable, respectivel, alog ith their properties i sstematic samplig. Some other remarkable ork i this area are Sigh ad Sigh (998), Sigh et al. (0), Verma et al. (0), Sigh ad Jata (0), Sigh ad Solaki (0), ad Verma et al. (0). I liear sstematic samplig, give a sample size, samplig is possible ol if populatio size is divisible b. Eve he this coditio is satisfied, the scheme caot provide estimate of variace of the sample mea. This scheme has to drabacks amel, give, has limited choice ad variace of the sample mea is ot estimable. The first limitatio could be removed through circular sstematic samplig () as suggested b Lahiri (9). The procedure cosists i th selectig a uit, b a radom start, from to ad the thereafter selectig ever k uit, k beig a iteger earest to /, i a circular maer, util a sample of uits is obtaied. Suppose that a uit ith radom umber i is selected. The sample ill the cosists of the uits correspodig to the serial umbers i jk, i jk, Label for i 0,,..., ( ) i jk, i jk. (for details see Sigh ad haudhar (986), pp8) I the folloig maer, e ma dra circular sstematic samples, each of size as displaed i table. Table Possible Samples usig ircular Sstematic Samplig Sample umber i u u u i u uk uk uk i u u(-)k u(-)k u(-)k i u 7

3 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp From this possible sample, a sample of size is selected radoml to observe ad.. Termiolog used i ircular Sstematic Samplig * Let us suppose that U be a fiite populatio cosists of distict labelled uits i.e. U * (U,U....U ) ad be a fied sample size. lso, let ad be stud ad auiliar variables takig values ad i (,,..., ), j (,,..., ). The sample meas meas / ad / are ubiased estimates of populatio j j / ad / respectivel. j The variace of ad here V j ad uder desig is ritte as- ( ) V( S ( ) S () S ) ( ) S (), S i j ( ) ( ), S i j ( ) ( ) ith ( )( )S i ju ( )( iu ) ad ( )( )S i ju ( )( iu ) 8

4 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp here, ) represets itraclass correlatio coefficiets betee pairs of uits ithi the ( for ad, respectivel. also, here ov( S / / ( ) ( ) S, ) i j ( )( ( ) ). Estimators i Literature. I literature, geerall, e use ratio, product ad regressio estimators for estimatig the populatio mea he e have iformatio o auiliar variables. Thus, e cosider ratio, product ad regressio estimators based o as stadard result for makig compariso ith our suggested class of estimators. The ratio estimator of the populatio mea based o ith ko is defied as R () S The product estimator of the populatio mea based o ith ko is defied as P () The liear regressio estimator of the populatio mea based o ith ko is defied as lr ˆ ( ) () s here ˆ is a estimator for populatio regressio coefficiet ith s s j ( ) ad s ( ) j ( )( ) ( ) 9

5 The bias epressios of estimators Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp R, P ad lr are give as Bias R (6) Bias P (7) Bias lr 0 (8) ad the MSE epressios of estimators MSE MSE MSE R R, P ad lr are give as (9) (0) P lr here, S S S () Double samplig scheme is applicable he populatio mea of auiliar variable, is uko. Uder double samplig scheme, first e divide the populatio ito clusters of size, each accordig to, ad select radoml m distict clusters (< m <k) to estimate ol. I secod phase, a cluster is selected radoml from m s to estimate. Hece, the epressios for ad lr, ith uko, are give as R () P () lr R, P ˆ ( ), () here m i j m 60

6 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp Defied, S S V() ov(, ) ( ) () m m ad ov / S S (6) m m /, ( ) ( ) The bias epressios of estimators R, P ad lr are give as Bias R (7) Bias P (8) Bias lr 0 (9) ad the MSE epressios of estimators R, P ad lr are give as MSE R (0) MSE P () MSE lr () here, S S S, such that m S S m ith S ad ad S S m S m () 6

7 . Proposed Estimator Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp Motivated b Koucu (0), e propose the folloig estimator for estimatig populatio mea of a stud variable uder assumig is ko as t p ep () Where is a suitable real umber, ad are either real umbers or the fuctios of the ko parameters associated ith a auiliar attribute. (, ) are suitabl chose scalars to be properl determied for miimum mea square error (MSE) of suggested estimators ad. (See Sharma ad Sigh 0a) Epadig equatio () i terms of e s up to the first order of approimatio, e have, t p e 0 e e e e e 8 8 e e e 0 e () here, e 0, e ad To obtai the bias ad MSE of the estimator t p to the first degree of approimatio, e rite Such that, E(e i ) 0; i 0,. also, 0, E e, E e ad Ee e, 0 Takig epectatio both sides of equatio (), e get the bias epressio of estimator t P as Bias t P 8 8 (6) Squarig both sides of equatio () ad takig epectatio e get the MSE epressio of estimator t P as 6

8 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp P t MSE (7) here, 8 8 Partiall differetiatig equatio (7) ith respect to ad ad equatig to zero, e get the optimum value of ad as (opt) (opt) o suppose, is uko, the aalogue of p t becomes ep t p (8) here, the otatios used here are alread defied earlier. To obtai the bias ad MSE of the proposed class of estimators T, e defie e Such that

9 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp m e E e ad m,e E e E e 0, E e 0 (9) The epressios for bias ad MSE of the proposed estimator P t usig are give respectivel as P 8 8 t Bias (0) P W t MSE () here, 8 8 Partiall differetiatig equatio () ith respect to ad ad equatig to zero, e get the optimum value of ad as (opt) (opt) ote: It ca be observed from equatio (6), (7) ad (0), () that the bias ad MSE of P t ad P t look similar. Hoever, due to sigle ad double samplig desig the dissimilarit eists ol i terms, ad,.

10 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp Table Members of class of estimators t P Estimators ostats t P t P ep 0 tp ep - t P ep - S S t P ep - t P ep - t P6 ep 0 t P7 ep 0 S S S t P8 ep - S. Empirical Stud I order to check the efficiec of proposed estimators, e take a data set hich is earlier cosidered b Koucu ad Kadilar (009) ad Sigh ad Solaki (0). The data cocers primar ad secodar schools of 9 districts of Turke i 007. The descriptio of variables is give belo 6

11 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp = umber of teachers i both primar ad secodar school; = umber of studets i both primar ad secodar school. =9 =60 =80 m = =0. =6. S =79.9 S =. =0.9 =-0.00 = For to-phases, oe ca select < m < (as e metioed earlier < m < k). ll possible values of m are cosidered. Here i this problem e have take m=. Folloig table shos the variace/mse ad PRE of all the estimators cosidered here. here, PRE., V 00. MSE. Table Variace/ Miimum MSE/PRE s of cosidered estimators uder sigle ad to phase samplig (usig usual otatio i to-phase as is ad so o) Estimators (Sigle phase) (To phase) V/MSE PRE V/MSE PRE P R lr t P t P t P t P t P t P t P t P

12 6. oclusio Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp I this paper, e proposed a efficiet class of epoetial estimators t P usig auiliar variable i circular sstematic samplig (). The Variace/MSE/miimum MSE ad PRE of differet estimators have bee sho i table ad it has bee observed that the estimator 7, hich is a member of proposed famil of estimators t P has miimum MSE amog all the estimators cosidered here. Thus, from empirical stud, e coclude that estimator 7 is more efficiet tha other eistig estimators i for sigle-phase ad to-phase samplig for the give data set. Refereces ochra, W. G. (96). Relative efficiec of sstematic ad stratified radom samples for a certai class of populatios.. Math. Stat., 7, Griffth,. L The efficiec of eumeratios. Forest-Research Istitute, Dehra Du. Idia Forest Leaflets,8 9. Hajeck, J., (99). Optimum strateg ad other problems i probabilit samplig.osopis pro Pestovai Mathematik, 8, 87. Hasel,.. (9): Estimatio of volume i timber stads b strip samplig, MS,, Koucu,. (0): Efficiet estimators of populatio mea usig auiliar attributes. M, 8, Koucu,., Kadilar,., (009). Efficiet estimators for the populatio mea. Hacettepe Joural of Mathematics ad Statistics, 8, 7. Lahiri, D. B., (9). method for selectio providig ubiased estimates. Bulleti of the Iteratioal Statistical Istitute, (), 0. Shabbir, J. ad S. Gupta, 007. O estimatig the fiite populatio mea ith ko populatio proportio of a auiliar variable. Pakista Joural of Statistics, (): -9. Sharma, P. ad Sigh, R (0). Improved Estimators for Simple radom samplig ad Stratified radom samplig Uder Secod order of pproimatio. Statistics I Trasitio- e series, (), Sharma, P. d Sigh, R. (0a). Efficiet Estimators of Populatio Mea i Stratified Radom Samplig Usig uiliar ttribute. World pplied Scieces Joural, 7(), Sharma, P. ad Sigh, R. (0). Geeralized class of estimators for populatio media usig auiliar iformatio, Hecettepe Joural of Statistics ad Mathematics ( I Press) Shukla,. D. (97). Sstematic samplig ad product method of estimatio, Proceedigs of all Idia Semiar o Demograph ad Statistics, B.H.U.,Varaasi, Idia. Sigh, D. ad haudhar, F. S. (986). Theor ad aalsis of sample surve desig. Wile Easter Limited. Sigh, R. ad Kumar, M. (0): ote o trasformatios o auiliar variable i surve samplig. MS, 6:, 7-9. Sigh, H. P., Jata,. K., (0). class of epoetial-tpe estimators i sstematic samplig. Ecoomic Qualit otrol, 7, Sigh, H. P., Solaki, R. S., (0). efficiet class of estimators for the populatio mea usig auiliar iformatio i sstematic samplig. Joural of Statistical Theor ad Practice, 6, Sigh, H. P., Solaki, R. S. (0). efficiet class of estimators for the populatio mea usig auiliar iformatio. ommuicatio i Statistics- Theor ad Methods,, t P t P

13 Hemat K. Verm ad Rajesh Sigh / Joural of dvaced omputig (0) Vol. o. pp Sigh, R., P. auha,. Saa ad F.Smaradache, (007). uiliar iformatio ad a priori values i costructio of improved estimators. Reaissace High Press. Sigh, R., Malik, S., haudr, M. K., Verma, H. K., deara,.., (0). geeral famil of ratio-tpe estimators i sstematic samplig. Joural of Reliabilit ad Statistical Studies, (), 7 8. Sigh, R., Sigh, H. P., (998). lmost ubiased ratio ad product-tpe estimators i sstematic samplig. QUESTIIO, (), 0 6. Sai,. K. P.. (96). The use of sstematic samplig i ratio estimate. J. Id. Stat. ssoc., (), Verma, H. K, Sigh, R. D. ad Sigh, R. (0). Geeral lass of Regressio Tpe Estimators i Sstematic Samplig Uder o-respose, OTOGO Mathematical Magazie, 0(), -0. Verma, H. K., Sigh, R. D. ad Sigh, R (0). Some Improved Estimators i Sstematic Samplig Uder o- Respose, atl. cad. Sci. Lett., 7(),