Estimation of Finite Population Variance Under Systematic Sampling Using Auxiliary Information

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1 Statistics Applications {ISS (online)} Volume 6 os 08 (ew Series) pp Estimation of Finite Population Variance Under Sstematic Samplin Usin Auiliar Information Uzma Yasmeen Muhammad oor-ul-amin Muhammad Hanif 3 3 ational Collee of Business Administration Economics Comsats Institute of Information Technolo Lahore Received: April 7 08; Revised: Ma 6 08; Accepted: Ma 8 08 Abstract This paper is based on the proposal of eneralized estimator for finite population variance usin an auiliar variable under sstematic samplin desin The epressions of bias mean square error are obtained for the proposed estimator The conditions are obtained for which the proposed estimators are better than the usual estimatoran empirical simulation stud is conducted to prove the superiorit of the proposed estimator Ke words: Auiliar variables Relative efficienc Eponential estimator Introduction In the surve samplin it is well reconized that the efficienc of an estimator of the population parameter can be increased b the use of auiliar information associated to auiliar variable which is related with the response variable At plannin stae or at desin stae auiliar variables ma be efficientl emploed to arrive at an enhanced estimator compared to those estimators not utilizin information of auiliar variable We shall use information of the auiliar variable under the contet of sstematic samplin technique Sstematic samplin is the simplest tpe of samplin desin that needs onl one rom start It offer ood results in some situations like forest reions for assessin the volume of the timber field operations of an surve ariculture meteorolo etc Due to its simplicit sstematic samplin technique is the frequentl used samplin technique in surves of finite population Apart from its simplicit sstematic samplin provides estimators which are more efficient than simple rom samplin or stratified rom samplin for certain tpes of populations Various authors includin Cochran (96) Gautschi (957) Hajeck (959) Sinh Sinh (998) Kadilar Cini (006) Kouncu Kadilar (009) Sinh et al (0) Tailor et al (03) Yasmeen et al ( ) oor-ul-amin et al(07) have discussed the estimation of population parameters usin auiliar information under different samplin desins Correspondin Author: Muhammad oor-ul-amin nooraminstats@mailcom

2 366 UZMA YASMEE MUHAMMAD OOR-UL-AMI AD MUHAMMAD HAIF [Volume 6 os In this paper variance estimators have been proposed under sstematic samplin technique Section presents some notations samplin process to follow the sstematic samplin technique In section 3 recommended estimators for the estimation of population variance have been presented Further the bias mean square error (MSE) of proposed estimator is obtained The simulation stud for evaluatin the proposed estimator is arraned in section The conclusion is iven in section 5 otations Consider a finite population P { P havin distinct identifiable units P P } seriall numbered from to It is assumed that nm where n m are positive inteers A sstematic sample of size n units is selected from the population b selectin the first unit at rom from to m then selectin ever subsequent mth unit The response variable the auiliar variable are observed for each ever selected unit in the sample Suppose for j m i n define the value of i th unit in the j th ( ji ji ) ( ) ( ) sample Then the respective sstematic sample variances for the variables ji - ji - * i * i s are the estimators of the population variance of s ( n -) ( n -) respectivel Suppose sss - S s e ss - S ss e ss S S so that where n n å ( ) å( ) ( ss ) Ee ( ss ) 0 ( ) Ee Eess A A a - Ee ( ss ) B B[ a -] ( -) Ee ( ssess ) C BA J - n ( ji Y) å - /( -) ê {( ji Y) /( ) } êå - - i i a ( ji X) å - /( -) ê {( ji X) /( ) } êå - - i i a ( ji Y) ( ji X) /( ) å ê i å{ ( ji -Y) /( -)} å ( ji -X) /( -) J ê ê { } ( )( ) i i { + ( - ) } ( ) A n r r ( ji -X)( ' -X ji ) ( ji - X) E ji -Y ' -Y ji B { + n- r } r E -Y ( ji ) are

3 08] SYSTEMATIC SAMPLIG USIG AUXILIARY IFORMATIO 367 are the correspondin intra-class correlation coefficients for the response variable variable respectivel 3 Proposed Estimators auiliar In this section under the framework of sstematic samplin technique the two estimators are developed usin information of auiliar variable for the estimation of finite population variance The proposed-i variance estimator is iven b æ ws + h ö ssg sss ç wsss + h (3) where w h are the parameters of auiliar variable such as the coefficient of correlation the coefficient of variation the coefficient of Skewness kurtosis Median the Tri-mean the uartile Deviation The proposed-ii eponential tpe estimator is iven b æs - s ö ss sss ep ç S ssg (3) 3 Bias mean square error of proposed estimators æ wse ö ss ssg S ( + ess ) - ç ws + h After simplification we have - ( ) S e e ssg ( + ss ) +Wi ss where ws W i ws + h i0 35 (33) (3) Ependin nelectin the hiher order terms we obtained - S» S e -S We e -S W e + S W e ssg ss i ss ss i ss i ss (35) After takin epectations we obtain epression of bias ( a J ) ( ) [ ] Bias ssg» S B Wi - -Wi BA - (36)

4 368 UZMA YASMEE MUHAMMAD OOR-UL-AMI AD MUHAMMAD HAIF [Volume 6 os In order to obtain mean square error of ssg aain usin (35) inorin the terms of power two reater we obtained E S SE e e e e ( ssg - )» ss +Wi ss -Wi ss ss (37) b After applin epectations on both sides the epression of mean square error is iven MSE( ssg) S êa ( a - ) +Wi B ( a -) - WiBA( J -) (38) 3 Bias Mean Square Error of Proposed Estimator-II B usin the notations iven in section it can be written (3) as ( ) æs - S + e ö ss ssg S ( + ess ) ep ç S Ependin nelectin the hiher order terms we obtained e ssg - S» S êess -essess - ess + ê After takin epectations we obtain epression of bias æ ö Bias ( ssg)» S ç B [ a -] -BAJ - After simplification the epression of mean square error is iven b MSE( ssg)» S êa ( a - ) + B ( a -) - BA( J -) ss (39) (30) (3) (3) Some Special Cases of Suested Estimator The followin special cases of the proposed-i estimator are summarized as i If w r b b h 0 then the form of suested estimator (3) is C w æ S ssg sss ç s ss ö () The respective approimate bias epression of mean square error of estimator b ssg is iven

5 08] SYSTEMATIC SAMPLIG USIG AUXILIARY IFORMATIO 369 æ ö Bias ( ssg )» S ç B W[ a -] -W AB J - MSE( ssg )» S êa ( a - ) +WB ( a -) - WBA ( J -) where W ii If w h 0 then the form of suested estimator (3) is C æ CS + ö ç + ssg sss Cs ss The respective approimate bias epression of mean square error of estimator b æ ö Bias ( ssg )» S ç B W[ a -] -W AB J - MSE( ssg )» S êa ( a - ) +WB ( a -) - WBA( J -) CS where W CS + ssg () is iven iii If h then the form of suested estimator (3) is w b æ b S + ö ç + 6 ssg sss bsss (3) The respective approimate bias epression of mean square error of estimator is iven b 6 æ ö Bias ( ssg )» S ç B W6[ a -] -W6 AB J - 6 MSE( ssg )» S êa ( a - ) +W6B ( a -) - W6BA( J -) bs where W 6 bs + iv If w h D then the form of suested estimator (3) is 6 ssg

6 370 UZMA YASMEE MUHAMMAD OOR-UL-AMI AD MUHAMMAD HAIF [Volume 6 os æ S + D ö 7 ssg sss ç sss + D () is iven b The respective approimate bias epression of mean square error of estimator 7 ( ssg)» ( W7[ a - ]-W7 J - ) Bias S B AB 7 MSE( ssg )» S êa ( a - ) +W7B ( a -) - W7BA ( J -) S where W 7 S + D w b h D v If then the form of suested estimator (3) is 7 ssg æ b S + D ö ç + D 0 ssg sss bsss (5) is iven b The respective approimate bias epression of mean square error of estimator 0 ( ssg)» ( W0[ a - ]-W0 J - ) Bias S B BA 0 MSE( ssg )» S êa ( a - ) +W0B ( a -) - W0BA( J -) bs where W 0 b S + D 5 umerical Illustration 0 ssg A simulation stud is conducted to demonstrate the efficienc of developed estimators usin the information on sinle auiliar variable under sstematic samplin technique In this section the performance of the proposed estimators is evaluated for real population The comparison is presented in the form of percentae relative efficienc The percentae relative efficiencies (PRE) iven in Table are computed usin the followin formulae var( 0 ) PRE 00 var( * ) (5)

7 08] SYSTEMATIC SAMPLIG USIG AUXILIARY IFORMATIO s * 0 ssg ssg ssg ssg ssg ssg 6 where s ss is denoted b usual sample variance ssg ssg ssg 7 0 ssg ssg ssg are denoted b developed estimators used in the formulae of the percent relative efficiencies For this stud consider the population iven in the Table (See Appendi) for summarizin the simulation procedures The procedure is used to find the efficienc of the developed estimators over usual sample variance The followin steps are used in R-Lanuae to perform simulation: Step : From the described population the sample size is considered as n 3 The procedure is repeated times to calculate the several values of estimators under sstematic samplin technique Step : Usin the sample obtained in step the values of sss ssg ssg separatel are obtained usin (3) (3) respectivel Step 3: Usin the values found in step the values of percent relative efficienc (PRE) is computed b (5) reported in Table This simulation stud involves the evaluation of the percent relative efficiencies of the proposed estimators usual sample variance for sample size n 3The results obtained from the simulation stud indicate that the developed variance estimators are more efficient than the usual estimator Table : Percent Relative Efficiencies Relative Biases of Different Estimators with respect to s ss ssg ssg 6 ssg 7 ssg 0 ssg ssg Estimator PRE RB (proposed) (proposed) (proposed) (proposed) (proposed) (proposed) It is concluded From Table that proposed estimators are more efficient than the usual sample variance for the sample size used in the Eample It is clear that proposed estimators are more useful than usual variance estimator as the performance of suested estimators are better than the usual variance estimator We have suested ratio estimators for estimatin variance of

8 37 UZMA YASMEE MUHAMMAD OOR-UL-AMI AD MUHAMMAD HAIF [Volume 6 os a finite population From the simulation results iven in Table we infer that the suested estimators are more efficient estimators than the usual sample variance for the finite population Hence the suested estimators are recommended for its practical use for estimatin variance of a finite population when sinle auiliar variable is available The simulated efficiencies for the estimators are obtained in Table for the sample size 3 The superiorit of proposed estimators ma be concluded from the information of Table However the simulation studies ma be etended with different population characteristics different sample sizes References Cochran WG (96) Relative accurac of sstematic stratified rom samples for a certain class of populations The Annals of Mathematical Statistics Gautschi W (957) Some remarks on sstematic samplin The Annals of Mathematical Statistics Hajeck J (959) Optimum strate other problems in probabilit samplin Casopis pro Pestovani Matematik Sinh HP Sinh R (998) Almost unbiased ratio product tpe estimators in sstematic samplin uestiio (3) 03 6 Kadilar C Cini H (006) An improvement in estimatin the population mean b usin the correlation coefficient Hacettepe Journal of Mathematics Statistics 35() Kouncu Kadilar C (009) Famil of estimators of population mean usin two auiliar variables in stratified rom samplin Communications in Statistics: Theor Methods 38(3 5) Sinh H P Tailor R Jatwa K (0) Modified ratio product estimators for population mean in sstematic samplin Journal of Modern Applied Statistical Methods 0() 35 Tailor T Jatwa Sinh H P (03) A ratio-cum-product estimator of finite population mean in sstematic samplin Statistics in Transition (3) Khan M Sinh R (05) Estimation of population mean in chain ratio-tpe estimator under sstematic samplin Journal of Probabilit Statistics Article ID paes Khan M Abdullah H (06) A note on a difference tpe estimator for population mean under two phase samplin desin Spriner Plus 5: oor-ul-amin M Javaid A Hanif M (07) Estimation of population mean in sstematic rom samplin usin auiliar information Journal of Statistics Manaement Sstems 0(6) Yasmeen U oor ul Amin M Hanif M (05) Generalized eponential estimators of finite population mean usin transformed auiliar variables International Journal of Applied Computational Mathematics () 0 Yasmeen U oor ul Amin M Hanif M (06) Eponential ratio product tpe estimators of finite population mean Journal of Statistics Manaement Sstems

9 08] SYSTEMATIC SAMPLIG USIG AUXILIARY IFORMATIO 373 Yasmeen U oor-ul-amin M Hanif M (08) Eponential estimators of finite population variance usin transformed auiliar variables Proceedins of the ational Academ of Sciences India Sect A Phsical Sciences: Appendi We consider the data iven b Cochran (977) to eamine the performances of different estimators of finite population variance Description of population parameters Population Y X z Food cost Size Income Value of population parameters Parameter Population 33 n n Y X Z C C C z r r z r z l 00 l 00 l 00 l 0 l 0 l 0 l 0 l